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Mirrors > Home > MPE Home > Th. List > grpodivf | Structured version Visualization version GIF version |
Description: Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpdivf.1 | ⊢ 𝑋 = ran 𝐺 |
grpdivf.3 | ⊢ 𝐷 = ( /𝑔 ‘𝐺) |
Ref | Expression |
---|---|
grpodivf | ⊢ (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpdivf.1 | . . . . . . . 8 ⊢ 𝑋 = ran 𝐺 | |
2 | eqid 2728 | . . . . . . . 8 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
3 | 1, 2 | grpoinvcl 30354 | . . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋) |
4 | 3 | 3adant2 1128 | . . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((inv‘𝐺)‘𝑦) ∈ 𝑋) |
5 | 1 | grpocl 30330 | . . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝑦) ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋) |
6 | 4, 5 | syld3an3 1406 | . . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋) |
7 | 6 | 3expib 1119 | . . . 4 ⊢ (𝐺 ∈ GrpOp → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋)) |
8 | 7 | ralrimivv 3196 | . . 3 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋) |
9 | eqid 2728 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))) | |
10 | 9 | fmpo 8078 | . . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺((inv‘𝐺)‘𝑦)) ∈ 𝑋 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋) |
11 | 8, 10 | sylib 217 | . 2 ⊢ (𝐺 ∈ GrpOp → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋) |
12 | grpdivf.3 | . . . 4 ⊢ 𝐷 = ( /𝑔 ‘𝐺) | |
13 | 1, 2, 12 | grpodivfval 30364 | . . 3 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))) |
14 | 13 | feq1d 6712 | . 2 ⊢ (𝐺 ∈ GrpOp → (𝐷:(𝑋 × 𝑋)⟶𝑋 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦))):(𝑋 × 𝑋)⟶𝑋)) |
15 | 11, 14 | mpbird 256 | 1 ⊢ (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3058 × cxp 5680 ran crn 5683 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 GrpOpcgr 30319 invcgn 30321 /𝑔 cgs 30322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-grpo 30323 df-gid 30324 df-ginv 30325 df-gdiv 30326 |
This theorem is referenced by: grpodivcl 30369 |
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