![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | β’ πΊ = (freeGrpβπΌ) |
frgpval.b | β’ π = (freeMndβ(πΌ Γ 2o)) |
frgpval.r | β’ βΌ = ( ~FG βπΌ) |
Ref | Expression |
---|---|
frgpval | β’ (πΌ β π β πΊ = (π /s βΌ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgpval.m | . 2 β’ πΊ = (freeGrpβπΌ) | |
2 | elex 3487 | . . 3 β’ (πΌ β π β πΌ β V) | |
3 | xpeq1 5683 | . . . . . . 7 β’ (π = πΌ β (π Γ 2o) = (πΌ Γ 2o)) | |
4 | 3 | fveq2d 6888 | . . . . . 6 β’ (π = πΌ β (freeMndβ(π Γ 2o)) = (freeMndβ(πΌ Γ 2o))) |
5 | frgpval.b | . . . . . 6 β’ π = (freeMndβ(πΌ Γ 2o)) | |
6 | 4, 5 | eqtr4di 2784 | . . . . 5 β’ (π = πΌ β (freeMndβ(π Γ 2o)) = π) |
7 | fveq2 6884 | . . . . . 6 β’ (π = πΌ β ( ~FG βπ) = ( ~FG βπΌ)) | |
8 | frgpval.r | . . . . . 6 β’ βΌ = ( ~FG βπΌ) | |
9 | 7, 8 | eqtr4di 2784 | . . . . 5 β’ (π = πΌ β ( ~FG βπ) = βΌ ) |
10 | 6, 9 | oveq12d 7422 | . . . 4 β’ (π = πΌ β ((freeMndβ(π Γ 2o)) /s ( ~FG βπ)) = (π /s βΌ )) |
11 | df-frgp 19627 | . . . 4 β’ freeGrp = (π β V β¦ ((freeMndβ(π Γ 2o)) /s ( ~FG βπ))) | |
12 | ovex 7437 | . . . 4 β’ (π /s βΌ ) β V | |
13 | 10, 11, 12 | fvmpt 6991 | . . 3 β’ (πΌ β V β (freeGrpβπΌ) = (π /s βΌ )) |
14 | 2, 13 | syl 17 | . 2 β’ (πΌ β π β (freeGrpβπΌ) = (π /s βΌ )) |
15 | 1, 14 | eqtrid 2778 | 1 β’ (πΌ β π β πΊ = (π /s βΌ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 Γ cxp 5667 βcfv 6536 (class class class)co 7404 2oc2o 8458 /s cqus 17457 freeMndcfrmd 18769 ~FG cefg 19623 freeGrpcfrgp 19624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-frgp 19627 |
This theorem is referenced by: frgp0 19677 frgpeccl 19678 frgpadd 19680 frgpupf 19690 frgpup1 19692 frgpup3lem 19694 frgpnabllem2 19791 |
Copyright terms: Public domain | W3C validator |