Theorem nnmord 3189

Description: Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers.

Assertion

Ref	Expression
nnmord	⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((A ∈ B ∧ ∅ ∈ C) ↔ (C ·o A) ∈ (C ·o B)))

Proof of Theorem nnmord

Step	Hyp	Ref	Expression
1		nnmordi 3188	. 2 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((A ∈ B ∧ ∅ ∈ C) → (C ·o A) ∈ (C ·o B)))
2		opreq1 3006	. . . . . . . . . 10 ⊢ (C = ∅ → (C ·o B) = (∅ ·o B))
3	2	cleq1d 1109	. . . . . . . . 9 ⊢ (C = ∅ → ((C ·o B) = ∅ ↔ (∅ ·o B) = ∅))
4		nnm0r 3171	. . . . . . . . 9 ⊢ (B ∈ ω → (∅ ·o B) = ∅)
5	3, 4	syl5bir 184	. . . . . . . 8 ⊢ (C = ∅ → (B ∈ ω → (C ·o B) = ∅))
6	5	com12 13	. . . . . . 7 ⊢ (B ∈ ω → (C = ∅ → (C ·o B) = ∅))
7		n0i 1712	. . . . . . 7 ⊢ ((C ·o A) ∈ (C ·o B) → ¬ (C ·o B) = ∅)
8	6, 7	nsyli 106	. . . . . 6 ⊢ (B ∈ ω → ((C ·o A) ∈ (C ·o B) → ¬ C = ∅))
9	8	adantr 306	. . . . 5 ⊢ ((B ∈ ω ∧ C ∈ ω) → ((C ·o A) ∈ (C ·o B) → ¬ C = ∅))
10		nnord 2381	. . . . . . 7 ⊢ (C ∈ ω → Ord C)
11		ord0eln0 2278	. . . . . . 7 ⊢ (Ord C → (∅ ∈ C ↔ ¬ C = ∅))
12	10, 11	syl 12	. . . . . 6 ⊢ (C ∈ ω → (∅ ∈ C ↔ ¬ C = ∅))
13	12	adantl 305	. . . . 5 ⊢ ((B ∈ ω ∧ C ∈ ω) → (∅ ∈ C ↔ ¬ C = ∅))
14	9, 13	sylibrd 179	. . . 4 ⊢ ((B ∈ ω ∧ C ∈ ω) → ((C ·o A) ∈ (C ·o B) → ∅ ∈ C))
15	14	3adant1 597	. . 3 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((C ·o A) ∈ (C ·o B) → ∅ ∈ C))
16		opreq2 3007	. . . . . . . . . 10 ⊢ (A = B → (C ·o A) = (C ·o B))
17	16	a1i 7	. . . . . . . . 9 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ C) → (A = B → (C ·o A) = (C ·o B)))
18		nnmordi 3188	. . . . . . . . . . . . 13 ⊢ ((B ∈ ω ∧ A ∈ ω ∧ C ∈ ω) → ((B ∈ A ∧ ∅ ∈ C) → (C ·o B) ∈ (C ·o A)))
19	18	exp3a 292	. . . . . . . . . . . 12 ⊢ ((B ∈ ω ∧ A ∈ ω ∧ C ∈ ω) → (B ∈ A → (∅ ∈ C → (C ·o B) ∈ (C ·o A))))
20	19	com23 32	. . . . . . . . . . 11 ⊢ ((B ∈ ω ∧ A ∈ ω ∧ C ∈ ω) → (∅ ∈ C → (B ∈ A → (C ·o B) ∈ (C ·o A))))
21	20	3com12 614	. . . . . . . . . 10 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → (∅ ∈ C → (B ∈ A → (C ·o B) ∈ (C ·o A))))
22	21	imp 277	. . . . . . . . 9 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ C) → (B ∈ A → (C ·o B) ∈ (C ·o A)))
23	17, 22	orim12d 436	. . . . . . . 8 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ C) → ((A = B ∨ B ∈ A) → ((C ·o A) = (C ·o B) ∨ (C ·o B) ∈ (C ·o A))))
24	23	con3d 87	. . . . . . 7 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ C) → (¬ ((C ·o A) = (C ·o B) ∨ (C ·o B) ∈ (C ·o A)) → ¬ (A = B ∨ B ∈ A)))
25		pm3.26 256	. . . . . . . . 9 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ C) → (A ∈ ω ∧ B ∈ ω ∧ C ∈ ω))
26		df-3an 583	. . . . . . . . . 10 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ↔ ((A ∈ ω ∧ B ∈ ω) ∧ C ∈ ω))
27		ancom 333	. . . . . . . . . 10 ⊢ (((A ∈ ω ∧ B ∈ ω) ∧ C ∈ ω) ↔ (C ∈ ω ∧ (A ∈ ω ∧ B ∈ ω)))
28		anandi 392	. . . . . . . . . 10 ⊢ ((C ∈ ω ∧ (A ∈ ω ∧ B ∈ ω)) ↔ ((C ∈ ω ∧ A ∈ ω) ∧ (C ∈ ω ∧ B ∈ ω)))
29	26, 27, 28	3bitr 155	. . . . . . . . 9 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ↔ ((C ∈ ω ∧ A ∈ ω) ∧ (C ∈ ω ∧ B ∈ ω)))
30	25, 29	sylib 173	. . . . . . . 8 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ C) → ((C ∈ ω ∧ A ∈ ω) ∧ (C ∈ ω ∧ B ∈ ω)))
31		nnmcl 3173	. . . . . . . . . 10 ⊢ ((C ∈ ω ∧ A ∈ ω) → (C ·o A) ∈ ω)
32		nnord 2381	. . . . . . . . . 10 ⊢ ((C ·o A) ∈ ω → Ord (C ·o A))
33	31, 32	syl 12	. . . . . . . . 9 ⊢ ((C ∈ ω ∧ A ∈ ω) → Ord (C ·o A))
34		nnmcl 3173	. . . . . . . . . 10 ⊢ ((C ∈ ω ∧ B ∈ ω) → (C ·o B) ∈ ω)
35		nnord 2381	. . . . . . . . . 10 ⊢ ((C ·o B) ∈ ω → Ord (C ·o B))
36	34, 35	syl 12	. . . . . . . . 9 ⊢ ((C ∈ ω ∧ B ∈ ω) → Ord (C ·o B))
37	33, 36	anim12i 268	. . . . . . . 8 ⊢ (((C ∈ ω ∧ A ∈ ω) ∧ (C ∈ ω ∧ B ∈ ω)) → (Ord (C ·o A) ∧ Ord (C ·o B)))
38		ordtri2 2233	. . . . . . . 8 ⊢ ((Ord (C ·o A) ∧ Ord (C ·o B)) → ((C ·o A) ∈ (C ·o B) ↔ ¬ ((C ·o A) = (C ·o B) ∨ (C ·o B) ∈ (C ·o A))))
39	30, 37, 38	3syl 21	. . . . . . 7 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ C) → ((C ·o A) ∈ (C ·o B) ↔ ¬ ((C ·o A) = (C ·o B) ∨ (C ·o B) ∈ (C ·o A))))
40		3simpa 591	. . . . . . . . 9 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → (A ∈ ω ∧ B ∈ ω))
41		nnord 2381	. . . . . . . . . 10 ⊢ (A ∈ ω → Ord A)
42		nnord 2381	. . . . . . . . . 10 ⊢ (B ∈ ω → Ord B)
43	41, 42	anim12i 268	. . . . . . . . 9 ⊢ ((A ∈ ω ∧ B ∈ ω) → (Ord A ∧ Ord B))
44	25, 40, 43	3syl 21	. . . . . . . 8 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ C) → (Ord A ∧ Ord B))
45		ordtri2 2233	. . . . . . . 8 ⊢ ((Ord A ∧ Ord B) → (A ∈ B ↔ ¬ (A = B ∨ B ∈ A)))
46	44, 45	syl 12	. . . . . . 7 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ C) → (A ∈ B ↔ ¬ (A = B ∨ B ∈ A)))
47	24, 39, 46	3imtr4d 421	. . . . . 6 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ C) → ((C ·o A) ∈ (C ·o B) → A ∈ B))
48	47	exp 291	. . . . 5 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → (∅ ∈ C → ((C ·o A) ∈ (C ·o B) → A ∈ B)))
49	48	com23 32	. . . 4 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((C ·o A) ∈ (C ·o B) → (∅ ∈ C → A ∈ B)))
50		ancr 243	. . . 4 ⊢ ((∅ ∈ C → A ∈ B) → (∅ ∈ C → (A ∈ B ∧ ∅ ∈ C)))
51	49, 50	syl6 23	. . 3 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((C ·o A) ∈ (C ·o B) → (∅ ∈ C → (A ∈ B ∧ ∅ ∈ C))))
52	15, 51	mpdd 47	. 2 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((C ·o A) ∈ (C ·o B) → (A ∈ B ∧ ∅ ∈ C)))
53	1, 52	impbid 397	1 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((A ∈ B ∧ ∅ ∈ C) ↔ (C ·o A) ∈ (C ·o B)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  ∅c0 1707  Ord word 2198  ωcom 2372  (class class class)co 3001   ·_o comu 3102

This theorem is referenced by:  ltmpi 3825

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

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-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  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-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-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106  df-omul 3107

metamath.org