Theorem mulcanpi 10860

Description: Multiplication cancellation law for positive integers. (Contributed by NM, 4-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
mulcanpi	⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem mulcanpi

Step	Hyp	Ref	Expression
1		mulclpi 10853	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N)
2		eleq1 2817	. . . . . . . . . 10 ⊢ ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴 ·N 𝐵) ∈ N ↔ (𝐴 ·N 𝐶) ∈ N))
3	1, 2	imbitrid 244	. . . . . . . . 9 ⊢ ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐶) ∈ N))
4	3	imp 406	. . . . . . . 8 ⊢ (((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴 ∈ N ∧ 𝐵 ∈ N)) → (𝐴 ·N 𝐶) ∈ N)
5		dmmulpi 10851	. . . . . . . . 9 ⊢ dom ·N = (N × N)
6		0npi 10842	. . . . . . . . 9 ⊢ ¬ ∅ ∈ N
7	5, 6	ndmovrcl 7578	. . . . . . . 8 ⊢ ((𝐴 ·N 𝐶) ∈ N → (𝐴 ∈ N ∧ 𝐶 ∈ N))
8		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → 𝐶 ∈ N)
9	4, 7, 8	3syl 18	. . . . . . 7 ⊢ (((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴 ∈ N ∧ 𝐵 ∈ N)) → 𝐶 ∈ N)
10		mulpiord 10845	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
11	10	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
12		mulpiord 10845	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
13	12	adantlr 715	. . . . . . . . 9 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
14	11, 13	eqeq12d 2746	. . . . . . . 8 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶)))
15		pinn 10838	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
16		pinn 10838	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω)
17		pinn 10838	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω)
18		elni2 10837	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
19	18	simprbi 496	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ N → ∅ ∈ 𝐴)
20		nnmcan 8601	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))
21	20	biimpd 229	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
22	19, 21	sylan2 593	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ 𝐴 ∈ N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
23	22	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
24	15, 16, 17, 23	syl3an 1160	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
25	24	3exp 1119	. . . . . . . . . . 11 ⊢ (𝐴 ∈ N → (𝐵 ∈ N → (𝐶 ∈ N → (𝐴 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))))
26	25	com4r 94	. . . . . . . . . 10 ⊢ (𝐴 ∈ N → (𝐴 ∈ N → (𝐵 ∈ N → (𝐶 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))))
27	26	pm2.43i 52	. . . . . . . . 9 ⊢ (𝐴 ∈ N → (𝐵 ∈ N → (𝐶 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))))
28	27	imp31 417	. . . . . . . 8 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
29	14, 28	sylbid 240	. . . . . . 7 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
30	9, 29	sylan2 593	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴 ∈ N ∧ 𝐵 ∈ N))) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
31	30	exp32 420	. . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))))
32	31	imp4b 421	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶)) → (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶)) → 𝐵 = 𝐶))
33	32	pm2.43i 52	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶)) → 𝐵 = 𝐶)
34	33	ex 412	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
35		oveq2 7398	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶))
36	34, 35	impbid1 225	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∅c0 4299  (class class class)co 7390  ωcom 7845   ·_o comu 8435  Ncnpi 10804   ·_N cmi 10806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-oadd 8441  df-omul 8442  df-ni 10832  df-mi 10834

This theorem is referenced by:  enqer  10881  nqereu  10889  adderpqlem  10914  mulerpqlem  10915