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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 18664 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) |
4 | 3 | fveq1d 6498 | . 2 ⊢ (𝐼 ∈ 𝑉 → (𝑈‘𝐴) = ((𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )‘𝐴)) |
5 | opeq1 4673 | . . . . 5 ⊢ (𝑗 = 𝐴 → 〈𝑗, ∅〉 = 〈𝐴, ∅〉) | |
6 | 5 | s1eqd 13762 | . . . 4 ⊢ (𝑗 = 𝐴 → 〈“〈𝑗, ∅〉”〉 = 〈“〈𝐴, ∅〉”〉) |
7 | 6 | eceq1d 8126 | . . 3 ⊢ (𝑗 = 𝐴 → [〈“〈𝑗, ∅〉”〉] ∼ = [〈“〈𝐴, ∅〉”〉] ∼ ) |
8 | eqid 2771 | . . 3 ⊢ (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ ) = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ ) | |
9 | 1 | fvexi 6510 | . . . 4 ⊢ ∼ ∈ V |
10 | ecexg 8091 | . . . 4 ⊢ ( ∼ ∈ V → [〈“〈𝐴, ∅〉”〉] ∼ ∈ V) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ [〈“〈𝐴, ∅〉”〉] ∼ ∈ V |
12 | 7, 8, 11 | fvmpt 6593 | . 2 ⊢ (𝐴 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
13 | 4, 12 | sylan9eq 2827 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 Vcvv 3408 ∅c0 4172 〈cop 4441 ↦ cmpt 5004 ‘cfv 6185 [cec 8085 〈“cs1 13756 ~FG cefg 18602 varFGrpcvrgp 18604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ec 8089 df-s1 13757 df-vrgp 18607 |
This theorem is referenced by: vrgpinv 18667 frgpup2 18674 frgpup3lem 18675 frgpnabllem1 18761 |
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