![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > grpodivid | Structured version Visualization version GIF version |
Description: Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpdivf.1 | ⊢ 𝑋 = ran 𝐺 |
grpdivf.3 | ⊢ 𝐷 = ( /𝑔 ‘𝐺) |
grpdivid.3 | ⊢ 𝑈 = (GId‘𝐺) |
Ref | Expression |
---|---|
grpodivid | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpdivf.1 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
2 | eqid 2726 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
3 | grpdivf.3 | . . . 4 ⊢ 𝐷 = ( /𝑔 ‘𝐺) | |
4 | 1, 2, 3 | grpodivval 30465 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = (𝐴𝐺((inv‘𝐺)‘𝐴))) |
5 | 4 | 3anidm23 1418 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = (𝐴𝐺((inv‘𝐺)‘𝐴))) |
6 | grpdivid.3 | . . 3 ⊢ 𝑈 = (GId‘𝐺) | |
7 | 1, 6, 2 | grporinv 30457 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺((inv‘𝐺)‘𝐴)) = 𝑈) |
8 | 5, 7 | eqtrd 2766 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ran crn 5675 ‘cfv 6546 (class class class)co 7416 GrpOpcgr 30419 GIdcgi 30420 invcgn 30421 /𝑔 cgs 30422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-grpo 30423 df-gid 30424 df-ginv 30425 df-gdiv 30426 |
This theorem is referenced by: ablonncan 30486 grpoeqdivid 37595 |
Copyright terms: Public domain | W3C validator |