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Theorem subgga 17925
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 17790 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
3 subgga.1 . . . 4 𝑋 = (Base‘𝐺)
4 fvex 6354 . . . 4 (Base‘𝐺) ∈ V
53, 4eqeltri 2827 . . 3 𝑋 ∈ V
62, 5jctir 562 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝐻 ∈ Grp ∧ 𝑋 ∈ V))
7 subgrcl 17792 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
87adantr 472 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝐺 ∈ Grp)
93subgss 17788 . . . . . . . . 9 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
109sselda 3736 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑌) → 𝑥𝑋)
1110adantrr 755 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝑥𝑋)
12 simprr 813 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → 𝑦𝑋)
13 subgga.2 . . . . . . . 8 + = (+g𝐺)
143, 13grpcl 17623 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝑦𝑋) → (𝑥 + 𝑦) ∈ 𝑋)
158, 11, 12, 14syl3anc 1473 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥𝑌𝑦𝑋)) → (𝑥 + 𝑦) ∈ 𝑋)
1615ralrimivva 3101 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → ∀𝑥𝑌𝑦𝑋 (𝑥 + 𝑦) ∈ 𝑋)
17 subgga.4 . . . . . 6 𝐹 = (𝑥𝑌, 𝑦𝑋 ↦ (𝑥 + 𝑦))
1817fmpt2 7397 . . . . 5 (∀𝑥𝑌𝑦𝑋 (𝑥 + 𝑦) ∈ 𝑋𝐹:(𝑌 × 𝑋)⟶𝑋)
1916, 18sylib 208 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:(𝑌 × 𝑋)⟶𝑋)
201subgbas 17791 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 = (Base‘𝐻))
2120xpeq1d 5287 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑌 × 𝑋) = ((Base‘𝐻) × 𝑋))
2221feq2d 6184 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝐹:(𝑌 × 𝑋)⟶𝑋𝐹:((Base‘𝐻) × 𝑋)⟶𝑋))
2319, 22mpbid 222 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → 𝐹:((Base‘𝐻) × 𝑋)⟶𝑋)
24 eqid 2752 . . . . . . . 8 (0g𝐺) = (0g𝐺)
2524subg0cl 17795 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑌)
26 oveq12 6814 . . . . . . . 8 ((𝑥 = (0g𝐺) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((0g𝐺) + 𝑢))
27 ovex 6833 . . . . . . . 8 ((0g𝐺) + 𝑢) ∈ V
2826, 17, 27ovmpt2a 6948 . . . . . . 7 (((0g𝐺) ∈ 𝑌𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐺) + 𝑢))
2925, 28sylan 489 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐺) + 𝑢))
301, 24subg0 17793 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
3130oveq1d 6820 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → ((0g𝐺)𝐹𝑢) = ((0g𝐻)𝐹𝑢))
3231adantr 472 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺)𝐹𝑢) = ((0g𝐻)𝐹𝑢))
333, 13, 24grplid 17645 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑢𝑋) → ((0g𝐺) + 𝑢) = 𝑢)
347, 33sylan 489 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐺) + 𝑢) = 𝑢)
3529, 32, 343eqtr3d 2794 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ((0g𝐻)𝐹𝑢) = 𝑢)
367ad2antrr 764 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝐺 ∈ Grp)
379ad2antrr 764 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑌𝑋)
38 simprl 811 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑣𝑌)
3937, 38sseldd 3737 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑣𝑋)
40 simprr 813 . . . . . . . . . . 11 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑤𝑌)
4137, 40sseldd 3737 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑤𝑋)
42 simplr 809 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → 𝑢𝑋)
433, 13grpass 17624 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (𝑣𝑋𝑤𝑋𝑢𝑋)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
4436, 39, 41, 42, 43syl13anc 1475 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣 + (𝑤 + 𝑢)))
453, 13grpcl 17623 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝑢𝑋) → (𝑤 + 𝑢) ∈ 𝑋)
4636, 41, 42, 45syl3anc 1473 . . . . . . . . . 10 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑤 + 𝑢) ∈ 𝑋)
47 oveq12 6814 . . . . . . . . . . 11 ((𝑥 = 𝑣𝑦 = (𝑤 + 𝑢)) → (𝑥 + 𝑦) = (𝑣 + (𝑤 + 𝑢)))
48 ovex 6833 . . . . . . . . . . 11 (𝑣 + (𝑤 + 𝑢)) ∈ V
4947, 17, 48ovmpt2a 6948 . . . . . . . . . 10 ((𝑣𝑌 ∧ (𝑤 + 𝑢) ∈ 𝑋) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
5038, 46, 49syl2anc 696 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣𝐹(𝑤 + 𝑢)) = (𝑣 + (𝑤 + 𝑢)))
5144, 50eqtr4d 2789 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤) + 𝑢) = (𝑣𝐹(𝑤 + 𝑢)))
5213subgcl 17797 . . . . . . . . . . 11 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑣𝑌𝑤𝑌) → (𝑣 + 𝑤) ∈ 𝑌)
53523expb 1113 . . . . . . . . . 10 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
5453adantlr 753 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣 + 𝑤) ∈ 𝑌)
55 oveq12 6814 . . . . . . . . . 10 ((𝑥 = (𝑣 + 𝑤) ∧ 𝑦 = 𝑢) → (𝑥 + 𝑦) = ((𝑣 + 𝑤) + 𝑢))
56 ovex 6833 . . . . . . . . . 10 ((𝑣 + 𝑤) + 𝑢) ∈ V
5755, 17, 56ovmpt2a 6948 . . . . . . . . 9 (((𝑣 + 𝑤) ∈ 𝑌𝑢𝑋) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
5854, 42, 57syl2anc 696 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣 + 𝑤) + 𝑢))
59 oveq12 6814 . . . . . . . . . . 11 ((𝑥 = 𝑤𝑦 = 𝑢) → (𝑥 + 𝑦) = (𝑤 + 𝑢))
60 ovex 6833 . . . . . . . . . . 11 (𝑤 + 𝑢) ∈ V
6159, 17, 60ovmpt2a 6948 . . . . . . . . . 10 ((𝑤𝑌𝑢𝑋) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
6240, 42, 61syl2anc 696 . . . . . . . . 9 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑤𝐹𝑢) = (𝑤 + 𝑢))
6362oveq2d 6821 . . . . . . . 8 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → (𝑣𝐹(𝑤𝐹𝑢)) = (𝑣𝐹(𝑤 + 𝑢)))
6451, 58, 633eqtr4d 2796 . . . . . . 7 (((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) ∧ (𝑣𝑌𝑤𝑌)) → ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
6564ralrimivva 3101 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ∀𝑣𝑌𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
661, 13ressplusg 16187 . . . . . . . . . . . 12 (𝑌 ∈ (SubGrp‘𝐺) → + = (+g𝐻))
6766oveqd 6822 . . . . . . . . . . 11 (𝑌 ∈ (SubGrp‘𝐺) → (𝑣 + 𝑤) = (𝑣(+g𝐻)𝑤))
6867oveq1d 6820 . . . . . . . . . 10 (𝑌 ∈ (SubGrp‘𝐺) → ((𝑣 + 𝑤)𝐹𝑢) = ((𝑣(+g𝐻)𝑤)𝐹𝑢))
6968eqeq1d 2754 . . . . . . . . 9 (𝑌 ∈ (SubGrp‘𝐺) → (((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7020, 69raleqbidv 3283 . . . . . . . 8 (𝑌 ∈ (SubGrp‘𝐺) → (∀𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7120, 70raleqbidv 3283 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (∀𝑣𝑌𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)) ↔ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7271biimpa 502 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ ∀𝑣𝑌𝑤𝑌 ((𝑣 + 𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))) → ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
7365, 72syldan 488 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢)))
7435, 73jca 555 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑢𝑋) → (((0g𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7574ralrimiva 3096 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → ∀𝑢𝑋 (((0g𝐻)𝐹𝑢) = 𝑢 ∧ ∀𝑣 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑣(+g𝐻)𝑤)𝐹𝑢) = (𝑣𝐹(𝑤𝐹𝑢))))
7623, 75jca 555 . 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)))))
77 eqid 2752 . . 3 (Base‘𝐻) = (Base‘𝐻)
78 eqid 2752 . . 3 (+g𝐻) = (+g𝐻)
79 eqid 2752 . . 3 (0g𝐻) = (0g𝐻)
8077, 78, 79isga 17916 . 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))))))
816, 76, 80sylanbrc 701 1 (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332  wss 3707   × cxp 5256  wf 6037  cfv 6041  (class class class)co 6805  cmpt2 6807  Basecbs 16051  s cress 16052  +gcplusg 16135  0gc0g 16294  Grpcgrp 17615  SubGrpcsubg 17781   GrpAct cga 17914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-ress 16059  df-plusg 16148  df-0g 16296  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-grp 17618  df-subg 17784  df-ga 17915
This theorem is referenced by:  gaid2  17928
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