Theorem mulcanpig 7646

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 7628	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
2	1	adantr 276	. . . . 5 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
3		mulpiord 7628	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
4	3	adantlr 477	. . . . 5 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
5	2, 4	eqeq12d 2247	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶)))
6		pinn 7620	. . . . . . . . 9 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
7		pinn 7620	. . . . . . . . 9 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω)
8		pinn 7620	. . . . . . . . 9 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω)
9		elni2 7625	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
10	9	simprbi 275	. . . . . . . . . . 11 ⊢ (𝐴 ∈ N → ∅ ∈ 𝐴)
11		nnmcan 6751	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))
12	11	biimpd 144	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
13	10, 12	sylan2 286	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ 𝐴 ∈ N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
14	13	ex 115	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
15	6, 7, 8, 14	syl3an 1316	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
16	15	3exp 1229	. . . . . . 7 ⊢ (𝐴 ∈ N → (𝐵 ∈ N → (𝐶 ∈ N → (𝐴 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))))
17	16	com4r 86	. . . . . 6 ⊢ (𝐴 ∈ N → (𝐴 ∈ N → (𝐵 ∈ N → (𝐶 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))))
18	17	pm2.43i 49	. . . . 5 ⊢ (𝐴 ∈ N → (𝐵 ∈ N → (𝐶 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))))
19	18	imp31 256	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
20	5, 19	sylbid 150	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
21	20	3impa 1221	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
22		oveq2 6057	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 ·N 𝐵) = (𝐴 ·N 𝐶))
23	21, 22	impbid1 142	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∅c0 3507  ωcom 4711  (class class class)co 6049   ·_o comu 6644  Ncnpi 7583   ·_N cmi 7585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-oadd 6650  df-omul 6651  df-ni 7615  df-mi 7617

This theorem is referenced by:  enqer  7669