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

Step	Hyp	Ref	Expression
1		mulpiord 6301	. . . . . 6 ⊢ ((A ∈ N ∧ B ∈ N) → (A ·N B) = (A ·𝑜 B))
2	1	adantr 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))
4	3	adantlr 446	. . . . 5 ⊢ (((A ∈ N ∧ B ∈ N) ∧ 𝐶 ∈ N) → (A ·N 𝐶) = (A ·𝑜 𝐶))
5	2, 4	eqeq12d 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 ∈ N → A ∈ 𝜔)
7		pinn 6293	. . . . . . . . 9 ⊢ (B ∈ N → B ∈ 𝜔)
8		pinn 6293	. . . . . . . . 9 ⊢ (𝐶 ∈ N → 𝐶 ∈ 𝜔)
9		elni2 6298	. . . . . . . . . . . 12 ⊢ (A ∈ N ↔ (A ∈ 𝜔 ∧ ∅ ∈ A))
10	9	simprbi 260	. . . . . . . . . . 11 ⊢ (A ∈ N → ∅ ∈ A)
11		nnmcan 6028	. . . . . . . . . . . 12 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ A) → ((A ·𝑜 B) = (A ·𝑜 𝐶) ↔ B = 𝐶))
12	11	biimpd 132	. . . . . . . . . . 11 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ A) → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))
13	10, 12	sylan2 270	. . . . . . . . . 10 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ A ∈ N) → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))
14	13	ex 108	. . . . . . . . 9 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (A ∈ N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)))
15	6, 7, 8, 14	syl3an 1176	. . . . . . . 8 ⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈ N) → (A ∈ N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)))
16	15	3exp 1102	. . . . . . 7 ⊢ (A ∈ N → (B ∈ N → (𝐶 ∈ N → (A ∈ N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)))))
17	16	com4r 80	. . . . . 6 ⊢ (A ∈ N → (A ∈ N → (B ∈ N → (𝐶 ∈ N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)))))
18	17	pm2.43i 43	. . . . 5 ⊢ (A ∈ N → (B ∈ N → (𝐶 ∈ N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))))
19	18	imp31 243	. . . 4 ⊢ (((A ∈ N ∧ B ∈ N) ∧ 𝐶 ∈ N) → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))
20	5, 19	sylbid 139	. . 3 ⊢ (((A ∈ N ∧ B ∈ N) ∧ 𝐶 ∈ N) → ((A ·N B) = (A ·N 𝐶) → B = 𝐶))
21	20	3impa 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))
23	21, 22	impbid1 130	1 ⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈ N) → ((A ·N B) = (A ·N 𝐶) ↔ B = 𝐶))

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-bndl 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