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Mirrors > Home > ILE Home > Th. List > grpinvfvalg | GIF version |
Description: The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (Revised by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
grpinvval.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinvval.p | ⊢ + = (+g‘𝐺) |
grpinvval.o | ⊢ 0 = (0g‘𝐺) |
grpinvval.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvfvalg | ⊢ (𝐺 ∈ 𝑉 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvval.n | . 2 ⊢ 𝑁 = (invg‘𝐺) | |
2 | df-minusg 12921 | . . 3 ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)))) | |
3 | fveq2 5530 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
4 | grpinvval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 3, 4 | eqtr4di 2240 | . . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
6 | fveq2 5530 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
7 | grpinvval.p | . . . . . . . 8 ⊢ + = (+g‘𝐺) | |
8 | 6, 7 | eqtr4di 2240 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
9 | 8 | oveqd 5908 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥)) |
10 | fveq2 5530 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺)) | |
11 | grpinvval.o | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
12 | 10, 11 | eqtr4di 2240 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 ) |
13 | 9, 12 | eqeq12d 2204 | . . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑦(+g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 + 𝑥) = 0 )) |
14 | 5, 13 | riotaeqbidv 5850 | . . . 4 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
15 | 5, 14 | mpteq12dv 4100 | . . 3 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
16 | elex 2763 | . . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V) | |
17 | basfn 12544 | . . . . . 6 ⊢ Base Fn V | |
18 | funfvex 5547 | . . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V) | |
19 | 18 | funfni 5331 | . . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V) |
20 | 17, 16, 19 | sylancr 414 | . . . . 5 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V) |
21 | 4, 20 | eqeltrid 2276 | . . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐵 ∈ V) |
22 | 21 | mptexd 5759 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) ∈ V) |
23 | 2, 15, 16, 22 | fvmptd3 5625 | . 2 ⊢ (𝐺 ∈ 𝑉 → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
24 | 1, 23 | eqtrid 2234 | 1 ⊢ (𝐺 ∈ 𝑉 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ↦ cmpt 4079 Fn wfn 5226 ‘cfv 5231 ℩crio 5846 (class class class)co 5891 Basecbs 12486 +gcplusg 12561 0gc0g 12733 invgcminusg 12918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-cnex 7921 ax-resscn 7922 ax-1re 7924 ax-addrcl 7927 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-inn 8939 df-ndx 12489 df-slot 12490 df-base 12492 df-minusg 12921 |
This theorem is referenced by: grpinvval 12959 grpinvfng 12960 grpsubval 12962 grpinvf 12963 grpinvpropdg 12991 opprnegg 13400 |
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