Theorem nnmcan 8570

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 8569	. . . . 5 ⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐵 ⊆ 𝐶 ↔ (𝐴 ·o 𝐵) ⊆ (𝐴 ·o 𝐶)))
3	1, 2	sylanb 582	. . . 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 8569	. . . . 5 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐶 ⊆ 𝐵 ↔ (𝐴 ·o 𝐶) ⊆ (𝐴 ·o 𝐵)))
6	4, 5	sylanb 582	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐶 ⊆ 𝐵 ↔ (𝐴 ·o 𝐶) ⊆ (𝐴 ·o 𝐵)))
7	3, 6	anbi12d 633	. . 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 3937	. 2 ⊢ ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ ((𝐴 ·o 𝐵) ⊆ (𝐴 ·o 𝐶) ∧ (𝐴 ·o 𝐶) ⊆ (𝐴 ·o 𝐵)))
10		eqss 3937	. 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 1542   ∈ wcel 2114   ⊆ wss 3889  ∅c0 4273  (class class class)co 7367  ωcom 7817   ·_o comu 8403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-oadd 8409  df-omul 8410

This theorem is referenced by:  mulcanpi  10823