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Theorem mulcanpi 9760
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 9753 . . . . . . . . . 10 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
2 eleq1 2718 . . . . . . . . . 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 9751 . . . . . . . . 9 dom ·N = (N × N)
6 0npi 9742 . . . . . . . . 9 ¬ ∅ ∈ N
75, 6ndmovrcl 6862 . . . . . . . 8 ((𝐴 ·N 𝐶) ∈ N → (𝐴N𝐶N))
8 simpr 476 . . . . . . . 8 ((𝐴N𝐶N) → 𝐶N)
94, 7, 83syl 18 . . . . . . 7 (((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N)) → 𝐶N)
10 mulpiord 9745 . . . . . . . . . 10 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
1110adantr 480 . . . . . . . . 9 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
12 mulpiord 9745 . . . . . . . . . 10 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·𝑜 𝐶))
1312adantlr 751 . . . . . . . . 9 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·𝑜 𝐶))
1411, 13eqeq12d 2666 . . . . . . . 8 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶)))
15 pinn 9738 . . . . . . . . . . . . 13 (𝐴N𝐴 ∈ ω)
16 pinn 9738 . . . . . . . . . . . . 13 (𝐵N𝐵 ∈ ω)
17 pinn 9738 . . . . . . . . . . . . 13 (𝐶N𝐶 ∈ ω)
18 elni2 9737 . . . . . . . . . . . . . . . 16 (𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
1918simprbi 479 . . . . . . . . . . . . . . 15 (𝐴N → ∅ ∈ 𝐴)
20 nnmcan 7759 . . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . 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 1408 . . . . . . . . . . . 12 ((𝐴N𝐵N𝐶N) → (𝐴N → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶)))
25243exp 1283 . . . . . . . . . . 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 490 . . . . . 6 (((𝐴N𝐵N) ∧ ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ∧ (𝐴N𝐵N))) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
3130exp32 630 . . . . 5 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))))
3231imp4b 612 . . . 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 6698 . 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 1054   = wceq 1523  wcel 2030  c0 3948  (class class class)co 6690  ωcom 7107   ·𝑜 comu 7603  Ncnpi 9704   ·N cmi 9706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-omul 7610  df-ni 9732  df-mi 9734
This theorem is referenced by:  enqer  9781  nqereu  9789  adderpqlem  9814  mulerpqlem  9815
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