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Mirrors > Home > MPE Home > Th. List > grpodivdiv | Structured version Visualization version GIF version |
Description: Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpdivf.1 | ⊢ 𝑋 = ran 𝐺 |
grpdivf.3 | ⊢ 𝐷 = ( /𝑔 ‘𝐺) |
Ref | Expression |
---|---|
grpodivdiv | ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺(𝐶𝐷𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ GrpOp) | |
2 | simpr1 1193 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑋) | |
3 | grpdivf.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
4 | grpdivf.3 | . . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺) | |
5 | 3, 4 | grpodivcl 28897 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐷𝐶) ∈ 𝑋) |
6 | 5 | 3adant3r1 1181 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐷𝐶) ∈ 𝑋) |
7 | eqid 2740 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
8 | 3, 7, 4 | grpodivval 28893 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝐵𝐷𝐶) ∈ 𝑋) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺((inv‘𝐺)‘(𝐵𝐷𝐶)))) |
9 | 1, 2, 6, 8 | syl3anc 1370 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺((inv‘𝐺)‘(𝐵𝐷𝐶)))) |
10 | 3, 7, 4 | grpoinvdiv 28895 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((inv‘𝐺)‘(𝐵𝐷𝐶)) = (𝐶𝐷𝐵)) |
11 | 10 | 3adant3r1 1181 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘(𝐵𝐷𝐶)) = (𝐶𝐷𝐵)) |
12 | 11 | oveq2d 7287 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐺((inv‘𝐺)‘(𝐵𝐷𝐶))) = (𝐴𝐺(𝐶𝐷𝐵))) |
13 | 9, 12 | eqtrd 2780 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺(𝐶𝐷𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ran crn 5591 ‘cfv 6432 (class class class)co 7271 GrpOpcgr 28847 invcgn 28849 /𝑔 cgs 28850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-1st 7824 df-2nd 7825 df-grpo 28851 df-gid 28852 df-ginv 28853 df-gdiv 28854 |
This theorem is referenced by: ablodivdiv 28911 |
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