Theorem ltmpq 3871

Description: Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120.

Hypotheses

Ref	Expression
ltapq.1	⊢ A ∈ V
ltapq.2	⊢ B ∈ V

Assertion

Ref	Expression
ltmpq	⊢ (C ∈ Q → (A <Q B ↔ (C ·Q A) <Q (C ·Q B)))

Proof of Theorem ltmpq

Step	Hyp	Ref	Expression
1		ltapq.2	. 2 ⊢ B ∈ V
2		dmmulpq 3855	. 2 ⊢ dom ·Q = (Q × Q)
3		ltapq.1	. 2 ⊢ A ∈ V
4		ltrelpq 3845	. 2 ⊢ <Q ⊆ (Q × Q)
5		0npq 3844	. 2 ⊢ ¬ ∅ ∈ Q
6		df-nq 3832	. . 3 ⊢ Q = ((N × N) / ~Q )
7		breq1 2065	. . . 4 ⊢ ([⟨x, y⟩] ~Q = A → ([⟨x, y⟩] ~Q <Q [⟨z, w⟩] ~Q ↔ A <Q [⟨z, w⟩] ~Q ))
8		opreq2 3007	. . . . 5 ⊢ ([⟨x, y⟩] ~Q = A → ([⟨v, u⟩] ~Q ·Q [⟨x, y⟩] ~Q ) = ([⟨v, u⟩] ~Q ·Q A))
9	8	breq1d 2071	. . . 4 ⊢ ([⟨x, y⟩] ~Q = A → (([⟨v, u⟩] ~Q ·Q [⟨x, y⟩] ~Q ) <Q ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q ) ↔ ([⟨v, u⟩] ~Q ·Q A) <Q ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q )))
10	7, 9	bibi12d 477	. . 3 ⊢ ([⟨x, y⟩] ~Q = A → (([⟨x, y⟩] ~Q <Q [⟨z, w⟩] ~Q ↔ ([⟨v, u⟩] ~Q ·Q [⟨x, y⟩] ~Q ) <Q ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q )) ↔ (A <Q [⟨z, w⟩] ~Q ↔ ([⟨v, u⟩] ~Q ·Q A) <Q ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q ))))
11		breq2 2066	. . . 4 ⊢ ([⟨z, w⟩] ~Q = B → (A <Q [⟨z, w⟩] ~Q ↔ A <Q B))
12		opreq2 3007	. . . . 5 ⊢ ([⟨z, w⟩] ~Q = B → ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = ([⟨v, u⟩] ~Q ·Q B))
13	12	breq2d 2072	. . . 4 ⊢ ([⟨z, w⟩] ~Q = B → (([⟨v, u⟩] ~Q ·Q A) <Q ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q ) ↔ ([⟨v, u⟩] ~Q ·Q A) <Q ([⟨v, u⟩] ~Q ·Q B)))
14	11, 13	bibi12d 477	. . 3 ⊢ ([⟨z, w⟩] ~Q = B → ((A <Q [⟨z, w⟩] ~Q ↔ ([⟨v, u⟩] ~Q ·Q A) <Q ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q )) ↔ (A <Q B ↔ ([⟨v, u⟩] ~Q ·Q A) <Q ([⟨v, u⟩] ~Q ·Q B))))
15		opreq1 3006	. . . . 5 ⊢ ([⟨v, u⟩] ~Q = C → ([⟨v, u⟩] ~Q ·Q A) = (C ·Q A))
16		opreq1 3006	. . . . 5 ⊢ ([⟨v, u⟩] ~Q = C → ([⟨v, u⟩] ~Q ·Q B) = (C ·Q B))
17	15, 16	breq12d 2073	. . . 4 ⊢ ([⟨v, u⟩] ~Q = C → (([⟨v, u⟩] ~Q ·Q A) <Q ([⟨v, u⟩] ~Q ·Q B) ↔ (C ·Q A) <Q (C ·Q B)))
18	17	bibi2d 470	. . 3 ⊢ ([⟨v, u⟩] ~Q = C → ((A <Q B ↔ ([⟨v, u⟩] ~Q ·Q A) <Q ([⟨v, u⟩] ~Q ·Q B)) ↔ (A <Q B ↔ (C ·Q A) <Q (C ·Q B))))
19		mulclpi 3815	. . . . . . . . 9 ⊢ ((v ∈ N ∧ u ∈ N) → (v ·N u) ∈ N)
20		oprex 3018	. . . . . . . . . . 11 ⊢ (x ·N w) ∈ V
21		oprex 3018	. . . . . . . . . . 11 ⊢ (y ·N z) ∈ V
22	20, 21	ltmpi 3825	. . . . . . . . . 10 ⊢ ((v ·N u) ∈ N → ((x ·N w) <N (y ·N z) ↔ ((v ·N u) ·N (x ·N w)) <N ((v ·N u) ·N (y ·N z))))
23		visset 1350	. . . . . . . . . . . 12 ⊢ v ∈ V
24		visset 1350	. . . . . . . . . . . 12 ⊢ x ∈ V
25		visset 1350	. . . . . . . . . . . 12 ⊢ u ∈ V
26		visset 1350	. . . . . . . . . . . . 13 ⊢ f ∈ V
27		visset 1350	. . . . . . . . . . . . 13 ⊢ g ∈ V
28	26, 27	mulcompi 3818	. . . . . . . . . . . 12 ⊢ (f ·N g) = (g ·N f)
29		visset 1350	. . . . . . . . . . . . 13 ⊢ h ∈ V
30	27, 29	mulasspi 3819	. . . . . . . . . . . 12 ⊢ ((f ·N g) ·N h) = (f ·N (g ·N h))
31		visset 1350	. . . . . . . . . . . 12 ⊢ w ∈ V
32	23, 24, 25, 28, 30, 31	caopr4 3078	. . . . . . . . . . 11 ⊢ ((v ·N x) ·N (u ·N w)) = ((v ·N u) ·N (x ·N w))
33		visset 1350	. . . . . . . . . . . . 13 ⊢ y ∈ V
34		visset 1350	. . . . . . . . . . . . 13 ⊢ z ∈ V
35	25, 33, 23, 28, 30, 34	caopr4 3078	. . . . . . . . . . . 12 ⊢ ((u ·N y) ·N (v ·N z)) = ((u ·N v) ·N (y ·N z))
36	25, 23	mulcompi 3818	. . . . . . . . . . . . 13 ⊢ (u ·N v) = (v ·N u)
37	36	opreq1i 3009	. . . . . . . . . . . 12 ⊢ ((u ·N v) ·N (y ·N z)) = ((v ·N u) ·N (y ·N z))
38	35, 37	eqtr 1119	. . . . . . . . . . 11 ⊢ ((u ·N y) ·N (v ·N z)) = ((v ·N u) ·N (y ·N z))
39	32, 38	breq12i 2070	. . . . . . . . . 10 ⊢ (((v ·N x) ·N (u ·N w)) <N ((u ·N y) ·N (v ·N z)) ↔ ((v ·N u) ·N (x ·N w)) <N ((v ·N u) ·N (y ·N z)))
40	22, 39	syl6bbr 416	. . . . . . . . 9 ⊢ ((v ·N u) ∈ N → ((x ·N w) <N (y ·N z) ↔ ((v ·N x) ·N (u ·N w)) <N ((u ·N y) ·N (v ·N z))))
41	19, 40	syl 12	. . . . . . . 8 ⊢ ((v ∈ N ∧ u ∈ N) → ((x ·N w) <N (y ·N z) ↔ ((v ·N x) ·N (u ·N w)) <N ((u ·N y) ·N (v ·N z))))
42	24, 33, 34, 31	ordpipq 3850	. . . . . . . 8 ⊢ ([⟨x, y⟩] ~Q <Q [⟨z, w⟩] ~Q ↔ (x ·N w) <N (y ·N z))
43		oprex 3018	. . . . . . . . 9 ⊢ (v ·N x) ∈ V
44		oprex 3018	. . . . . . . . 9 ⊢ (u ·N y) ∈ V
45		oprex 3018	. . . . . . . . 9 ⊢ (v ·N z) ∈ V
46		oprex 3018	. . . . . . . . 9 ⊢ (u ·N w) ∈ V
47	43, 44, 45, 46	ordpipq 3850	. . . . . . . 8 ⊢ ([⟨(v ·N x), (u ·N y)⟩] ~Q <Q [⟨(v ·N z), (u ·N w)⟩] ~Q ↔ ((v ·N x) ·N (u ·N w)) <N ((u ·N y) ·N (v ·N z)))
48	41, 42, 47	3bitr4g 428	. . . . . . 7 ⊢ ((v ∈ N ∧ u ∈ N) → ([⟨x, y⟩] ~Q <Q [⟨z, w⟩] ~Q ↔ [⟨(v ·N x), (u ·N y)⟩] ~Q <Q [⟨(v ·N z), (u ·N w)⟩] ~Q ))
49	48	adantr 306	. . . . . 6 ⊢ (((v ∈ N ∧ u ∈ N) ∧ ((x ∈ N ∧ y ∈ N) ∧ (z ∈ N ∧ w ∈ N))) → ([⟨x, y⟩] ~Q <Q [⟨z, w⟩] ~Q ↔ [⟨(v ·N x), (u ·N y)⟩] ~Q <Q [⟨(v ·N z), (u ·N w)⟩] ~Q ))
50		mulpipq 3849	. . . . . . . 8 ⊢ (((v ∈ N ∧ u ∈ N) ∧ (x ∈ N ∧ y ∈ N)) → ([⟨v, u⟩] ~Q ·Q [⟨x, y⟩] ~Q ) = [⟨(v ·N x), (u ·N y)⟩] ~Q )
51	50	adantrr 312	. . . . . . 7 ⊢ (((v ∈ N ∧ u ∈ N) ∧ ((x ∈ N ∧ y ∈ N) ∧ (z ∈ N ∧ w ∈ N))) → ([⟨v, u⟩] ~Q ·Q [⟨x, y⟩] ~Q ) = [⟨(v ·N x), (u ·N y)⟩] ~Q )
52		mulpipq 3849	. . . . . . . 8 ⊢ (((v ∈ N ∧ u ∈ N) ∧ (z ∈ N ∧ w ∈ N)) → ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨(v ·N z), (u ·N w)⟩] ~Q )
53	52	adantrl 311	. . . . . . 7 ⊢ (((v ∈ N ∧ u ∈ N) ∧ ((x ∈ N ∧ y ∈ N) ∧ (z ∈ N ∧ w ∈ N))) → ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q ) = [⟨(v ·N z), (u ·N w)⟩] ~Q )
54	51, 53	breq12d 2073	. . . . . 6 ⊢ (((v ∈ N ∧ u ∈ N) ∧ ((x ∈ N ∧ y ∈ N) ∧ (z ∈ N ∧ w ∈ N))) → (([⟨v, u⟩] ~Q ·Q [⟨x, y⟩] ~Q ) <Q ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q ) ↔ [⟨(v ·N x), (u ·N y)⟩] ~Q <Q [⟨(v ·N z), (u ·N w)⟩] ~Q ))
55	49, 54	bitr4d 409	. . . . 5 ⊢ (((v ∈ N ∧ u ∈ N) ∧ ((x ∈ N ∧ y ∈ N) ∧ (z ∈ N ∧ w ∈ N))) → ([⟨x, y⟩] ~Q <Q [⟨z, w⟩] ~Q ↔ ([⟨v, u⟩] ~Q ·Q [⟨x, y⟩] ~Q ) <Q ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q )))
56	55	3impb 610	. . . 4 ⊢ (((v ∈ N ∧ u ∈ N) ∧ (x ∈ N ∧ y ∈ N) ∧ (z ∈ N ∧ w ∈ N)) → ([⟨x, y⟩] ~Q <Q [⟨z, w⟩] ~Q ↔ ([⟨v, u⟩] ~Q ·Q [⟨x, y⟩] ~Q ) <Q ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q )))
57	56	3coml 617	. . 3 ⊢ (((x ∈ N ∧ y ∈ N) ∧ (z ∈ N ∧ w ∈ N) ∧ (v ∈ N ∧ u ∈ N)) → ([⟨x, y⟩] ~Q <Q [⟨z, w⟩] ~Q ↔ ([⟨v, u⟩] ~Q ·Q [⟨x, y⟩] ~Q ) <Q ([⟨v, u⟩] ~Q ·Q [⟨z, w⟩] ~Q )))
58	6, 10, 14, 18, 57	3ecoptocl 3241	. 2 ⊢ ((A ∈ Q ∧ B ∈ Q ∧ C ∈ Q) → (A <Q B ↔ (C ·Q A) <Q (C ·Q B)))
59	1, 2, 3, 4, 5, 58	ndmord 3064	1 ⊢ (C ∈ Q → (A <Q B ↔ (C ·Q A) <Q (C ·Q B)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  (class class class)co 3001  [cec 3198  Ncnpi 3766   ·_N cmi 3768   <_N clti 3769   ~_Q ceq 3772  Qcnq 3773   ·_Q cmq 3776   <_Q cltq 3778

This theorem is referenced by:  ltaddpq 3873  ltrpq 3879  addclprlem1 3912  mulclprlem 3915  mulclpr 3916  distrlem4pr 3924  1idpr 3927  prlem936a 3947  prlem936 3949  reclem3pr 3952  reclem4pr 3953

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-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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