Theorem mulcanpig 7696

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 7678	. . . . . 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 7678	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
4	3	adantlr 481	. . . . 5 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
5	2, 4	eqeq12d 2253	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) ↔ (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶)))
6		pinn 7670	. . . . . . . . 9 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
7		pinn 7670	. . . . . . . . 9 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω)
8		pinn 7670	. . . . . . . . 9 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω)
9		elni2 7675	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴))
10	9	simprbi 275	. . . . . . . . . . 11 ⊢ (𝐴 ∈ N → ∅ ∈ 𝐴)
11		nnmcan 6786	. . . . . . . . . . . 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 1320	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ∈ N → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶)))
16	15	3exp 1233	. . . . . . 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 1225	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) = (𝐴 ·N 𝐶) → 𝐵 = 𝐶))
22		oveq2 6087	. 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 1009   = wceq 1402   ∈ wcel 2209  ∅c0 3520  ωcom 4735  (class class class)co 6079   ·_o comu 6679  Ncnpi 7633   ·_N cmi 7635

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-oadd 6685  df-omul 6686  df-ni 7665  df-mi 7667

This theorem is referenced by:  enqer  7719