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Theorem subgga 19088
Description: A subgroup acts on its parent group. (Contributed by Jeff Hankins, 13-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
subgga.1 𝑋 = (Base‘𝐺)
subgga.2 + = (+g𝐺)
subgga.3 𝐻 = (𝐺s 𝑌)
subgga.4 𝐹 = (𝑥𝑌, 𝑦𝑋 ↦ (𝑥 + 𝑦))
Assertion
Ref Expression
subgga (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥, + ,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem subgga
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subgga.3 . . . 4 𝐻 = (𝐺s 𝑌)
21subggrp 18939 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
3 subgga.1 . . . 4 𝑋 = (Base‘𝐺)
43fvexi 6860 . . 3 𝑋 ∈ V
52, 4jctir 522 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝐻 ∈ Grp ∧ 𝑋 ∈ V))
6 subgrcl 18941 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
76adantr 482 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝐺 ∈ Grp)
83subgss 18937 . . . . . . . . 9 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
98sselda 3948 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑌) → 𝑥𝑋)
109adantrr 716 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝑥𝑋)
11 simprr 772 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝑦𝑋)
12 subgga.2 . . . . . . . 8 + = (+g𝐺)
133, 12grpcl 18764 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝑦𝑋) → (𝑥 + 𝑦) ∈ 𝑋)
147, 10, 11, 13syl3anc 1372 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → (𝑥 + 𝑦) ∈ 𝑋)
1514ralrimivva 3194 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → ∀𝑥𝑌𝑦𝑋 (𝑥 + 𝑦) ∈ 𝑋)
16 subgga.4 . . . . . 6 𝐹 = (𝑥𝑌, 𝑦𝑋 ↦ (𝑥 + 𝑦))
1716fmpo 8004 . . . . 5 (∀𝑥𝑌𝑦𝑋 (𝑥 + 𝑦) ∈ 𝑋𝐹:(𝑌 × 𝑋)⟶𝑋)
1815, 17sylib 217 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:(𝑌 × 𝑋)⟶𝑋)
191subgbas 18940 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 = (Base‘𝐻))
2019xpeq1d 5666 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑌 × 𝑋) = ((Base‘𝐻) × 𝑋))
2120feq2d 6658 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝐹:(𝑌 × 𝑋)⟶𝑋𝐹:((Base‘𝐻) × 𝑋)⟶𝑋))
2218, 21mpbid 231 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:((Base‘𝐻) × 𝑋)⟶𝑋)
23 eqid 2733 . . . . . . . 8 (0g𝐺) = (0g𝐺)
2423subg0cl 18944 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑌)
25 oveq12 7370 . . . . . . . 8 ((𝑥 = (0g𝐺) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((0g𝐺) + 𝑢))
26 ovex 7394 . . . . . . . 8 ((0g𝐺) + 𝑢) ∈ V
2725, 16, 26ovmpoa 7514 . . . . . . 7 (((0g𝐺) ∈ 𝑌𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐺) + 𝑢))
2824, 27sylan 581 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐺) + 𝑢))
291, 23subg0 18942 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
3029oveq1d 7376 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → ((0g𝐺)𝐹𝑢) = ((0g𝐻)𝐹𝑢))
3130adantr 482 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐻)𝐹𝑢))
323, 12, 23grplid 18788 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑢𝑋) → ((0g𝐺) + 𝑢) = 𝑢)
336, 32sylan 581 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺) + 𝑢) = 𝑢)
3428, 31, 333eqtr3d 2781 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐻)𝐹𝑢) = 𝑢)
356ad2antrr 725 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝐺 ∈ Grp)
368ad2antrr 725 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑌𝑋)
37 simprl 770 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑣𝑌)
3836, 37sseldd 3949 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑣𝑋)
39 simprr 772 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑤𝑌)
4036, 39sseldd 3949 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑤𝑋)
41 simplr 768 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑢𝑋)
423, 12grpass 18765 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑣𝑋𝑤𝑋𝑢𝑋)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
4335, 38, 40, 41, 42syl13anc 1373 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
443, 12grpcl 18764 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝑢𝑋) → (𝑤 + 𝑢) ∈ 𝑋)
4535, 40, 41, 44syl3anc 1372 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑤 + 𝑢) ∈ 𝑋)
46 oveq12 7370 . . . . . . . . . . 11 ((𝑥 = 𝑣𝑦 = (𝑤 + 𝑢)) → (𝑥 + 𝑦) = (𝑣 + (𝑤 + 𝑢)))
47 ovex 7394 . . . . . . . . . . 11 (𝑣 + (𝑤 + 𝑢)) ∈ V
4846, 16, 47ovmpoa 7514 . . . . . . . . . 10 ((𝑣𝑌 ∧ (𝑤 + 𝑢) ∈ 𝑋) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
4937, 45, 48syl2anc 585 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
5043, 49eqtr4d 2776 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣𝐹(𝑤 + 𝑢)))
5112subgcl 18946 . . . . . . . . . . 11 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑣𝑌𝑤𝑌) → (𝑣 + 𝑤) ∈ 𝑌)
52513expb 1121 . . . . . . . . . 10 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
5352adantlr 714 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
54 oveq12 7370 . . . . . . . . . 10 ((𝑥 = (𝑣 + 𝑤) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((𝑣 + 𝑤) + 𝑢))
55 ovex 7394 . . . . . . . . . 10 ((𝑣 + 𝑤) + 𝑢) ∈ V
5654, 16, 55ovmpoa 7514 . . . . . . . . 9 (((𝑣 + 𝑤) ∈ 𝑌𝑢𝑋) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
5753, 41, 56syl2anc 585 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
58 oveq12 7370 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑢) → (𝑥 + 𝑦) = (𝑤 + 𝑢))
59 ovex 7394 . . . . . . . . . . 11 (𝑤 + 𝑢) ∈ V
6058, 16, 59ovmpoa 7514 . . . . . . . . . 10 ((𝑤𝑌𝑢𝑋) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
6139, 41, 60syl2anc 585 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
6261oveq2d 7377 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣𝐹(𝑤𝐹𝑢)) = (𝑣𝐹(𝑤 + 𝑢)))
6350, 57, 623eqtr4d 2783 . . . . . . 7 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
6463ralrimivva 3194 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ∀𝑣𝑌𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
651, 12ressplusg 17179 . . . . . . . . . . . 12 (𝑌 ∈ (SubGrp‘𝐺) → + = (+g𝐻))
6665oveqd 7378 . . . . . . . . . . 11 (𝑌 ∈ (SubGrp‘𝐺) → (𝑣 + 𝑤) = (𝑣(+g𝐻)𝑤))
6766oveq1d 7376 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣(+g𝐻)𝑤)𝐹𝑢))
6867eqeq1d 2735 . . . . . . . . 9 (𝑌 ∈ (SubGrp‘𝐺) → (((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
6919, 68raleqbidv 3318 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → (∀𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7019, 69raleqbidv 3318 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (∀𝑣𝑌𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7170biimpa 478 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ ∀𝑣𝑌𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))) → ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
7264, 71syldan 592 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
7334, 72jca 513 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → (((0g𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7473ralrimiva 3140 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → ∀𝑢𝑋 (((0g𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7522, 74jca 513 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝐹:((Base‘𝐻) × 𝑋)⟶𝑋 ∧ ∀𝑢𝑋 (((0g𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))))
76 eqid 2733 . . 3 (Base‘𝐻) = (Base‘𝐻)
77 eqid 2733 . . 3 (+g𝐻) = (+g𝐻)
78 eqid 2733 . . 3 (0g𝐻) = (0g𝐻)
7976, 77, 78isga 19079 . 2 (𝐹 ∈ (𝐻 GrpAct 𝑋) ↔ ((𝐻 ∈ Grp ∧ 𝑋 ∈ V) ∧ (𝐹:((Base‘𝐻) × 𝑋)⟶𝑋 ∧ ∀𝑢𝑋 (((0g𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))))
805, 75, 79sylanbrc 584 1 (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3447  wss 3914   × cxp 5635  wf 6496  cfv 6500  (class class class)co 7361  cmpo 7363  Basecbs 17091  s cress 17120  +gcplusg 17141  0gc0g 17329  Grpcgrp 18756  SubGrpcsubg 18930   GrpAct cga 19077
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-0g 17331  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-grp 18759  df-subg 18933  df-ga 19078
This theorem is referenced by:  gaid2  19091
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