Theorem nnmcan 3190

Description: Cancellation law for multiplication of natural numbers.

Assertion

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

Proof of Theorem nnmcan

Step	Hyp	Ref	Expression
1		nnmordi 3188	. . . . . . . . . 10 ⊢ ((B ∈ ω ∧ C ∈ ω ∧ A ∈ ω) → ((B ∈ C ∧ ∅ ∈ A) → (A ·o B) ∈ (A ·o C)))
2	1	3comr 618	. . . . . . . . 9 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((B ∈ C ∧ ∅ ∈ A) → (A ·o B) ∈ (A ·o C)))
3		nnmordi 3188	. . . . . . . . . 10 ⊢ ((C ∈ ω ∧ B ∈ ω ∧ A ∈ ω) → ((C ∈ B ∧ ∅ ∈ A) → (A ·o C) ∈ (A ·o B)))
4	3	3com13 615	. . . . . . . . 9 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((C ∈ B ∧ ∅ ∈ A) → (A ·o C) ∈ (A ·o B)))
5	2, 4	orim12d 436	. . . . . . . 8 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → (((B ∈ C ∧ ∅ ∈ A) ∨ (C ∈ B ∧ ∅ ∈ A)) → ((A ·o B) ∈ (A ·o C) ∨ (A ·o C) ∈ (A ·o B))))
6		andir 457	. . . . . . . 8 ⊢ (((B ∈ C ∨ C ∈ B) ∧ ∅ ∈ A) ↔ ((B ∈ C ∧ ∅ ∈ A) ∨ (C ∈ B ∧ ∅ ∈ A)))
7	5, 6	syl5ib 181	. . . . . . 7 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → (((B ∈ C ∨ C ∈ B) ∧ ∅ ∈ A) → ((A ·o B) ∈ (A ·o C) ∨ (A ·o C) ∈ (A ·o B))))
8	7	exp3a 292	. . . . . 6 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((B ∈ C ∨ C ∈ B) → (∅ ∈ A → ((A ·o B) ∈ (A ·o C) ∨ (A ·o C) ∈ (A ·o B)))))
9	8	com23 32	. . . . 5 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → (∅ ∈ A → ((B ∈ C ∨ C ∈ B) → ((A ·o B) ∈ (A ·o C) ∨ (A ·o C) ∈ (A ·o B)))))
10	9	imp 277	. . . 4 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ A) → ((B ∈ C ∨ C ∈ B) → ((A ·o B) ∈ (A ·o C) ∨ (A ·o C) ∈ (A ·o B))))
11	10	con3d 87	. . 3 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ A) → (¬ ((A ·o B) ∈ (A ·o C) ∨ (A ·o C) ∈ (A ·o B)) → ¬ (B ∈ C ∨ C ∈ B)))
12		ordtri3 2234	. . . . . 6 ⊢ ((Ord (A ·o B) ∧ Ord (A ·o C)) → ((A ·o B) = (A ·o C) ↔ ¬ ((A ·o B) ∈ (A ·o C) ∨ (A ·o C) ∈ (A ·o B))))
13		nnmcl 3173	. . . . . . 7 ⊢ ((A ∈ ω ∧ B ∈ ω) → (A ·o B) ∈ ω)
14		nnord 2381	. . . . . . 7 ⊢ ((A ·o B) ∈ ω → Ord (A ·o B))
15	13, 14	syl 12	. . . . . 6 ⊢ ((A ∈ ω ∧ B ∈ ω) → Ord (A ·o B))
16		nnmcl 3173	. . . . . . 7 ⊢ ((A ∈ ω ∧ C ∈ ω) → (A ·o C) ∈ ω)
17		nnord 2381	. . . . . . 7 ⊢ ((A ·o C) ∈ ω → Ord (A ·o C))
18	16, 17	syl 12	. . . . . 6 ⊢ ((A ∈ ω ∧ C ∈ ω) → Ord (A ·o C))
19	12, 15, 18	syl2an 349	. . . . 5 ⊢ (((A ∈ ω ∧ B ∈ ω) ∧ (A ∈ ω ∧ C ∈ ω)) → ((A ·o B) = (A ·o C) ↔ ¬ ((A ·o B) ∈ (A ·o C) ∨ (A ·o C) ∈ (A ·o B))))
20	19	3impdi 630	. . . 4 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((A ·o B) = (A ·o C) ↔ ¬ ((A ·o B) ∈ (A ·o C) ∨ (A ·o C) ∈ (A ·o B))))
21	20	adantr 306	. . 3 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ A) → ((A ·o B) = (A ·o C) ↔ ¬ ((A ·o B) ∈ (A ·o C) ∨ (A ·o C) ∈ (A ·o B))))
22		ordtri3 2234	. . . . . 6 ⊢ ((Ord B ∧ Ord C) → (B = C ↔ ¬ (B ∈ C ∨ C ∈ B)))
23		nnord 2381	. . . . . 6 ⊢ (B ∈ ω → Ord B)
24		nnord 2381	. . . . . 6 ⊢ (C ∈ ω → Ord C)
25	22, 23, 24	syl2an 349	. . . . 5 ⊢ ((B ∈ ω ∧ C ∈ ω) → (B = C ↔ ¬ (B ∈ C ∨ C ∈ B)))
26	25	3adant1 597	. . . 4 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → (B = C ↔ ¬ (B ∈ C ∨ C ∈ B)))
27	26	adantr 306	. . 3 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ A) → (B = C ↔ ¬ (B ∈ C ∨ C ∈ B)))
28	11, 21, 27	3imtr4d 421	. 2 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ A) → ((A ·o B) = (A ·o C) → B = C))
29		opreq2 3007	. . 3 ⊢ (B = C → (A ·o B) = (A ·o C))
30	29	a1i 7	. 2 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ A) → (B = C → (A ·o B) = (A ·o C)))
31	28, 30	impbid 397	1 ⊢ (((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) ∧ ∅ ∈ A) → ((A ·o B) = (A ·o C) ↔ B = C))

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:  mulcanpi 3821

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

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