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Mirrors > Home > ILE Home > Th. List > fn0g | GIF version |
Description: The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
Ref | Expression |
---|---|
fn0g | ⊢ 0g Fn V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-riota 5873 | . . 3 ⊢ (℩𝑒 ∈ (Base‘𝑔)∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)) = (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))) | |
2 | basfn 12676 | . . . . 5 ⊢ Base Fn V | |
3 | vex 2763 | . . . . 5 ⊢ 𝑔 ∈ V | |
4 | funfvex 5571 | . . . . . 6 ⊢ ((Fun Base ∧ 𝑔 ∈ dom Base) → (Base‘𝑔) ∈ V) | |
5 | 4 | funfni 5354 | . . . . 5 ⊢ ((Base Fn V ∧ 𝑔 ∈ V) → (Base‘𝑔) ∈ V) |
6 | 2, 3, 5 | mp2an 426 | . . . 4 ⊢ (Base‘𝑔) ∈ V |
7 | riotaexg 5877 | . . . 4 ⊢ ((Base‘𝑔) ∈ V → (℩𝑒 ∈ (Base‘𝑔)∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)) ∈ V) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ (℩𝑒 ∈ (Base‘𝑔)∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)) ∈ V |
9 | 1, 8 | eqeltrri 2267 | . 2 ⊢ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))) ∈ V |
10 | df-0g 12869 | . 2 ⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)))) | |
11 | 9, 10 | fnmpti 5382 | 1 ⊢ 0g Fn V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ∈ wcel 2164 ∀wral 2472 Vcvv 2760 ℩cio 5213 Fn wfn 5249 ‘cfv 5254 ℩crio 5872 (class class class)co 5918 Basecbs 12618 +gcplusg 12695 0gc0g 12867 |
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 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-cnex 7963 ax-resscn 7964 ax-1re 7966 ax-addrcl 7969 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-sbc 2986 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-iota 5215 df-fun 5256 df-fn 5257 df-fv 5262 df-riota 5873 df-inn 8983 df-ndx 12621 df-slot 12622 df-base 12624 df-0g 12869 |
This theorem is referenced by: fngsum 12971 igsumvalx 12972 gsumfzval 12974 gsum0g 12979 0mhm 13058 mulgval 13192 mulgfng 13194 issrg 13461 isdomn 13765 |
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