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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 3499 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
3 | xpeq1 5703 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o)) | |
4 | 3 | fveq2d 6911 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o))) |
5 | frgpval.b | . . . . . 6 ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o)) | |
6 | 4, 5 | eqtr4di 2793 | . . . . 5 ⊢ (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀) |
7 | fveq2 6907 | . . . . . 6 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ( ~FG ‘𝐼)) | |
8 | frgpval.r | . . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼) | |
9 | 7, 8 | eqtr4di 2793 | . . . . 5 ⊢ (𝑖 = 𝐼 → ( ~FG ‘𝑖) = ∼ ) |
10 | 6, 9 | oveq12d 7449 | . . . 4 ⊢ (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)) = (𝑀 /s ∼ )) |
11 | df-frgp 19743 | . . . 4 ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖))) | |
12 | ovex 7464 | . . . 4 ⊢ (𝑀 /s ∼ ) ∈ V | |
13 | 10, 11, 12 | fvmpt 7016 | . . 3 ⊢ (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ∼ )) |
14 | 2, 13 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (freeGrp‘𝐼) = (𝑀 /s ∼ )) |
15 | 1, 14 | eqtrid 2787 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 × cxp 5687 ‘cfv 6563 (class class class)co 7431 2oc2o 8499 /s cqus 17552 freeMndcfrmd 18873 ~FG cefg 19739 freeGrpcfrgp 19740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-frgp 19743 |
This theorem is referenced by: frgp0 19793 frgpeccl 19794 frgpadd 19796 frgpupf 19806 frgpup1 19808 frgpup3lem 19810 frgpnabllem2 19907 |
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