![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > vrgpval | Structured version Visualization version GIF version |
Description: The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
vrgpfval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
vrgpfval.u | ⊢ 𝑈 = (varFGrp‘𝐼) |
Ref | Expression |
---|---|
vrgpval | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vrgpfval.r | . . . 4 ⊢ ∼ = ( ~FG ‘𝐼) | |
2 | vrgpfval.u | . . . 4 ⊢ 𝑈 = (varFGrp‘𝐼) | |
3 | 1, 2 | vrgpfval 18884 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
4 | 3 | fveq1d 6647 | . 2 ⊢ (𝐼 ∈ 𝑉 → (𝑈‘𝐴) = ((𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )‘𝐴)) |
5 | opeq1 4763 | . . . . 5 ⊢ (𝑗 = 𝐴 → 〈𝑗, ∅〉 = 〈𝐴, ∅〉) | |
6 | 5 | s1eqd 13946 | . . . 4 ⊢ (𝑗 = 𝐴 → 〈“〈𝑗, ∅〉”〉 = 〈“〈𝐴, ∅〉”〉) |
7 | 6 | eceq1d 8311 | . . 3 ⊢ (𝑗 = 𝐴 → [〈“〈𝑗, ∅〉”〉] ∼ = [〈“〈𝐴, ∅〉”〉] ∼ ) |
8 | eqid 2798 | . . 3 ⊢ (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ ) = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ ) | |
9 | 1 | fvexi 6659 | . . . 4 ⊢ ∼ ∈ V |
10 | ecexg 8276 | . . . 4 ⊢ ( ∼ ∈ V → [〈“〈𝐴, ∅〉”〉] ∼ ∈ V) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ [〈“〈𝐴, ∅〉”〉] ∼ ∈ V |
12 | 7, 8, 11 | fvmpt 6745 | . 2 ⊢ (𝐴 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
13 | 4, 12 | sylan9eq 2853 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 〈cop 4531 ↦ cmpt 5110 ‘cfv 6324 [cec 8270 〈“cs1 13940 ~FG cefg 18824 varFGrpcvrgp 18826 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ec 8274 df-s1 13941 df-vrgp 18829 |
This theorem is referenced by: vrgpinv 18887 frgpup2 18894 frgpup3lem 18895 frgpnabllem1 18986 |
Copyright terms: Public domain | W3C validator |