Theorem mulcanpi 10848

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 10841	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N)
2		eleq1 2844	. . . . . . . . . 10 ⊢ ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴 ·N 𝐵) ∈ N ↔ (𝐴 ·N 𝐶) ∈ N))
3	1, 2	imbitrid 246	. . . . . . . . 9 ⊢ ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐶) ∈ N))
4	3	imp 409	. . . . . . . 8 ⊢ (((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴 ∈ N ∧ 𝐵 ∈ N)) → (𝐴 ·N 𝐶) ∈ N)
5		dmmulpi 10839	. . . . . . . . 9 ⊢ dom ·N = (N × N)
6		0npi 10830	. . . . . . . . 9 ⊢ ¬ ∅ ∈ N
7	5, 6	ndmovrcl 7571	. . . . . . . 8 ⊢ ((𝐴 ·N 𝐶) ∈ N → (𝐴 ∈ N ∧ 𝐶 ∈ N))
8		simpr 487	. . . . . . . 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 10833	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
11	10	adantr 483	. . . . . . . . 9 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
12		mulpiord 10833	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
13	12	adantlr 723	. . . . . . . . 9 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
14	11, 13	eqeq12d 2772	. . . . . . . 8 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶)))
15		pinn 10826	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
16		pinn 10826	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω)
17		pinn 10826	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω)
18		elni2 10825	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
19	18	simprbi 500	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ N → ∅ ∈ 𝐴)
20		nnmcan 8592	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))
21	20	biimpd 231	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
22	19, 21	sylan2 601	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ 𝐴 ∈ N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
23	22	ex 415	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
24	15, 16, 17, 23	syl3an 1169	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
25	24	3exp 1128	. . . . . . . . . . 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 420	. . . . . . . 8 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
29	14, 28	sylbid 242	. . . . . . 7 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
30	9, 29	sylan2 601	. . . . . 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 423	. . . . 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 424	. . . 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 415	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
35		oveq2 7393	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶))
36	34, 35	impbid1 227	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ∅c0 4280  (class class class)co 7385  ωcom 7835   ·_o comu 8423  Ncnpi 10792   ·_N cmi 10794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-oadd 8429  df-omul 8430  df-ni 10820  df-mi 10822

This theorem is referenced by:  enqer  10869  nqereu  10877  adderpqlem  10902  mulerpqlem  10903