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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 6990 | . . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V) | |
4 | 2, 3 | syl5eqel 2692 | . . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V) |
5 | mpt2exga 7135 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) | |
6 | 4, 4, 5 | syl2anc 691 | . . 3 ⊢ (𝐺 ∈ GrpOp → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) |
7 | rneq 5272 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺) | |
8 | 7, 2 | syl6eqr 2662 | . . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋) |
9 | id 22 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑔 = 𝐺) | |
10 | eqidd 2611 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥) | |
11 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = (inv‘𝐺)) | |
12 | grpdiv.2 | . . . . . . . 8 ⊢ 𝑁 = (inv‘𝐺) | |
13 | 11, 12 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = 𝑁) |
14 | 13 | fveq1d 6105 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((inv‘𝑔)‘𝑦) = (𝑁‘𝑦)) |
15 | 9, 10, 14 | oveq123d 6570 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥𝑔((inv‘𝑔)‘𝑦)) = (𝑥𝐺(𝑁‘𝑦))) |
16 | 8, 8, 15 | mpt2eq123dv 6615 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
17 | df-gdiv 26734 | . . . 4 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) | |
18 | 16, 17 | fvmptg 6189 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
19 | 6, 18 | mpdan 699 | . 2 ⊢ (𝐺 ∈ GrpOp → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
20 | 1, 19 | syl5eq 2656 | 1 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 GrpOpcgr 26727 invcgn 26729 /𝑔 cgs 26730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-gdiv 26734 |
This theorem is referenced by: grpodivval 26773 grpodivf 26776 nvmfval 26883 |
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