ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mulcanpig GIF version

Theorem mulcanpig 6955
Description: Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
Assertion
Ref Expression
mulcanpig ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem mulcanpig
StepHypRef Expression
1 mulpiord 6937 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
21adantr 271 . . . . 5 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
3 mulpiord 6937 . . . . . 6 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
43adantlr 462 . . . . 5 (((𝐴N𝐵N) ∧ 𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
52, 4eqeq12d 2103 . . . 4 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶)))
6 pinn 6929 . . . . . . . . 9 (𝐴N𝐴 ∈ ω)
7 pinn 6929 . . . . . . . . 9 (𝐵N𝐵 ∈ ω)
8 pinn 6929 . . . . . . . . 9 (𝐶N𝐶 ∈ ω)
9 elni2 6934 . . . . . . . . . . . 12 (𝐴N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
109simprbi 270 . . . . . . . . . . 11 (𝐴N → ∅ ∈ 𝐴)
11 nnmcan 6292 . . . . . . . . . . . 12 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))
1211biimpd 143 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
1310, 12sylan2 281 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ 𝐴N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
1413ex 114 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
156, 7, 8, 14syl3an 1217 . . . . . . . 8 ((𝐴N𝐵N𝐶N) → (𝐴N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
16153exp 1143 . . . . . . 7 (𝐴N → (𝐵N → (𝐶N → (𝐴N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))))
1716com4r 86 . . . . . 6 (𝐴N → (𝐴N → (𝐵N → (𝐶N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))))
1817pm2.43i 49 . . . . 5 (𝐴N → (𝐵N → (𝐶N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))))
1918imp31 253 . . . 4 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
205, 19sylbid 149 . . 3 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
21203impa 1139 . 2 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
22 oveq2 5674 . 2 (𝐵 = 𝐶 → (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶))
2321, 22impbid1 141 1 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 925   = wceq 1290  wcel 1439  c0 3287  ωcom 4418  (class class class)co 5666   ·o comu 6193  Ncnpi 6892   ·N cmi 6894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-oadd 6199  df-omul 6200  df-ni 6924  df-mi 6926
This theorem is referenced by:  enqer  6978
  Copyright terms: Public domain W3C validator