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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 | 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 3466 | . . 3 β’ (πΌ β π β πΌ β V) | |
3 | xpeq1 5652 | . . . . . . 7 β’ (π = πΌ β (π Γ 2o) = (πΌ Γ 2o)) | |
4 | 3 | fveq2d 6851 | . . . . . 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 2795 | . . . . 5 β’ (π = πΌ β (freeMndβ(π Γ 2o)) = π) |
7 | fveq2 6847 | . . . . . 6 β’ (π = πΌ β ( ~FG βπ) = ( ~FG βπΌ)) | |
8 | frgpval.r | . . . . . 6 β’ βΌ = ( ~FG βπΌ) | |
9 | 7, 8 | eqtr4di 2795 | . . . . 5 β’ (π = πΌ β ( ~FG βπ) = βΌ ) |
10 | 6, 9 | oveq12d 7380 | . . . 4 β’ (π = πΌ β ((freeMndβ(π Γ 2o)) /s ( ~FG βπ)) = (π /s βΌ )) |
11 | df-frgp 19499 | . . . 4 β’ freeGrp = (π β V β¦ ((freeMndβ(π Γ 2o)) /s ( ~FG βπ))) | |
12 | ovex 7395 | . . . 4 β’ (π /s βΌ ) β V | |
13 | 10, 11, 12 | fvmpt 6953 | . . 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 2789 | 1 β’ (πΌ β π β πΊ = (π /s βΌ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3448 Γ cxp 5636 βcfv 6501 (class class class)co 7362 2oc2o 8411 /s cqus 17394 freeMndcfrmd 18664 ~FG cefg 19495 freeGrpcfrgp 19496 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-frgp 19499 |
This theorem is referenced by: frgp0 19549 frgpeccl 19550 frgpadd 19552 frgpupf 19562 frgpup1 19564 frgpup3lem 19566 frgpnabllem2 19659 |
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