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Theorem subgga 19201
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 19041 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
3 subgga.1 . . . 4 𝑋 = (Base‘𝐺)
43fvexi 6895 . . 3 𝑋 ∈ V
52, 4jctir 520 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝐻 ∈ Grp ∧ 𝑋 ∈ V))
6 subgrcl 19043 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
76adantr 480 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝐺 ∈ Grp)
83subgss 19039 . . . . . . . . 9 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
98sselda 3974 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑌) → 𝑥𝑋)
109adantrr 714 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝑥𝑋)
11 simprr 770 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝑦𝑋)
12 subgga.2 . . . . . . . 8 + = (+g𝐺)
133, 12grpcl 18858 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝑦𝑋) → (𝑥 + 𝑦) ∈ 𝑋)
147, 10, 11, 13syl3anc 1368 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → (𝑥 + 𝑦) ∈ 𝑋)
1514ralrimivva 3192 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → ∀𝑥𝑌𝑦𝑋 (𝑥 + 𝑦) ∈ 𝑋)
16 subgga.4 . . . . . 6 𝐹 = (𝑥𝑌, 𝑦𝑋 ↦ (𝑥 + 𝑦))
1716fmpo 8047 . . . . 5 (∀𝑥𝑌𝑦𝑋 (𝑥 + 𝑦) ∈ 𝑋𝐹:(𝑌 × 𝑋)⟶𝑋)
1815, 17sylib 217 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:(𝑌 × 𝑋)⟶𝑋)
191subgbas 19042 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 = (Base‘𝐻))
2019xpeq1d 5695 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑌 × 𝑋) = ((Base‘𝐻) × 𝑋))
2120feq2d 6693 . . . 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 2724 . . . . . . . 8 (0g𝐺) = (0g𝐺)
2423subg0cl 19046 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑌)
25 oveq12 7410 . . . . . . . 8 ((𝑥 = (0g𝐺) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((0g𝐺) + 𝑢))
26 ovex 7434 . . . . . . . 8 ((0g𝐺) + 𝑢) ∈ V
2725, 16, 26ovmpoa 7555 . . . . . . 7 (((0g𝐺) ∈ 𝑌𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐺) + 𝑢))
2824, 27sylan 579 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐺) + 𝑢))
291, 23subg0 19044 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
3029oveq1d 7416 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → ((0g𝐺)𝐹𝑢) = ((0g𝐻)𝐹𝑢))
3130adantr 480 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐻)𝐹𝑢))
323, 12, 23grplid 18884 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑢𝑋) → ((0g𝐺) + 𝑢) = 𝑢)
336, 32sylan 579 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺) + 𝑢) = 𝑢)
3428, 31, 333eqtr3d 2772 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐻)𝐹𝑢) = 𝑢)
356ad2antrr 723 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝐺 ∈ Grp)
368ad2antrr 723 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑌𝑋)
37 simprl 768 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑣𝑌)
3836, 37sseldd 3975 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑣𝑋)
39 simprr 770 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑤𝑌)
4036, 39sseldd 3975 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑤𝑋)
41 simplr 766 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑢𝑋)
423, 12grpass 18859 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑣𝑋𝑤𝑋𝑢𝑋)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
4335, 38, 40, 41, 42syl13anc 1369 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
443, 12grpcl 18858 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝑢𝑋) → (𝑤 + 𝑢) ∈ 𝑋)
4535, 40, 41, 44syl3anc 1368 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑤 + 𝑢) ∈ 𝑋)
46 oveq12 7410 . . . . . . . . . . 11 ((𝑥 = 𝑣𝑦 = (𝑤 + 𝑢)) → (𝑥 + 𝑦) = (𝑣 + (𝑤 + 𝑢)))
47 ovex 7434 . . . . . . . . . . 11 (𝑣 + (𝑤 + 𝑢)) ∈ V
4846, 16, 47ovmpoa 7555 . . . . . . . . . 10 ((𝑣𝑌 ∧ (𝑤 + 𝑢) ∈ 𝑋) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
4937, 45, 48syl2anc 583 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
5043, 49eqtr4d 2767 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣𝐹(𝑤 + 𝑢)))
5112subgcl 19048 . . . . . . . . . . 11 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑣𝑌𝑤𝑌) → (𝑣 + 𝑤) ∈ 𝑌)
52513expb 1117 . . . . . . . . . 10 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
5352adantlr 712 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
54 oveq12 7410 . . . . . . . . . 10 ((𝑥 = (𝑣 + 𝑤) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((𝑣 + 𝑤) + 𝑢))
55 ovex 7434 . . . . . . . . . 10 ((𝑣 + 𝑤) + 𝑢) ∈ V
5654, 16, 55ovmpoa 7555 . . . . . . . . 9 (((𝑣 + 𝑤) ∈ 𝑌𝑢𝑋) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
5753, 41, 56syl2anc 583 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
58 oveq12 7410 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑢) → (𝑥 + 𝑦) = (𝑤 + 𝑢))
59 ovex 7434 . . . . . . . . . . 11 (𝑤 + 𝑢) ∈ V
6058, 16, 59ovmpoa 7555 . . . . . . . . . 10 ((𝑤𝑌𝑢𝑋) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
6139, 41, 60syl2anc 583 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
6261oveq2d 7417 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣𝐹(𝑤𝐹𝑢)) = (𝑣𝐹(𝑤 + 𝑢)))
6350, 57, 623eqtr4d 2774 . . . . . . 7 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
6463ralrimivva 3192 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ∀𝑣𝑌𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
651, 12ressplusg 17231 . . . . . . . . . . . 12 (𝑌 ∈ (SubGrp‘𝐺) → + = (+g𝐻))
6665oveqd 7418 . . . . . . . . . . 11 (𝑌 ∈ (SubGrp‘𝐺) → (𝑣 + 𝑤) = (𝑣(+g𝐻)𝑤))
6766oveq1d 7416 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣(+g𝐻)𝑤)𝐹𝑢))
6867eqeq1d 2726 . . . . . . . . 9 (𝑌 ∈ (SubGrp‘𝐺) → (((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
6919, 68raleqbidv 3334 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → (∀𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7019, 69raleqbidv 3334 . . . . . . 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 590 . . . . 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 3138 . . 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 2724 . . 3 (Base‘𝐻) = (Base‘𝐻)
77 eqid 2724 . . 3 (+g𝐻) = (+g𝐻)
78 eqid 2724 . . 3 (0g𝐻) = (0g𝐻)
7976, 77, 78isga 19192 . 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 582 1 (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3053  Vcvv 3466  wss 3940   × cxp 5664  wf 6529  cfv 6533  (class class class)co 7401  cmpo 7403  Basecbs 17140  s cress 17169  +gcplusg 17193  0gc0g 17381  Grpcgrp 18850  SubGrpcsubg 19032   GrpAct cga 19190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8698  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-0g 17383  df-mgm 18560  df-sgrp 18639  df-mnd 18655  df-grp 18853  df-subg 19035  df-ga 19191
This theorem is referenced by:  gaid2  19204
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