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Mirrors > Home > MPE Home > Th. List > gaid2 | Structured version Visualization version GIF version |
Description: A group operation is a left group action of the group on itself. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
gaid2.1 | ⊢ 𝑋 = (Base‘𝐺) |
gaid2.2 | ⊢ + = (+g‘𝐺) |
gaid2.3 | ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) |
Ref | Expression |
---|---|
gaid2 | ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpAct 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gaid2.1 | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
2 | 1 | subgid 18935 | . . 3 ⊢ (𝐺 ∈ Grp → 𝑋 ∈ (SubGrp‘𝐺)) |
3 | gaid2.2 | . . . 4 ⊢ + = (+g‘𝐺) | |
4 | eqid 2733 | . . . 4 ⊢ (𝐺 ↾s 𝑋) = (𝐺 ↾s 𝑋) | |
5 | gaid2.3 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦)) | |
6 | 1, 3, 4, 5 | subgga 19085 | . . 3 ⊢ (𝑋 ∈ (SubGrp‘𝐺) → 𝐹 ∈ ((𝐺 ↾s 𝑋) GrpAct 𝑋)) |
7 | 2, 6 | syl 17 | . 2 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ ((𝐺 ↾s 𝑋) GrpAct 𝑋)) |
8 | 1 | ressid 17130 | . . 3 ⊢ (𝐺 ∈ Grp → (𝐺 ↾s 𝑋) = 𝐺) |
9 | 8 | oveq1d 7373 | . 2 ⊢ (𝐺 ∈ Grp → ((𝐺 ↾s 𝑋) GrpAct 𝑋) = (𝐺 GrpAct 𝑋)) |
10 | 7, 9 | eleqtrd 2836 | 1 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpAct 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 ∈ cmpo 7360 Basecbs 17088 ↾s cress 17117 +gcplusg 17138 Grpcgrp 18753 SubGrpcsubg 18927 GrpAct cga 19074 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-subg 18930 df-ga 19075 |
This theorem is referenced by: cayleylem1 19199 |
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