Theorem prlem934b 3932

Description: Sublemma for Lemma 9-3.4 of [Gleason] p. 122.

Assertion

Ref	Expression
prlem934b	⊢ (((u ∈ N ∧ w ∈ N) ∧ (v ∈ N ∧ z ∈ N)) → (([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨v, u⟩] ~Q ∨ [⟨v, u⟩] ~Q <Q ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q )))

Distinct variable group(s):   z,w,v,u

Proof of Theorem prlem934b

Step	Hyp	Ref	Expression
1		mulclpi 3815	. . . . . . 7 ⊢ ((u ∈ N ∧ z ∈ N) → (u ·N z) ∈ N)
2		nlt1pi 3827	. . . . . . . 8 ⊢ ¬ (u ·N z) <N 1o
3		1pi 3805	. . . . . . . . . 10 ⊢ 1o ∈ N
4		ltsopi 3810	. . . . . . . . . . 11 ⊢ <N Or N
5		sotric 2148	. . . . . . . . . . 11 ⊢ (( <N Or N ∧ ((u ·N z) ∈ N ∧ 1o ∈ N)) → ((u ·N z) <N 1o ↔ ¬ ((u ·N z) = 1o ∨ 1o <N (u ·N z))))
6	4, 5	mpan 518	. . . . . . . . . 10 ⊢ (((u ·N z) ∈ N ∧ 1o ∈ N) → ((u ·N z) <N 1o ↔ ¬ ((u ·N z) = 1o ∨ 1o <N (u ·N z))))
7	3, 6	mpan2 519	. . . . . . . . 9 ⊢ ((u ·N z) ∈ N → ((u ·N z) <N 1o ↔ ¬ ((u ·N z) = 1o ∨ 1o <N (u ·N z))))
8	7	bicon2d 404	. . . . . . . 8 ⊢ ((u ·N z) ∈ N → (((u ·N z) = 1o ∨ 1o <N (u ·N z)) ↔ ¬ (u ·N z) <N 1o))
9	2, 8	mpbiri 169	. . . . . . 7 ⊢ ((u ·N z) ∈ N → ((u ·N z) = 1o ∨ 1o <N (u ·N z)))
10	1, 9	syl 12	. . . . . 6 ⊢ ((u ∈ N ∧ z ∈ N) → ((u ·N z) = 1o ∨ 1o <N (u ·N z)))
11	10	adantl 305	. . . . 5 ⊢ ((v ∈ N ∧ (u ∈ N ∧ z ∈ N)) → ((u ·N z) = 1o ∨ 1o <N (u ·N z)))
12		enqeceq 3841	. . . . . . . . . . . 12 ⊢ ((((v ·N z) ∈ N ∧ 1o ∈ N) ∧ (v ∈ N ∧ u ∈ N)) → ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ↔ ((v ·N z) ·N u) = (1o ·N v)))
13	12	ancoms 334	. . . . . . . . . . 11 ⊢ (((v ∈ N ∧ u ∈ N) ∧ ((v ·N z) ∈ N ∧ 1o ∈ N)) → ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ↔ ((v ·N z) ·N u) = (1o ·N v)))
14	3, 13	mpan22 532	. . . . . . . . . 10 ⊢ (((v ∈ N ∧ u ∈ N) ∧ (v ·N z) ∈ N) → ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ↔ ((v ·N z) ·N u) = (1o ·N v)))
15		mulclpi 3815	. . . . . . . . . 10 ⊢ ((v ∈ N ∧ z ∈ N) → (v ·N z) ∈ N)
16	14, 15	sylan2 346	. . . . . . . . 9 ⊢ (((v ∈ N ∧ u ∈ N) ∧ (v ∈ N ∧ z ∈ N)) → ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ↔ ((v ·N z) ·N u) = (1o ·N v)))
17	16	an4s 390	. . . . . . . 8 ⊢ (((v ∈ N ∧ v ∈ N) ∧ (u ∈ N ∧ z ∈ N)) → ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ↔ ((v ·N z) ·N u) = (1o ·N v)))
18	17	anabsan 386	. . . . . . 7 ⊢ ((v ∈ N ∧ (u ∈ N ∧ z ∈ N)) → ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ↔ ((v ·N z) ·N u) = (1o ·N v)))
19		opreq1 3006	. . . . . . . 8 ⊢ ((u ·N z) = 1o → ((u ·N z) ·N v) = (1o ·N v))
20		visset 1350	. . . . . . . . 9 ⊢ v ∈ V
21		visset 1350	. . . . . . . . 9 ⊢ z ∈ V
22		visset 1350	. . . . . . . . 9 ⊢ u ∈ V
23		visset 1350	. . . . . . . . . 10 ⊢ f ∈ V
24		visset 1350	. . . . . . . . . 10 ⊢ g ∈ V
25	23, 24	mulcompi 3818	. . . . . . . . 9 ⊢ (f ·N g) = (g ·N f)
26		visset 1350	. . . . . . . . . 10 ⊢ h ∈ V
27	24, 26	mulasspi 3819	. . . . . . . . 9 ⊢ ((f ·N g) ·N h) = (f ·N (g ·N h))
28	20, 21, 22, 25, 27	caopr31 3076	. . . . . . . 8 ⊢ ((v ·N z) ·N u) = ((u ·N z) ·N v)
29	19, 28	syl5eq 1136	. . . . . . 7 ⊢ ((u ·N z) = 1o → ((v ·N z) ·N u) = (1o ·N v))
30	18, 29	syl5bir 184	. . . . . 6 ⊢ ((v ∈ N ∧ (u ∈ N ∧ z ∈ N)) → ((u ·N z) = 1o → [⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ))
31	3	elisseti 1355	. . . . . . . . . 10 ⊢ 1o ∈ V
32		oprex 3018	. . . . . . . . . 10 ⊢ (u ·N z) ∈ V
33	31, 32	ltmpi 3825	. . . . . . . . 9 ⊢ (v ∈ N → (1o <N (u ·N z) ↔ (v ·N 1o) <N (v ·N (u ·N z))))
34		oprex 3018	. . . . . . . . . . 11 ⊢ (v ·N z) ∈ V
35	20, 22, 34, 31	ordpipq 3850	. . . . . . . . . 10 ⊢ ([⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q ↔ (v ·N 1o) <N (u ·N (v ·N z)))
36	22, 20, 21, 25, 27	caopr12 3075	. . . . . . . . . . 11 ⊢ (u ·N (v ·N z)) = (v ·N (u ·N z))
37	36	breq2i 2069	. . . . . . . . . 10 ⊢ ((v ·N 1o) <N (u ·N (v ·N z)) ↔ (v ·N 1o) <N (v ·N (u ·N z)))
38	35, 37	bitr 151	. . . . . . . . 9 ⊢ ([⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q ↔ (v ·N 1o) <N (v ·N (u ·N z)))
39	33, 38	syl6bbr 416	. . . . . . . 8 ⊢ (v ∈ N → (1o <N (u ·N z) ↔ [⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q ))
40	39	biimpd 135	. . . . . . 7 ⊢ (v ∈ N → (1o <N (u ·N z) → [⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q ))
41	40	adantr 306	. . . . . 6 ⊢ ((v ∈ N ∧ (u ∈ N ∧ z ∈ N)) → (1o <N (u ·N z) → [⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q ))
42	30, 41	orim12d 436	. . . . 5 ⊢ ((v ∈ N ∧ (u ∈ N ∧ z ∈ N)) → (((u ·N z) = 1o ∨ 1o <N (u ·N z)) → ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ∨ [⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q )))
43	11, 42	mpd 46	. . . 4 ⊢ ((v ∈ N ∧ (u ∈ N ∧ z ∈ N)) → ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ∨ [⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q ))
44	43	an1s 372	. . 3 ⊢ ((u ∈ N ∧ (v ∈ N ∧ z ∈ N)) → ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ∨ [⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q ))
45	44	adantlr 310	. 2 ⊢ (((u ∈ N ∧ w ∈ N) ∧ (v ∈ N ∧ z ∈ N)) → ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ∨ [⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q ))
46		an42 389	. . . . . . 7 ⊢ (((w ∈ N ∧ w ∈ N) ∧ (v ∈ N ∧ z ∈ N)) ↔ ((w ∈ N ∧ v ∈ N) ∧ (z ∈ N ∧ w ∈ N)))
47		mulpipq 3849	. . . . . . . . 9 ⊢ ((((w ·N v) ∈ N ∧ 1o ∈ N) ∧ (z ∈ N ∧ w ∈ N)) → ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨((w ·N v) ·N z), (1o ·N w)⟩] ~Q )
48	3, 47	mpan12 530	. . . . . . . 8 ⊢ (((w ·N v) ∈ N ∧ (z ∈ N ∧ w ∈ N)) → ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨((w ·N v) ·N z), (1o ·N w)⟩] ~Q )
49		mulclpi 3815	. . . . . . . 8 ⊢ ((w ∈ N ∧ v ∈ N) → (w ·N v) ∈ N)
50	48, 49	sylan 343	. . . . . . 7 ⊢ (((w ∈ N ∧ v ∈ N) ∧ (z ∈ N ∧ w ∈ N)) → ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨((w ·N v) ·N z), (1o ·N w)⟩] ~Q )
51	46, 50	sylbi 174	. . . . . 6 ⊢ (((w ∈ N ∧ w ∈ N) ∧ (v ∈ N ∧ z ∈ N)) → ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨((w ·N v) ·N z), (1o ·N w)⟩] ~Q )
52	51	anabsan 386	. . . . 5 ⊢ ((w ∈ N ∧ (v ∈ N ∧ z ∈ N)) → ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨((w ·N v) ·N z), (1o ·N w)⟩] ~Q )
53		visset 1350	. . . . . . . . 9 ⊢ w ∈ V
54	53, 34, 31	distrpqlem 3860	. . . . . . . 8 ⊢ ((w ∈ N ∧ (v ·N z) ∈ N ∧ 1o ∈ N) → [⟨(w ·N (v ·N z)), (w ·N 1o)⟩] ~Q = [⟨(v ·N z), 1o⟩] ~Q )
55	3, 54	mp3an3 641	. . . . . . 7 ⊢ ((w ∈ N ∧ (v ·N z) ∈ N) → [⟨(w ·N (v ·N z)), (w ·N 1o)⟩] ~Q = [⟨(v ·N z), 1o⟩] ~Q )
56	55, 15	sylan2 346	. . . . . 6 ⊢ ((w ∈ N ∧ (v ∈ N ∧ z ∈ N)) → [⟨(w ·N (v ·N z)), (w ·N 1o)⟩] ~Q = [⟨(v ·N z), 1o⟩] ~Q )
57	20, 21	mulasspi 3819	. . . . . . . 8 ⊢ ((w ·N v) ·N z) = (w ·N (v ·N z))
58	31, 53	mulcompi 3818	. . . . . . . 8 ⊢ (1o ·N w) = (w ·N 1o)
59		opeq12 1878	. . . . . . . 8 ⊢ ((((w ·N v) ·N z) = (w ·N (v ·N z)) ∧ (1o ·N w) = (w ·N 1o)) → ⟨((w ·N v) ·N z), (1o ·N w)⟩ = ⟨(w ·N (v ·N z)), (w ·N 1o)⟩)
60	57, 58, 59	mp2an 520	. . . . . . 7 ⊢ ⟨((w ·N v) ·N z), (1o ·N w)⟩ = ⟨(w ·N (v ·N z)), (w ·N 1o)⟩
61		eceq2 3215	. . . . . . 7 ⊢ (⟨((w ·N v) ·N z), (1o ·N w)⟩ = ⟨(w ·N (v ·N z)), (w ·N 1o)⟩ → [⟨((w ·N v) ·N z), (1o ·N w)⟩] ~Q = [⟨(w ·N (v ·N z)), (w ·N 1o)⟩] ~Q )
62	60, 61	ax-mp 6	. . . . . 6 ⊢ [⟨((w ·N v) ·N z), (1o ·N w)⟩] ~Q = [⟨(w ·N (v ·N z)), (w ·N 1o)⟩] ~Q
63	56, 62	syl5eq 1136	. . . . 5 ⊢ ((w ∈ N ∧ (v ∈ N ∧ z ∈ N)) → [⟨((w ·N v) ·N z), (1o ·N w)⟩] ~Q = [⟨(v ·N z), 1o⟩] ~Q )
64	52, 63	eqtrd 1128	. . . 4 ⊢ ((w ∈ N ∧ (v ∈ N ∧ z ∈ N)) → ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨(v ·N z), 1o⟩] ~Q )
65	64	adantll 309	. . 3 ⊢ (((u ∈ N ∧ w ∈ N) ∧ (v ∈ N ∧ z ∈ N)) → ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨(v ·N z), 1o⟩] ~Q )
66		cleq1 1107	. . . 4 ⊢ (([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨(v ·N z), 1o⟩] ~Q → (([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨v, u⟩] ~Q ↔ [⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ))
67		breq2 2066	. . . 4 ⊢ (([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨(v ·N z), 1o⟩] ~Q → ([⟨v, u⟩] ~Q <Q ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) ↔ [⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q ))
68	66, 67	orbi12d 475	. . 3 ⊢ (([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨(v ·N z), 1o⟩] ~Q → ((([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨v, u⟩] ~Q ∨ [⟨v, u⟩] ~Q <Q ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q )) ↔ ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ∨ [⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q )))
69	65, 68	syl 12	. 2 ⊢ (((u ∈ N ∧ w ∈ N) ∧ (v ∈ N ∧ z ∈ N)) → ((([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨v, u⟩] ~Q ∨ [⟨v, u⟩] ~Q <Q ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q )) ↔ ([⟨(v ·N z), 1o⟩] ~Q = [⟨v, u⟩] ~Q ∨ [⟨v, u⟩] ~Q <Q [⟨(v ·N z), 1o⟩] ~Q )))
70	45, 69	mpbird 171	1 ⊢ (((u ∈ N ∧ w ∈ N) ∧ (v ∈ N ∧ z ∈ N)) → (([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨v, u⟩] ~Q ∨ [⟨v, u⟩] ~Q <Q ([⟨(w ·N v), 1o⟩] ~Q ·Q [⟨z, w⟩] ~Q )))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054   Or wor 2059  (class class class)co 3001  1_oc1o 3099  [cec 3198  Ncnpi 3766   ·_N cmi 3768   <_N clti 3769   ~_Q ceq 3772   ·_Q cmq 3776   <_Q cltq 3778

This theorem is referenced by:  prlem934 3933

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-mi 3796  df-lti 3797  df-mpq 3830  df-enq 3831  df-nq 3832  df-mq 3834  df-ltq 3836

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