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Mirrors > Home > MPE Home > Th. List > frgpval | Structured version Visualization version GIF version |
Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.) |
Ref | Expression |
---|---|
frgpval.m | ⊢ 𝐺 = (freeGrp‘𝐼) |
frgpval.b | ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o)) |
frgpval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
Ref | Expression |
---|---|
frgpval | ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgpval.m | . 2 ⊢ 𝐺 = (freeGrp‘𝐼) | |
2 | elex 3461 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
3 | xpeq1 5645 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o)) | |
4 | 3 | fveq2d 6843 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o))) |
5 | frgpval.b | . . . . . 6 ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o)) | |
6 | 4, 5 | eqtr4di 2795 | . . . . 5 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀) |
7 | fveq2 6839 | . . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼)) | |
8 | frgpval.r | . . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼) | |
9 | 7, 8 | eqtr4di 2795 | . . . . 5 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ ) |
10 | 6, 9 | oveq12d 7369 | . . . 4 ⊢ (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)) = (𝑀 /s ∼ )) |
11 | df-frgp 19451 | . . . 4 ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖))) | |
12 | ovex 7384 | . . . 4 ⊢ (𝑀 /s ∼ ) ∈ V | |
13 | 10, 11, 12 | fvmpt 6945 | . . 3 ⊢ (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ∼ )) |
14 | 2, 13 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (freeGrp‘𝐼) = (𝑀 /s ∼ )) |
15 | 1, 14 | eqtrid 2789 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3443 × cxp 5629 ‘cfv 6493 (class class class)co 7351 2oc2o 8398 /s cqus 17347 freeMndcfrmd 18617 ~FG cefg 19447 freeGrpcfrgp 19448 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 df-ov 7354 df-frgp 19451 |
This theorem is referenced by: frgp0 19501 frgpeccl 19502 frgpadd 19504 frgpupf 19514 frgpup1 19516 frgpup3lem 19518 frgpnabllem2 19611 |
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