Theorem nnmcan 8597

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 1111	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ (𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω))
2		nnmword 8596	. . . . 5 ⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐵 ⊆ 𝐶 ↔ (𝐴 ·o 𝐵) ⊆ (𝐴 ·o 𝐶)))
3	1, 2	sylanb 590	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐵 ⊆ 𝐶 ↔ (𝐴 ·o 𝐵) ⊆ (𝐴 ·o 𝐶)))
4		3anrev 1112	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ (𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω))
5		nnmword 8596	. . . . 5 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐶 ⊆ 𝐵 ↔ (𝐴 ·o 𝐶) ⊆ (𝐴 ·o 𝐵)))
6	4, 5	sylanb 590	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐶 ⊆ 𝐵 ↔ (𝐴 ·o 𝐶) ⊆ (𝐴 ·o 𝐵)))
7	3, 6	anbi12d 641	. . 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 225	. 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 3949	. 2 ⊢ ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ ((𝐴 ·o 𝐵) ⊆ (𝐴 ·o 𝐶) ∧ (𝐴 ·o 𝐶) ⊆ (𝐴 ·o 𝐵)))
10		eqss 3949	. 2 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
11	8, 9, 10	3bitr4g 316	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ⊆ wss 3902  ∅c0 4283  (class class class)co 7390  ωcom 7840   ·_o comu 8428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-oadd 8434  df-omul 8435

This theorem is referenced by:  mulcanpi  10851