Theorem nnmcan 8672

Description: Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnmcan	⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem nnmcan

Step	Hyp	Ref	Expression
1		3anrot 1100	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ (𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω))
2		nnmword 8671	. . . . 5 ⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐵 ⊆ 𝐶 ↔ (𝐴 ·o 𝐵) ⊆ (𝐴 ·o 𝐶)))
3	1, 2	sylanb 581	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐵 ⊆ 𝐶 ↔ (𝐴 ·o 𝐵) ⊆ (𝐴 ·o 𝐶)))
4		3anrev 1101	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ (𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω))
5		nnmword 8671	. . . . 5 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐶 ⊆ 𝐵 ↔ (𝐴 ·o 𝐶) ⊆ (𝐴 ·o 𝐵)))
6	4, 5	sylanb 581	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐶 ⊆ 𝐵 ↔ (𝐴 ·o 𝐶) ⊆ (𝐴 ·o 𝐵)))
7	3, 6	anbi12d 632	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ↔ ((𝐴 ·o 𝐵) ⊆ (𝐴 ·o 𝐶) ∧ (𝐴 ·o 𝐶) ⊆ (𝐴 ·o 𝐵))))
8	7	bicomd 223	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → (((𝐴 ·o 𝐵) ⊆ (𝐴 ·o 𝐶) ∧ (𝐴 ·o 𝐶) ⊆ (𝐴 ·o 𝐵)) ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵)))
9		eqss 3999	. 2 ⊢ ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ ((𝐴 ·o 𝐵) ⊆ (𝐴 ·o 𝐶) ∧ (𝐴 ·o 𝐶) ⊆ (𝐴 ·o 𝐵)))
10		eqss 3999	. 2 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
11	8, 9, 10	3bitr4g 314	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  ∅c0 4333  (class class class)co 7431  ωcom 7887   ·_o comu 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510  df-omul 8511

This theorem is referenced by:  mulcanpi  10940