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Theorem subgga 19331
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 19160 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
3 subgga.1 . . . 4 𝑋 = (Base‘𝐺)
43fvexi 6921 . . 3 𝑋 ∈ V
52, 4jctir 520 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝐻 ∈ Grp ∧ 𝑋 ∈ V))
6 subgrcl 19162 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
76adantr 480 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝐺 ∈ Grp)
83subgss 19158 . . . . . . . . 9 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
98sselda 3995 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑌) → 𝑥𝑋)
109adantrr 717 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝑥𝑋)
11 simprr 773 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝑦𝑋)
12 subgga.2 . . . . . . . 8 + = (+g𝐺)
133, 12grpcl 18972 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝑦𝑋) → (𝑥 + 𝑦) ∈ 𝑋)
147, 10, 11, 13syl3anc 1370 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → (𝑥 + 𝑦) ∈ 𝑋)
1514ralrimivva 3200 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → ∀𝑥𝑌𝑦𝑋 (𝑥 + 𝑦) ∈ 𝑋)
16 subgga.4 . . . . . 6 𝐹 = (𝑥𝑌, 𝑦𝑋 ↦ (𝑥 + 𝑦))
1716fmpo 8092 . . . . 5 (∀𝑥𝑌𝑦𝑋 (𝑥 + 𝑦) ∈ 𝑋𝐹:(𝑌 × 𝑋)⟶𝑋)
1815, 17sylib 218 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:(𝑌 × 𝑋)⟶𝑋)
191subgbas 19161 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 = (Base‘𝐻))
2019xpeq1d 5718 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑌 × 𝑋) = ((Base‘𝐻) × 𝑋))
2120feq2d 6723 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝐹:(𝑌 × 𝑋)⟶𝑋𝐹:((Base‘𝐻) × 𝑋)⟶𝑋))
2218, 21mpbid 232 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:((Base‘𝐻) × 𝑋)⟶𝑋)
23 eqid 2735 . . . . . . . 8 (0g𝐺) = (0g𝐺)
2423subg0cl 19165 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑌)
25 oveq12 7440 . . . . . . . 8 ((𝑥 = (0g𝐺) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((0g𝐺) + 𝑢))
26 ovex 7464 . . . . . . . 8 ((0g𝐺) + 𝑢) ∈ V
2725, 16, 26ovmpoa 7588 . . . . . . 7 (((0g𝐺) ∈ 𝑌𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐺) + 𝑢))
2824, 27sylan 580 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐺) + 𝑢))
291, 23subg0 19163 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
3029oveq1d 7446 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → ((0g𝐺)𝐹𝑢) = ((0g𝐻)𝐹𝑢))
3130adantr 480 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐻)𝐹𝑢))
323, 12, 23grplid 18998 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑢𝑋) → ((0g𝐺) + 𝑢) = 𝑢)
336, 32sylan 580 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺) + 𝑢) = 𝑢)
3428, 31, 333eqtr3d 2783 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐻)𝐹𝑢) = 𝑢)
356ad2antrr 726 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝐺 ∈ Grp)
368ad2antrr 726 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑌𝑋)
37 simprl 771 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑣𝑌)
3836, 37sseldd 3996 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑣𝑋)
39 simprr 773 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑤𝑌)
4036, 39sseldd 3996 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑤𝑋)
41 simplr 769 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑢𝑋)
423, 12grpass 18973 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑣𝑋𝑤𝑋𝑢𝑋)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
4335, 38, 40, 41, 42syl13anc 1371 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
443, 12grpcl 18972 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝑢𝑋) → (𝑤 + 𝑢) ∈ 𝑋)
4535, 40, 41, 44syl3anc 1370 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑤 + 𝑢) ∈ 𝑋)
46 oveq12 7440 . . . . . . . . . . 11 ((𝑥 = 𝑣𝑦 = (𝑤 + 𝑢)) → (𝑥 + 𝑦) = (𝑣 + (𝑤 + 𝑢)))
47 ovex 7464 . . . . . . . . . . 11 (𝑣 + (𝑤 + 𝑢)) ∈ V
4846, 16, 47ovmpoa 7588 . . . . . . . . . 10 ((𝑣𝑌 ∧ (𝑤 + 𝑢) ∈ 𝑋) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
4937, 45, 48syl2anc 584 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
5043, 49eqtr4d 2778 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣𝐹(𝑤 + 𝑢)))
5112subgcl 19167 . . . . . . . . . . 11 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑣𝑌𝑤𝑌) → (𝑣 + 𝑤) ∈ 𝑌)
52513expb 1119 . . . . . . . . . 10 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
5352adantlr 715 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
54 oveq12 7440 . . . . . . . . . 10 ((𝑥 = (𝑣 + 𝑤) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((𝑣 + 𝑤) + 𝑢))
55 ovex 7464 . . . . . . . . . 10 ((𝑣 + 𝑤) + 𝑢) ∈ V
5654, 16, 55ovmpoa 7588 . . . . . . . . 9 (((𝑣 + 𝑤) ∈ 𝑌𝑢𝑋) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
5753, 41, 56syl2anc 584 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
58 oveq12 7440 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑢) → (𝑥 + 𝑦) = (𝑤 + 𝑢))
59 ovex 7464 . . . . . . . . . . 11 (𝑤 + 𝑢) ∈ V
6058, 16, 59ovmpoa 7588 . . . . . . . . . 10 ((𝑤𝑌𝑢𝑋) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
6139, 41, 60syl2anc 584 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
6261oveq2d 7447 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣𝐹(𝑤𝐹𝑢)) = (𝑣𝐹(𝑤 + 𝑢)))
6350, 57, 623eqtr4d 2785 . . . . . . 7 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
6463ralrimivva 3200 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ∀𝑣𝑌𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
651, 12ressplusg 17336 . . . . . . . . . . . 12 (𝑌 ∈ (SubGrp‘𝐺) → + = (+g𝐻))
6665oveqd 7448 . . . . . . . . . . 11 (𝑌 ∈ (SubGrp‘𝐺) → (𝑣 + 𝑤) = (𝑣(+g𝐻)𝑤))
6766oveq1d 7446 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣(+g𝐻)𝑤)𝐹𝑢))
6867eqeq1d 2737 . . . . . . . . 9 (𝑌 ∈ (SubGrp‘𝐺) → (((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
6919, 68raleqbidv 3344 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → (∀𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7019, 69raleqbidv 3344 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (∀𝑣𝑌𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7170biimpa 476 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ ∀𝑣𝑌𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))) → ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
7264, 71syldan 591 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
7334, 72jca 511 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → (((0g𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7473ralrimiva 3144 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → ∀𝑢𝑋 (((0g𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7522, 74jca 511 . 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 2735 . . 3 (Base‘𝐻) = (Base‘𝐻)
77 eqid 2735 . . 3 (+g𝐻) = (+g𝐻)
78 eqid 2735 . . 3 (0g𝐻) = (0g𝐻)
7976, 77, 78isga 19322 . 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 583 1 (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  Basecbs 17245  s cress 17274  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964  SubGrpcsubg 19151   GrpAct cga 19320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-subg 19154  df-ga 19321
This theorem is referenced by:  gaid2  19334
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