Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ablonnncan1 | Structured version Visualization version GIF version |
Description: Cancellation law for group division. (nnncan1 10924 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
abldiv.1 | ⊢ 𝑋 = ran 𝐺 |
abldiv.3 | ⊢ 𝐷 = ( /𝑔 ‘𝐺) |
Ref | Expression |
---|---|
ablonnncan1 | ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷(𝐴𝐷𝐶)) = (𝐶𝐷𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1190 | . . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑋) | |
2 | simpr2 1191 | . . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋) | |
3 | ablogrpo 28326 | . . . . . 6 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) | |
4 | abldiv.1 | . . . . . . 7 ⊢ 𝑋 = ran 𝐺 | |
5 | abldiv.3 | . . . . . . 7 ⊢ 𝐷 = ( /𝑔 ‘𝐺) | |
6 | 4, 5 | grpodivcl 28318 | . . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋) |
7 | 3, 6 | syl3an1 1159 | . . . . 5 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷𝐶) ∈ 𝑋) |
8 | 7 | 3adant3r2 1179 | . . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐶) ∈ 𝑋) |
9 | 1, 2, 8 | 3jca 1124 | . . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋)) |
10 | 4, 5 | ablodiv32 28334 | . . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐶) ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷(𝐴𝐷𝐶)) = ((𝐴𝐷(𝐴𝐷𝐶))𝐷𝐵)) |
11 | 9, 10 | syldan 593 | . 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷(𝐴𝐷𝐶)) = ((𝐴𝐷(𝐴𝐷𝐶))𝐷𝐵)) |
12 | 4, 5 | ablonncan 28335 | . . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐷(𝐴𝐷𝐶)) = 𝐶) |
13 | 12 | 3adant3r2 1179 | . . 3 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐴𝐷𝐶)) = 𝐶) |
14 | 13 | oveq1d 7173 | . 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷(𝐴𝐷𝐶))𝐷𝐵) = (𝐶𝐷𝐵)) |
15 | 11, 14 | eqtrd 2858 | 1 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷(𝐴𝐷𝐶)) = (𝐶𝐷𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ran crn 5558 ‘cfv 6357 (class class class)co 7158 GrpOpcgr 28268 /𝑔 cgs 28271 AbelOpcablo 28323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 |
This theorem is referenced by: nvnnncan1 28426 |
Copyright terms: Public domain | W3C validator |