Theorem gaid 19506

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 3472	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2	1	anim2i 629	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (𝐺 ∈ Grp ∧ 𝑆 ∈ V))
3		gaid.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
4		eqid 2761	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
5	3, 4	grpidcl 19169	. . . . . . 7 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋)
6	5	adantr 486	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (0g‘𝐺) ∈ 𝑋)
7		ovres 7584	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((0g‘𝐺)2nd 𝑥))
8		df-ov 7421	. . . . . . . 8 ⊢ ((0g‘𝐺)2nd 𝑥) = (2nd ‘⟨(0g‘𝐺), 𝑥⟩)
9		fvex 6896	. . . . . . . . 9 ⊢ (0g‘𝐺) ∈ V
10		vex 3455	. . . . . . . . 9 ⊢ 𝑥 ∈ V
11	9, 10	op2nd 8008	. . . . . . . 8 ⊢ (2nd ‘⟨(0g‘𝐺), 𝑥⟩) = 𝑥
12	8, 11	eqtri 2784	. . . . . . 7 ⊢ ((0g‘𝐺)2nd 𝑥) = 𝑥
13	7, 12	eqtrdi 2812	. . . . . 6 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
14	6, 13	sylan 592	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
15		simprl 783	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
16		simplr 781	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ 𝑆)
17		ovres 7584	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦2nd 𝑥))
18		df-ov 7421	. . . . . . . . . 10 ⊢ (𝑦2nd 𝑥) = (2nd ‘⟨𝑦, 𝑥⟩)
19		vex 3455	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
20	19, 10	op2nd 8008	. . . . . . . . . 10 ⊢ (2nd ‘⟨𝑦, 𝑥⟩) = 𝑥
21	18, 20	eqtri 2784	. . . . . . . . 9 ⊢ (𝑦2nd 𝑥) = 𝑥
22	17, 21	eqtrdi 2812	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
23	15, 16, 22	syl2anc 596	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
24		simprr 785	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
25		ovres 7584	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑧2nd 𝑥))
26		df-ov 7421	. . . . . . . . . . 11 ⊢ (𝑧2nd 𝑥) = (2nd ‘⟨𝑧, 𝑥⟩)
27		vex 3455	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
28	27, 10	op2nd 8008	. . . . . . . . . . 11 ⊢ (2nd ‘⟨𝑧, 𝑥⟩) = 𝑥
29	26, 28	eqtri 2784	. . . . . . . . . 10 ⊢ (𝑧2nd 𝑥) = 𝑥
30	25, 29	eqtrdi 2812	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
31	24, 16, 30	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
32	31	oveq2d 7434	. . . . . . 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 2761	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
34	3, 33	grpcl 19145	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋)
35	34	3expb 1138	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋)
36	35	ad4ant14 765	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋)
37		ovres 7584	. . . . . . . . 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 7421	. . . . . . . . . 10 ⊢ ((𝑦(+g‘𝐺)𝑧)2nd 𝑥) = (2nd ‘⟨(𝑦(+g‘𝐺)𝑧), 𝑥⟩)
39		ovex 7451	. . . . . . . . . . 11 ⊢ (𝑦(+g‘𝐺)𝑧) ∈ V
40	39, 10	op2nd 8008	. . . . . . . . . 10 ⊢ (2nd ‘⟨(𝑦(+g‘𝐺)𝑧), 𝑥⟩) = 𝑥
41	38, 40	eqtri 2784	. . . . . . . . 9 ⊢ ((𝑦(+g‘𝐺)𝑧)2nd 𝑥) = 𝑥
42	37, 41	eqtrdi 2812	. . . . . . . 8 ⊢ (((𝑦(+g‘𝐺)𝑧) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥)
43	36, 16, 42	syl2anc 596	. . . . . . 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 2807	. . . . . 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 3206	. . . . 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 521	. . . 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 3155	. . 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 8024	. . 3 ⊢ (2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆
49	47, 48	jctil 529	. 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 19498	. 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 595	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ⟨cop 4590   × cxp 5649   ↾ cres 5653  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  2^nd c2nd 7998  Basecbs 17380  +_gcplusg 17421  0_gc0g 17603  Grpcgrp 19137   GrpAct cga 19496

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-2nd 8000  df-map 8842  df-0g 17605  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-grp 19140  df-ga 19497

This theorem is used by: (None)