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Mirrors > Home > MPE Home > Th. List > ga0 | Structured version Visualization version GIF version |
Description: The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
ga0 | β’ (πΊ β Grp β β β (πΊ GrpAct β )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5308 | . . 3 β’ β β V | |
2 | 1 | jctr 526 | . 2 β’ (πΊ β Grp β (πΊ β Grp β§ β β V)) |
3 | f0 6773 | . . . 4 β’ β :β βΆβ | |
4 | xp0 6158 | . . . . 5 β’ ((BaseβπΊ) Γ β ) = β | |
5 | 4 | feq2i 6710 | . . . 4 β’ (β :((BaseβπΊ) Γ β )βΆβ β β :β βΆβ ) |
6 | 3, 5 | mpbir 230 | . . 3 β’ β :((BaseβπΊ) Γ β )βΆβ |
7 | ral0 4513 | . . 3 β’ βπ₯ β β (((0gβπΊ)β π₯) = π₯ β§ βπ¦ β (BaseβπΊ)βπ§ β (BaseβπΊ)((π¦(+gβπΊ)π§)β π₯) = (π¦β (π§β π₯))) | |
8 | 6, 7 | pm3.2i 472 | . 2 β’ (β :((BaseβπΊ) Γ β )βΆβ β§ βπ₯ β β (((0gβπΊ)β π₯) = π₯ β§ βπ¦ β (BaseβπΊ)βπ§ β (BaseβπΊ)((π¦(+gβπΊ)π§)β π₯) = (π¦β (π§β π₯)))) |
9 | eqid 2733 | . . 3 β’ (BaseβπΊ) = (BaseβπΊ) | |
10 | eqid 2733 | . . 3 β’ (+gβπΊ) = (+gβπΊ) | |
11 | eqid 2733 | . . 3 β’ (0gβπΊ) = (0gβπΊ) | |
12 | 9, 10, 11 | isga 19155 | . 2 β’ (β β (πΊ GrpAct β ) β ((πΊ β Grp β§ β β V) β§ (β :((BaseβπΊ) Γ β )βΆβ β§ βπ₯ β β (((0gβπΊ)β π₯) = π₯ β§ βπ¦ β (BaseβπΊ)βπ§ β (BaseβπΊ)((π¦(+gβπΊ)π§)β π₯) = (π¦β (π§β π₯)))))) |
13 | 2, 8, 12 | sylanblrc 591 | 1 β’ (πΊ β Grp β β β (πΊ GrpAct β )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3062 Vcvv 3475 β c0 4323 Γ cxp 5675 βΆwf 6540 βcfv 6544 (class class class)co 7409 Basecbs 17144 +gcplusg 17197 0gc0g 17385 Grpcgrp 18819 GrpAct cga 19153 |
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-sep 5300 ax-nul 5307 ax-pow 5364 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-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-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-map 8822 df-ga 19154 |
This theorem is referenced by: (None) |
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