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Theorem mulcanpi 9910
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 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem mulcanpi
StepHypRef Expression
1 mulclpi 9903 . . . . . . . . . 10 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
2 eleq1 2823 . . . . . . . . . 10 ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴 ·N 𝐵) ∈ N ↔ (𝐴 ·N 𝐶) ∈ N))
31, 2syl5ib 234 . . . . . . . . 9 ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴N𝐵N) → (𝐴 ·N 𝐶) ∈ N))
43imp 444 . . . . . . . 8 (((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N)) → (𝐴 ·N 𝐶) ∈ N)
5 dmmulpi 9901 . . . . . . . . 9 dom ·N = (N × N)
6 0npi 9892 . . . . . . . . 9 ¬ ∅ ∈ N
75, 6ndmovrcl 6981 . . . . . . . 8 ((𝐴 ·N 𝐶) ∈ N → (𝐴N𝐶N))
8 simpr 479 . . . . . . . 8 ((𝐴N𝐶N) → 𝐶N)
94, 7, 83syl 18 . . . . . . 7 (((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N)) → 𝐶N)
10 mulpiord 9895 . . . . . . . . . 10 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
1110adantr 472 . . . . . . . . 9 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
12 mulpiord 9895 . . . . . . . . . 10 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·𝑜 𝐶))
1312adantlr 753 . . . . . . . . 9 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·𝑜 𝐶))
1411, 13eqeq12d 2771 . . . . . . . 8 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶)))
15 pinn 9888 . . . . . . . . . . . . 13 (𝐴N𝐴 ∈ ω)
16 pinn 9888 . . . . . . . . . . . . 13 (𝐵N𝐵 ∈ ω)
17 pinn 9888 . . . . . . . . . . . . 13 (𝐶N𝐶 ∈ ω)
18 elni2 9887 . . . . . . . . . . . . . . . 16 (𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
1918simprbi 483 . . . . . . . . . . . . . . 15 (𝐴N → ∅ ∈ 𝐴)
20 nnmcan 7879 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ 𝐵 = 𝐶))
2120biimpd 219 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶))
2219, 21sylan2 492 . . . . . . . . . . . . . 14 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ 𝐴N) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶))
2322ex 449 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴N → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶)))
2415, 16, 17, 23syl3an 1164 . . . . . . . . . . . 12 ((𝐴N𝐵N𝐶N) → (𝐴N → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶)))
25243exp 1113 . . . . . . . . . . 11 (𝐴N → (𝐵N → (𝐶N → (𝐴N → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶)))))
2625com4r 94 . . . . . . . . . 10 (𝐴N → (𝐴N → (𝐵N → (𝐶N → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶)))))
2726pm2.43i 52 . . . . . . . . 9 (𝐴N → (𝐵N → (𝐶N → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶))))
2827imp31 447 . . . . . . . 8 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶))
2914, 28sylbid 230 . . . . . . 7 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
309, 29sylan2 492 . . . . . 6 (((𝐴N𝐵N) ∧ ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N))) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
3130exp32 632 . . . . 5 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))))
3231imp4b 614 . . . 4 (((𝐴N𝐵N) ∧ (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶)) → (((𝐴N𝐵N) ∧ (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶)) → 𝐵 = 𝐶))
3332pm2.43i 52 . . 3 (((𝐴N𝐵N) ∧ (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶)) → 𝐵 = 𝐶)
3433ex 449 . 2 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
35 oveq2 6817 . 2 (𝐵 = 𝐶 → (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶))
3634, 35impbid1 215 1 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1628  wcel 2135  c0 4054  (class class class)co 6809  ωcom 7226   ·𝑜 comu 7723  Ncnpi 9854   ·N cmi 9856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-oadd 7729  df-omul 7730  df-ni 9882  df-mi 9884
This theorem is referenced by:  enqer  9931  nqereu  9939  adderpqlem  9964  mulerpqlem  9965
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