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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 19634 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )) |
4 | 3 | fveq1d 6894 | . 2 ⊢ (𝐼 ∈ 𝑉 → (𝑈‘𝐴) = ((𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )‘𝐴)) |
5 | opeq1 4874 | . . . . 5 ⊢ (𝑗 = 𝐴 → ⟨𝑗, ∅⟩ = ⟨𝐴, ∅⟩) | |
6 | 5 | s1eqd 14551 | . . . 4 ⊢ (𝑗 = 𝐴 → ⟨“⟨𝑗, ∅⟩”⟩ = ⟨“⟨𝐴, ∅⟩”⟩) |
7 | 6 | eceq1d 8742 | . . 3 ⊢ (𝑗 = 𝐴 → [⟨“⟨𝑗, ∅⟩”⟩] ∼ = [⟨“⟨𝐴, ∅⟩”⟩] ∼ ) |
8 | eqid 2733 | . . 3 ⊢ (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ) = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ) | |
9 | 1 | fvexi 6906 | . . . 4 ⊢ ∼ ∈ V |
10 | ecexg 8707 | . . . 4 ⊢ ( ∼ ∈ V → [⟨“⟨𝐴, ∅⟩”⟩] ∼ ∈ V) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ [⟨“⟨𝐴, ∅⟩”⟩] ∼ ∈ V |
12 | 7, 8, 11 | fvmpt 6999 | . 2 ⊢ (𝐴 ∈ 𝐼 → ((𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ )‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ ) |
13 | 4, 12 | sylan9eq 2793 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4323 ⟨cop 4635 ↦ cmpt 5232 ‘cfv 6544 [cec 8701 ⟨“cs1 14545 ~FG cefg 19574 varFGrpcvrgp 19576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ec 8705 df-s1 14546 df-vrgp 19579 |
This theorem is referenced by: vrgpinv 19637 frgpup2 19644 frgpup3lem 19645 frgpnabllem1 19741 |
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