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Theorem mulcanpi 10938
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 10931 . . . . . . . . . 10 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
2 eleq1 2827 . . . . . . . . . 10 ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴 ·N 𝐵) ∈ N ↔ (𝐴 ·N 𝐶) ∈ N))
31, 2imbitrid 244 . . . . . . . . 9 ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴N𝐵N) → (𝐴 ·N 𝐶) ∈ N))
43imp 406 . . . . . . . 8 (((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N)) → (𝐴 ·N 𝐶) ∈ N)
5 dmmulpi 10929 . . . . . . . . 9 dom ·N = (N × N)
6 0npi 10920 . . . . . . . . 9 ¬ ∅ ∈ N
75, 6ndmovrcl 7619 . . . . . . . 8 ((𝐴 ·N 𝐶) ∈ N → (𝐴N𝐶N))
8 simpr 484 . . . . . . . 8 ((𝐴N𝐶N) → 𝐶N)
94, 7, 83syl 18 . . . . . . 7 (((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N)) → 𝐶N)
10 mulpiord 10923 . . . . . . . . . 10 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
1110adantr 480 . . . . . . . . 9 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
12 mulpiord 10923 . . . . . . . . . 10 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
1312adantlr 715 . . . . . . . . 9 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
1411, 13eqeq12d 2751 . . . . . . . 8 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶)))
15 pinn 10916 . . . . . . . . . . . . 13 (𝐴N𝐴 ∈ ω)
16 pinn 10916 . . . . . . . . . . . . 13 (𝐵N𝐵 ∈ ω)
17 pinn 10916 . . . . . . . . . . . . 13 (𝐶N𝐶 ∈ ω)
18 elni2 10915 . . . . . . . . . . . . . . . 16 (𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
1918simprbi 496 . . . . . . . . . . . . . . 15 (𝐴N → ∅ ∈ 𝐴)
20 nnmcan 8671 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))
2120biimpd 229 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
2219, 21sylan2 593 . . . . . . . . . . . . . 14 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ 𝐴N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
2322ex 412 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
2415, 16, 17, 23syl3an 1159 . . . . . . . . . . . 12 ((𝐴N𝐵N𝐶N) → (𝐴N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
25243exp 1118 . . . . . . . . . . 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 417 . . . . . . . 8 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
2914, 28sylbid 240 . . . . . . 7 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
309, 29sylan2 593 . . . . . 6 (((𝐴N𝐵N) ∧ ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N))) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
3130exp32 420 . . . . 5 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))))
3231imp4b 421 . . . 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 412 . 2 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
35 oveq2 7439 . 2 (𝐵 = 𝐶 → (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶))
3634, 35impbid1 225 1 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  c0 4339  (class class class)co 7431  ωcom 7887   ·o comu 8503  Ncnpi 10882   ·N cmi 10884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509  df-omul 8510  df-ni 10910  df-mi 10912
This theorem is referenced by:  enqer  10959  nqereu  10967  adderpqlem  10992  mulerpqlem  10993
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