Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nnmcan | GIF version |
Description: Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnmcan | ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anrot 924 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ (𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω)) | |
2 | nnmword 6114 | . . . . 5 ⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐵 ⊆ 𝐶 ↔ (𝐴 ·𝑜 𝐵) ⊆ (𝐴 ·𝑜 𝐶))) | |
3 | 1, 2 | sylanb 278 | . . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐵 ⊆ 𝐶 ↔ (𝐴 ·𝑜 𝐵) ⊆ (𝐴 ·𝑜 𝐶))) |
4 | 3anrev 929 | . . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ (𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω)) | |
5 | nnmword 6114 | . . . . 5 ⊢ (((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐶 ⊆ 𝐵 ↔ (𝐴 ·𝑜 𝐶) ⊆ (𝐴 ·𝑜 𝐵))) | |
6 | 4, 5 | sylanb 278 | . . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → (𝐶 ⊆ 𝐵 ↔ (𝐴 ·𝑜 𝐶) ⊆ (𝐴 ·𝑜 𝐵))) |
7 | 3, 6 | anbi12d 456 | . . 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 139 | . 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 3014 | . 2 ⊢ ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ ((𝐴 ·𝑜 𝐵) ⊆ (𝐴 ·𝑜 𝐶) ∧ (𝐴 ·𝑜 𝐶) ⊆ (𝐴 ·𝑜 𝐵))) | |
10 | eqss 3014 | . 2 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵)) | |
11 | 8, 9, 10 | 3bitr4g 221 | 1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ 𝐵 = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 919 = wceq 1284 ∈ wcel 1433 ⊆ wss 2973 ∅c0 3251 ωcom 4331 (class class class)co 5532 ·𝑜 comu 6022 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 |
This theorem is referenced by: mulcanpig 6525 enq0tr 6624 |
Copyright terms: Public domain | W3C validator |