Theorem gaid 18905

Description: The trivial action of a group on any set. Each group element corresponds to the identity permutation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Hypothesis

Ref	Expression
gaid.1	⊢ 𝑋 = (Base‘𝐺)

Assertion

Ref	Expression
gaid	⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆))

Proof of Theorem gaid

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3450	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2	1	anim2i 617	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (𝐺 ∈ Grp ∧ 𝑆 ∈ V))
3		gaid.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
4		eqid 2738	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
5	3, 4	grpidcl 18607	. . . . . . 7 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
6	5	adantr 481	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (0g‘𝐺) ∈ 𝑋)
7		ovres 7438	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((0g‘𝐺)2nd 𝑥))
8		df-ov 7278	. . . . . . . 8 ⊢ ((0g‘𝐺)2nd 𝑥) = (2nd ‘⟨(0g‘𝐺), 𝑥⟩)
9		fvex 6787	. . . . . . . . 9 ⊢ (0g‘𝐺) ∈ V
10		vex 3436	. . . . . . . . 9 ⊢ 𝑥 ∈ V
11	9, 10	op2nd 7840	. . . . . . . 8 ⊢ (2nd ‘⟨(0g‘𝐺), 𝑥⟩) = 𝑥
12	8, 11	eqtri 2766	. . . . . . 7 ⊢ ((0g‘𝐺)2nd 𝑥) = 𝑥
13	7, 12	eqtrdi 2794	. . . . . 6 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
14	6, 13	sylan 580	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
15		simprl 768	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
16		simplr 766	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ 𝑆)
17		ovres 7438	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦2nd 𝑥))
18		df-ov 7278	. . . . . . . . . 10 ⊢ (𝑦2nd 𝑥) = (2nd ‘⟨𝑦, 𝑥⟩)
19		vex 3436	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
20	19, 10	op2nd 7840	. . . . . . . . . 10 ⊢ (2nd ‘⟨𝑦, 𝑥⟩) = 𝑥
21	18, 20	eqtri 2766	. . . . . . . . 9 ⊢ (𝑦2nd 𝑥) = 𝑥
22	17, 21	eqtrdi 2794	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
23	15, 16, 22	syl2anc 584	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
24		simprr 770	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
25		ovres 7438	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑧2nd 𝑥))
26		df-ov 7278	. . . . . . . . . . 11 ⊢ (𝑧2nd 𝑥) = (2nd ‘⟨𝑧, 𝑥⟩)
27		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
28	27, 10	op2nd 7840	. . . . . . . . . . 11 ⊢ (2nd ‘⟨𝑧, 𝑥⟩) = 𝑥
29	26, 28	eqtri 2766	. . . . . . . . . 10 ⊢ (𝑧2nd 𝑥) = 𝑥
30	25, 29	eqtrdi 2794	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
31	24, 16, 30	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
32	31	oveq2d 7291	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)) = (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥))
33		eqid 2738	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
34	3, 33	grpcl 18585	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋)
35	34	3expb 1119	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋)
36	35	ad4ant14 749	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋)
37		ovres 7438	. . . . . . . . 9 ⊢ (((𝑦(+g‘𝐺)𝑧) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((𝑦(+g‘𝐺)𝑧)2nd 𝑥))
38		df-ov 7278	. . . . . . . . . 10 ⊢ ((𝑦(+g‘𝐺)𝑧)2nd 𝑥) = (2nd ‘⟨(𝑦(+g‘𝐺)𝑧), 𝑥⟩)
39		ovex 7308	. . . . . . . . . . 11 ⊢ (𝑦(+g‘𝐺)𝑧) ∈ V
40	39, 10	op2nd 7840	. . . . . . . . . 10 ⊢ (2nd ‘⟨(𝑦(+g‘𝐺)𝑧), 𝑥⟩) = 𝑥
41	38, 40	eqtri 2766	. . . . . . . . 9 ⊢ ((𝑦(+g‘𝐺)𝑧)2nd 𝑥) = 𝑥
42	37, 41	eqtrdi 2794	. . . . . . . 8 ⊢ (((𝑦(+g‘𝐺)𝑧) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
43	36, 16, 42	syl2anc 584	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
44	23, 32, 43	3eqtr4rd 2789	. . . . . 6 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))
45	44	ralrimivva 3123	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))
46	14, 45	jca 512	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) → (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))))
47	46	ralrimiva 3103	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → ∀𝑥 ∈ 𝑆 (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))))
48		f2ndres 7856	. . 3 ⊢ (2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆
49	47, 48	jctil 520	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → ((2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑥 ∈ 𝑆 (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))))
50	3, 33, 4	isga 18897	. 2 ⊢ ((2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆) ↔ ((𝐺 ∈ Grp ∧ 𝑆 ∈ V) ∧ ((2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑥 ∈ 𝑆 (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))))))
51	2, 49, 50	sylanbrc 583	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432  ⟨cop 4567   × cxp 5587   ↾ cres 5591  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  2^nd c2nd 7830  Basecbs 16912  +_gcplusg 16962  0_gc0g 17150  Grpcgrp 18577   GrpAct cga 18895

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-2nd 7832  df-map 8617  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-ga 18896

This theorem is referenced by: (None)