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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 7751 | . . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V) | |
4 | 2, 3 | eqeltrid 2843 | . . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V) |
5 | mpoexga 7918 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) | |
6 | 4, 4, 5 | syl2anc 584 | . . 3 ⊢ (𝐺 ∈ GrpOp → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) |
7 | rneq 5845 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺) | |
8 | 7, 2 | eqtr4di 2796 | . . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋) |
9 | id 22 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑔 = 𝐺) | |
10 | eqidd 2739 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥) | |
11 | fveq2 6774 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = (inv‘𝐺)) | |
12 | grpdiv.2 | . . . . . . . 8 ⊢ 𝑁 = (inv‘𝐺) | |
13 | 11, 12 | eqtr4di 2796 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = 𝑁) |
14 | 13 | fveq1d 6776 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((inv‘𝑔)‘𝑦) = (𝑁‘𝑦)) |
15 | 9, 10, 14 | oveq123d 7296 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥𝑔((inv‘𝑔)‘𝑦)) = (𝑥𝐺(𝑁‘𝑦))) |
16 | 8, 8, 15 | mpoeq123dv 7350 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
17 | df-gdiv 28858 | . . . 4 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) | |
18 | 16, 17 | fvmptg 6873 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
19 | 6, 18 | mpdan 684 | . 2 ⊢ (𝐺 ∈ GrpOp → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
20 | 1, 19 | eqtrid 2790 | 1 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ran crn 5590 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 GrpOpcgr 28851 invcgn 28853 /𝑔 cgs 28854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-gdiv 28858 |
This theorem is referenced by: grpodivval 28897 grpodivf 28900 nvmfval 29006 |
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