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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 7311 | . 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V | |
2 | grpinvfn.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2737 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2737 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | grpinvfn.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
6 | 2, 3, 4, 5 | grpinvfval 18749 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
7 | 1, 6 | fnmpti 6641 | 1 ⊢ 𝑁 Fn 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 Fn wfn 6488 ‘cfv 6493 ℩crio 7306 (class class class)co 7351 Basecbs 17043 +gcplusg 17093 0gc0g 17281 invgcminusg 18709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-riota 7307 df-ov 7354 df-minusg 18712 |
This theorem is referenced by: grpinvfvi 18753 isgrpinv 18763 invrfval 20055 mplsubglem 21357 mhpinvcl 21494 |
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