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Mirrors > Home > MPE Home > Th. List > grpodivfval | Structured version Visualization version GIF version |
Description: Group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpdiv.1 | ⊢ 𝑋 = ran 𝐺 |
grpdiv.2 | ⊢ 𝑁 = (inv‘𝐺) |
grpdiv.3 | ⊢ 𝐷 = ( /𝑔 ‘𝐺) |
Ref | Expression |
---|---|
grpodivfval | ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpdiv.3 | . 2 ⊢ 𝐷 = ( /𝑔 ‘𝐺) | |
2 | grpdiv.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
3 | rnexg 7637 | . . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V) | |
4 | 2, 3 | eqeltrid 2837 | . . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V) |
5 | mpoexga 7803 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) | |
6 | 4, 4, 5 | syl2anc 587 | . . 3 ⊢ (𝐺 ∈ GrpOp → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) |
7 | rneq 5779 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺) | |
8 | 7, 2 | eqtr4di 2791 | . . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋) |
9 | id 22 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑔 = 𝐺) | |
10 | eqidd 2739 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥) | |
11 | fveq2 6676 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = (inv‘𝐺)) | |
12 | grpdiv.2 | . . . . . . . 8 ⊢ 𝑁 = (inv‘𝐺) | |
13 | 11, 12 | eqtr4di 2791 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = 𝑁) |
14 | 13 | fveq1d 6678 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((inv‘𝑔)‘𝑦) = (𝑁‘𝑦)) |
15 | 9, 10, 14 | oveq123d 7193 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥𝑔((inv‘𝑔)‘𝑦)) = (𝑥𝐺(𝑁‘𝑦))) |
16 | 8, 8, 15 | mpoeq123dv 7245 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
17 | df-gdiv 28433 | . . . 4 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) | |
18 | 16, 17 | fvmptg 6775 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
19 | 6, 18 | mpdan 687 | . 2 ⊢ (𝐺 ∈ GrpOp → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
20 | 1, 19 | syl5eq 2785 | 1 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ran crn 5526 ‘cfv 6339 (class class class)co 7172 ∈ cmpo 7174 GrpOpcgr 28426 invcgn 28428 /𝑔 cgs 28429 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7175 df-oprab 7176 df-mpo 7177 df-1st 7716 df-2nd 7717 df-gdiv 28433 |
This theorem is referenced by: grpodivval 28472 grpodivf 28475 nvmfval 28581 |
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