Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7614 | . . . . 5 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V) | |
4 | 2, 3 | eqeltrid 2917 | . . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V) |
5 | mpoexga 7775 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) | |
6 | 4, 4, 5 | syl2anc 586 | . . 3 ⊢ (𝐺 ∈ GrpOp → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) |
7 | rneq 5806 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺) | |
8 | 7, 2 | syl6eqr 2874 | . . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋) |
9 | id 22 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑔 = 𝐺) | |
10 | eqidd 2822 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥) | |
11 | fveq2 6670 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = (inv‘𝐺)) | |
12 | grpdiv.2 | . . . . . . . 8 ⊢ 𝑁 = (inv‘𝐺) | |
13 | 11, 12 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (inv‘𝑔) = 𝑁) |
14 | 13 | fveq1d 6672 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((inv‘𝑔)‘𝑦) = (𝑁‘𝑦)) |
15 | 9, 10, 14 | oveq123d 7177 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥𝑔((inv‘𝑔)‘𝑦)) = (𝑥𝐺(𝑁‘𝑦))) |
16 | 8, 8, 15 | mpoeq123dv 7229 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
17 | df-gdiv 28273 | . . . 4 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) | |
18 | 16, 17 | fvmptg 6766 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) ∈ V) → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
19 | 6, 18 | mpdan 685 | . 2 ⊢ (𝐺 ∈ GrpOp → ( /𝑔 ‘𝐺) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
20 | 1, 19 | syl5eq 2868 | 1 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ran crn 5556 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 GrpOpcgr 28266 invcgn 28268 /𝑔 cgs 28269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-gdiv 28273 |
This theorem is referenced by: grpodivval 28312 grpodivf 28315 nvmfval 28421 |
Copyright terms: Public domain | W3C validator |