Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > grpinvfn | Structured version Visualization version GIF version |
Description: Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
grpinvfn.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinvfn.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvfn | ⊢ 𝑁 Fn 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotaex 6515 | . 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V | |
2 | grpinvfn.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2610 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2610 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | grpinvfn.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
6 | 2, 3, 4, 5 | grpinvfval 17283 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
7 | 1, 6 | fnmpti 5935 | 1 ⊢ 𝑁 Fn 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 Fn wfn 5799 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 invgcminusg 17246 |
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 |
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-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-riota 6511 df-ov 6552 df-minusg 17249 |
This theorem is referenced by: grpinvfvi 17286 isgrpinv 17295 invrfval 18496 |
Copyright terms: Public domain | W3C validator |