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Theorem mulcanpi 10057
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 10050 . . . . . . . . . 10 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
2 eleq1 2847 . . . . . . . . . 10 ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴 ·N 𝐵) ∈ N ↔ (𝐴 ·N 𝐶) ∈ N))
31, 2syl5ib 236 . . . . . . . . 9 ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴N𝐵N) → (𝐴 ·N 𝐶) ∈ N))
43imp 397 . . . . . . . 8 (((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N)) → (𝐴 ·N 𝐶) ∈ N)
5 dmmulpi 10048 . . . . . . . . 9 dom ·N = (N × N)
6 0npi 10039 . . . . . . . . 9 ¬ ∅ ∈ N
75, 6ndmovrcl 7097 . . . . . . . 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 10042 . . . . . . . . . 10 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
1110adantr 474 . . . . . . . . 9 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
12 mulpiord 10042 . . . . . . . . . 10 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
1312adantlr 705 . . . . . . . . 9 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
1411, 13eqeq12d 2793 . . . . . . . 8 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶)))
15 pinn 10035 . . . . . . . . . . . . 13 (𝐴N𝐴 ∈ ω)
16 pinn 10035 . . . . . . . . . . . . 13 (𝐵N𝐵 ∈ ω)
17 pinn 10035 . . . . . . . . . . . . 13 (𝐶N𝐶 ∈ ω)
18 elni2 10034 . . . . . . . . . . . . . . . 16 (𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
1918simprbi 492 . . . . . . . . . . . . . . 15 (𝐴N → ∅ ∈ 𝐴)
20 nnmcan 7998 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))
2120biimpd 221 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
2219, 21sylan2 586 . . . . . . . . . . . . . 14 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ 𝐴N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
2322ex 403 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
2415, 16, 17, 23syl3an 1160 . . . . . . . . . . . 12 ((𝐴N𝐵N𝐶N) → (𝐴N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
25243exp 1109 . . . . . . . . . . 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 410 . . . . . . . 8 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
2914, 28sylbid 232 . . . . . . 7 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
309, 29sylan2 586 . . . . . 6 (((𝐴N𝐵N) ∧ ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N))) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
3130exp32 413 . . . . 5 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))))
3231imp4b 414 . . . 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 403 . 2 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
35 oveq2 6930 . 2 (𝐵 = 𝐶 → (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶))
3634, 35impbid1 217 1 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  c0 4141  (class class class)co 6922  ωcom 7343   ·o comu 7841  Ncnpi 10001   ·N cmi 10003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-oadd 7847  df-omul 7848  df-ni 10029  df-mi 10031
This theorem is referenced by:  enqer  10078  nqereu  10086  adderpqlem  10111  mulerpqlem  10112
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