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Theorem mulcanpig 6319
Description: Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
Assertion
Ref Expression
mulcanpig ((A N B N 𝐶 N) → ((A ·N B) = (A ·N 𝐶) ↔ B = 𝐶))

Proof of Theorem mulcanpig
StepHypRef Expression
1 mulpiord 6301 . . . . . 6 ((A N B N) → (A ·N B) = (A ·𝑜 B))
21adantr 261 . . . . 5 (((A N B N) 𝐶 N) → (A ·N B) = (A ·𝑜 B))
3 mulpiord 6301 . . . . . 6 ((A N 𝐶 N) → (A ·N 𝐶) = (A ·𝑜 𝐶))
43adantlr 446 . . . . 5 (((A N B N) 𝐶 N) → (A ·N 𝐶) = (A ·𝑜 𝐶))
52, 4eqeq12d 2051 . . . 4 (((A N B N) 𝐶 N) → ((A ·N B) = (A ·N 𝐶) ↔ (A ·𝑜 B) = (A ·𝑜 𝐶)))
6 pinn 6293 . . . . . . . . 9 (A NA 𝜔)
7 pinn 6293 . . . . . . . . 9 (B NB 𝜔)
8 pinn 6293 . . . . . . . . 9 (𝐶 N𝐶 𝜔)
9 elni2 6298 . . . . . . . . . . . 12 (A N ↔ (A 𝜔 A))
109simprbi 260 . . . . . . . . . . 11 (A N → ∅ A)
11 nnmcan 6028 . . . . . . . . . . . 12 (((A 𝜔 B 𝜔 𝐶 𝜔) A) → ((A ·𝑜 B) = (A ·𝑜 𝐶) ↔ B = 𝐶))
1211biimpd 132 . . . . . . . . . . 11 (((A 𝜔 B 𝜔 𝐶 𝜔) A) → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))
1310, 12sylan2 270 . . . . . . . . . 10 (((A 𝜔 B 𝜔 𝐶 𝜔) A N) → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))
1413ex 108 . . . . . . . . 9 ((A 𝜔 B 𝜔 𝐶 𝜔) → (A N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)))
156, 7, 8, 14syl3an 1176 . . . . . . . 8 ((A N B N 𝐶 N) → (A N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)))
16153exp 1102 . . . . . . 7 (A N → (B N → (𝐶 N → (A N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)))))
1716com4r 80 . . . . . 6 (A N → (A N → (B N → (𝐶 N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)))))
1817pm2.43i 43 . . . . 5 (A N → (B N → (𝐶 N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))))
1918imp31 243 . . . 4 (((A N B N) 𝐶 N) → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))
205, 19sylbid 139 . . 3 (((A N B N) 𝐶 N) → ((A ·N B) = (A ·N 𝐶) → B = 𝐶))
21203impa 1098 . 2 ((A N B N 𝐶 N) → ((A ·N B) = (A ·N 𝐶) → B = 𝐶))
22 oveq2 5463 . 2 (B = 𝐶 → (A ·N B) = (A ·N 𝐶))
2321, 22impbid1 130 1 ((A N B N 𝐶 N) → ((A ·N B) = (A ·N 𝐶) ↔ B = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  c0 3218  𝜔com 4256  (class class class)co 5455   ·𝑜 comu 5938  Ncnpi 6256   ·N cmi 6258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-ni 6288  df-mi 6290
This theorem is referenced by:  enqer  6342
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