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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 7107 | . 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)) ∈ V | |
2 | grpinvfn.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2818 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2818 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | grpinvfn.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
6 | 2, 3, 4, 5 | grpinvfval 18080 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))) |
7 | 1, 6 | fnmpti 6484 | 1 ⊢ 𝑁 Fn 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 Fn wfn 6343 ‘cfv 6348 ℩crio 7102 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 invgcminusg 18042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-riota 7103 df-ov 7148 df-minusg 18045 |
This theorem is referenced by: grpinvfvi 18084 isgrpinv 18094 invrfval 19352 |
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