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Mirrors > Home > ILE Home > Th. List > grpinvval | GIF version |
Description: The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) |
Ref | Expression |
---|---|
grpinvval.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinvval.p | ⊢ + = (+g‘𝐺) |
grpinvval.o | ⊢ 0 = (0g‘𝐺) |
grpinvval.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvval | ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | 1 | basmex 12515 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝐺 ∈ V) |
3 | grpinvval.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
4 | grpinvval.o | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
5 | grpinvval.n | . . . . 5 ⊢ 𝑁 = (invg‘𝐺) | |
6 | 1, 3, 4, 5 | grpinvfvalg 12869 | . . . 4 ⊢ (𝐺 ∈ V → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
7 | 2, 6 | syl 14 | . . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
8 | 7 | fveq1d 5517 | . 2 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))‘𝑋)) |
9 | eqid 2177 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) | |
10 | oveq2 5882 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋)) | |
11 | 10 | eqeq1d 2186 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 )) |
12 | 11 | riotabidv 5832 | . . 3 ⊢ (𝑥 = 𝑋 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
13 | id 19 | . . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵) | |
14 | basfn 12514 | . . . . . 6 ⊢ Base Fn V | |
15 | funfvex 5532 | . . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V) | |
16 | 15 | funfni 5316 | . . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V) |
17 | 14, 2, 16 | sylancr 414 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (Base‘𝐺) ∈ V) |
18 | 1, 17 | eqeltrid 2264 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝐵 ∈ V) |
19 | riotaexg 5834 | . . . 4 ⊢ (𝐵 ∈ V → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) ∈ V) | |
20 | 18, 19 | syl 14 | . . 3 ⊢ (𝑋 ∈ 𝐵 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) ∈ V) |
21 | 9, 12, 13, 20 | fvmptd3 5609 | . 2 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
22 | 8, 21 | eqtrd 2210 | 1 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ↦ cmpt 4064 Fn wfn 5211 ‘cfv 5216 ℩crio 5829 (class class class)co 5874 Basecbs 12456 +gcplusg 12530 0gc0g 12695 invgcminusg 12832 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-inn 8918 df-ndx 12459 df-slot 12460 df-base 12462 df-minusg 12835 |
This theorem is referenced by: grplinv 12876 isgrpinv 12880 |
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