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 18894 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
4 | 3 | fveq1d 6674 | . 2 ⊢ (𝐼 ∈ 𝑉 → (𝑈‘𝐴) = ((𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )‘𝐴)) |
5 | opeq1 4805 | . . . . 5 ⊢ (𝑗 = 𝐴 → 〈𝑗, ∅〉 = 〈𝐴, ∅〉) | |
6 | 5 | s1eqd 13957 | . . . 4 ⊢ (𝑗 = 𝐴 → 〈“〈𝑗, ∅〉”〉 = 〈“〈𝐴, ∅〉”〉) |
7 | 6 | eceq1d 8330 | . . 3 ⊢ (𝑗 = 𝐴 → [〈“〈𝑗, ∅〉”〉] ∼ = [〈“〈𝐴, ∅〉”〉] ∼ ) |
8 | eqid 2823 | . . 3 ⊢ (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ ) = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ ) | |
9 | 1 | fvexi 6686 | . . . 4 ⊢ ∼ ∈ V |
10 | ecexg 8295 | . . . 4 ⊢ ( ∼ ∈ V → [〈“〈𝐴, ∅〉”〉] ∼ ∈ V) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ [〈“〈𝐴, ∅〉”〉] ∼ ∈ V |
12 | 7, 8, 11 | fvmpt 6770 | . 2 ⊢ (𝐴 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
13 | 4, 12 | sylan9eq 2878 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 〈cop 4575 ↦ cmpt 5148 ‘cfv 6357 [cec 8289 〈“cs1 13951 ~FG cefg 18834 varFGrpcvrgp 18836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ec 8293 df-s1 13952 df-vrgp 18839 |
This theorem is referenced by: vrgpinv 18897 frgpup2 18904 frgpup3lem 18905 frgpnabllem1 18995 |
Copyright terms: Public domain | W3C validator |