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Theorem mulcanpi 10931
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 10924 . . . . . . . . . 10 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
2 eleq1 2817 . . . . . . . . . 10 ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴 ·N 𝐵) ∈ N ↔ (𝐴 ·N 𝐶) ∈ N))
31, 2imbitrid 243 . . . . . . . . 9 ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴N𝐵N) → (𝐴 ·N 𝐶) ∈ N))
43imp 405 . . . . . . . 8 (((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N)) → (𝐴 ·N 𝐶) ∈ N)
5 dmmulpi 10922 . . . . . . . . 9 dom ·N = (N × N)
6 0npi 10913 . . . . . . . . 9 ¬ ∅ ∈ N
75, 6ndmovrcl 7613 . . . . . . . 8 ((𝐴 ·N 𝐶) ∈ N → (𝐴N𝐶N))
8 simpr 483 . . . . . . . 8 ((𝐴N𝐶N) → 𝐶N)
94, 7, 83syl 18 . . . . . . 7 (((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N)) → 𝐶N)
10 mulpiord 10916 . . . . . . . . . 10 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
1110adantr 479 . . . . . . . . 9 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
12 mulpiord 10916 . . . . . . . . . 10 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
1312adantlr 713 . . . . . . . . 9 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
1411, 13eqeq12d 2744 . . . . . . . 8 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶)))
15 pinn 10909 . . . . . . . . . . . . 13 (𝐴N𝐴 ∈ ω)
16 pinn 10909 . . . . . . . . . . . . 13 (𝐵N𝐵 ∈ ω)
17 pinn 10909 . . . . . . . . . . . . 13 (𝐶N𝐶 ∈ ω)
18 elni2 10908 . . . . . . . . . . . . . . . 16 (𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
1918simprbi 495 . . . . . . . . . . . . . . 15 (𝐴N → ∅ ∈ 𝐴)
20 nnmcan 8661 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))
2120biimpd 228 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
2219, 21sylan2 591 . . . . . . . . . . . . . 14 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ 𝐴N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
2322ex 411 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
2415, 16, 17, 23syl3an 1157 . . . . . . . . . . . 12 ((𝐴N𝐵N𝐶N) → (𝐴N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
25243exp 1116 . . . . . . . . . . 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 416 . . . . . . . 8 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
2914, 28sylbid 239 . . . . . . 7 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
309, 29sylan2 591 . . . . . 6 (((𝐴N𝐵N) ∧ ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N))) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
3130exp32 419 . . . . 5 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))))
3231imp4b 420 . . . 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 411 . 2 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
35 oveq2 7434 . 2 (𝐵 = 𝐶 → (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶))
3634, 35impbid1 224 1 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  c0 4326  (class class class)co 7426  ωcom 7876   ·o comu 8491  Ncnpi 10875   ·N cmi 10877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-oadd 8497  df-omul 8498  df-ni 10903  df-mi 10905
This theorem is referenced by:  enqer  10952  nqereu  10960  adderpqlem  10985  mulerpqlem  10986
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