Theorem fxpsubm 33137

Description: Provided the group action 𝐴 induces monoid automorphisms, the set of fixed points of 𝐴 on a monoid 𝑊 is a submonoid, which could be called the fixed submonoid under 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.)

Hypotheses

Ref	Expression
fxpsubm.b	⊢ 𝐵 = (Base‘𝐺)
fxpsubg.c	⊢ 𝐶 = (Base‘𝑊)
fxpsubm.f	⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥))
fxpsubg.a	⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
fxpsubm.1	⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 MndHom 𝑊))

Assertion

Ref	Expression
fxpsubm	⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubMnd‘𝑊))

Distinct variable groups:   𝐴,𝑝,𝑥   𝐵,𝑝,𝑥   𝐶,𝑝,𝑥   𝐺,𝑝,𝑥   𝑊,𝑝,𝑥   𝜑,𝑝,𝑥

Allowed substitution hints:   𝐹(𝑥,𝑝)

Proof of Theorem fxpsubm

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

Step	Hyp	Ref	Expression
1		fxpsubm.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥))
2		oveq1 7401	. . . . . . 7 ⊢ (𝑝 = (0g‘𝐺) → (𝑝𝐴𝑥) = ((0g‘𝐺)𝐴𝑥))
3	2	mpteq2dv 5209	. . . . . 6 ⊢ (𝑝 = (0g‘𝐺) → (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥)) = (𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)))
4	1, 3	eqtrid 2777	. . . . 5 ⊢ (𝑝 = (0g‘𝐺) → 𝐹 = (𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)))
5	4	eleq1d 2814	. . . 4 ⊢ (𝑝 = (0g‘𝐺) → (𝐹 ∈ (𝑊 MndHom 𝑊) ↔ (𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)) ∈ (𝑊 MndHom 𝑊)))
6		fxpsubm.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 MndHom 𝑊))
7	6	ralrimiva 3127	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝐵 𝐹 ∈ (𝑊 MndHom 𝑊))
8		fxpsubg.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
9		gagrp 19230	. . . . . 6 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐺 ∈ Grp)
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
11		fxpsubm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
12		eqid 2730	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	11, 12	grpidcl 18903	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
14	10, 13	syl 17	. . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝐵)
15	5, 7, 14	rspcdva 3598	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)) ∈ (𝑊 MndHom 𝑊))
16		mhmrcl1 18720	. . 3 ⊢ ((𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)) ∈ (𝑊 MndHom 𝑊) → 𝑊 ∈ Mnd)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → 𝑊 ∈ Mnd)
18		gaset 19231	. . . 4 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐶 ∈ V)
19	8, 18	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
20	19, 8	fxpss 33131	. 2 ⊢ (𝜑 → (𝐶FixPts𝐴) ⊆ 𝐶)
21		oveq2 7402	. . . . . 6 ⊢ (𝑥 = (0g‘𝑊) → (𝑝𝐴𝑥) = (𝑝𝐴(0g‘𝑊)))
22		fxpsubg.c	. . . . . . . . 9 ⊢ 𝐶 = (Base‘𝑊)
23		eqid 2730	. . . . . . . . 9 ⊢ (0g‘𝑊) = (0g‘𝑊)
24	22, 23	mndidcl 18682	. . . . . . . 8 ⊢ (𝑊 ∈ Mnd → (0g‘𝑊) ∈ 𝐶)
25	17, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑊) ∈ 𝐶)
26	25	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (0g‘𝑊) ∈ 𝐶)
27		ovexd 7429	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(0g‘𝑊)) ∈ V)
28	1, 21, 26, 27	fvmptd3 6998	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝐹‘(0g‘𝑊)) = (𝑝𝐴(0g‘𝑊)))
29	23, 23	mhm0 18727	. . . . . 6 ⊢ (𝐹 ∈ (𝑊 MndHom 𝑊) → (𝐹‘(0g‘𝑊)) = (0g‘𝑊))
30	6, 29	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝐹‘(0g‘𝑊)) = (0g‘𝑊))
31	28, 30	eqtr3d 2767	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(0g‘𝑊)) = (0g‘𝑊))
32	31	ralrimiva 3127	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ 𝐵 (𝑝𝐴(0g‘𝑊)) = (0g‘𝑊))
33	11, 8, 25	isfxp 33133	. . 3 ⊢ (𝜑 → ((0g‘𝑊) ∈ (𝐶FixPts𝐴) ↔ ∀𝑝 ∈ 𝐵 (𝑝𝐴(0g‘𝑊)) = (0g‘𝑊)))
34	32, 33	mpbird 257	. 2 ⊢ (𝜑 → (0g‘𝑊) ∈ (𝐶FixPts𝐴))
35	6	ad4ant14 752	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 MndHom 𝑊))
36	20	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → (𝐶FixPts𝐴) ⊆ 𝐶)
37		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝑧 ∈ (𝐶FixPts𝐴))
38	36, 37	sseldd 3955	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝑧 ∈ 𝐶)
39	38	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑧 ∈ 𝐶)
40	20	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) → (𝐶FixPts𝐴) ⊆ 𝐶)
41	40	sselda 3954	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝑦 ∈ 𝐶)
42	41	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑦 ∈ 𝐶)
43		eqid 2730	. . . . . . . . 9 ⊢ (+g‘𝑊) = (+g‘𝑊)
44	22, 43, 43	mhmlin 18726	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑊 MndHom 𝑊) ∧ 𝑧 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝐹‘(𝑧(+g‘𝑊)𝑦)) = ((𝐹‘𝑧)(+g‘𝑊)(𝐹‘𝑦)))
45	35, 39, 42, 44	syl3anc 1373	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘(𝑧(+g‘𝑊)𝑦)) = ((𝐹‘𝑧)(+g‘𝑊)(𝐹‘𝑦)))
46		oveq2 7402	. . . . . . . 8 ⊢ (𝑥 = (𝑧(+g‘𝑊)𝑦) → (𝑝𝐴𝑥) = (𝑝𝐴(𝑧(+g‘𝑊)𝑦)))
47	17	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝑊 ∈ Mnd)
48	22, 43, 47, 38, 41	mndcld 32971	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → (𝑧(+g‘𝑊)𝑦) ∈ 𝐶)
49	48	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑧(+g‘𝑊)𝑦) ∈ 𝐶)
50		ovexd 7429	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(𝑧(+g‘𝑊)𝑦)) ∈ V)
51	1, 46, 49, 50	fvmptd3 6998	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘(𝑧(+g‘𝑊)𝑦)) = (𝑝𝐴(𝑧(+g‘𝑊)𝑦)))
52		oveq2 7402	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑝𝐴𝑥) = (𝑝𝐴𝑧))
53		ovexd 7429	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑧) ∈ V)
54	1, 52, 39, 53	fvmptd3 6998	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑧) = (𝑝𝐴𝑧))
55	8	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝐴 ∈ (𝐺 GrpAct 𝐶))
56	55	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝐴 ∈ (𝐺 GrpAct 𝐶))
57	37	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑧 ∈ (𝐶FixPts𝐴))
58		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑝 ∈ 𝐵)
59	11, 56, 57, 58	fxpgaeq 33134	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑧) = 𝑧)
60	54, 59	eqtrd 2765	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑧) = 𝑧)
61		oveq2 7402	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑝𝐴𝑥) = (𝑝𝐴𝑦))
62		ovexd 7429	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑦) ∈ V)
63	1, 61, 42, 62	fvmptd3 6998	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑦) = (𝑝𝐴𝑦))
64		simplr 768	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑦 ∈ (𝐶FixPts𝐴))
65	11, 56, 64, 58	fxpgaeq 33134	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑦) = 𝑦)
66	63, 65	eqtrd 2765	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑦) = 𝑦)
67	60, 66	oveq12d 7412	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → ((𝐹‘𝑧)(+g‘𝑊)(𝐹‘𝑦)) = (𝑧(+g‘𝑊)𝑦))
68	45, 51, 67	3eqtr3d 2773	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(𝑧(+g‘𝑊)𝑦)) = (𝑧(+g‘𝑊)𝑦))
69	68	ralrimiva 3127	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → ∀𝑝 ∈ 𝐵 (𝑝𝐴(𝑧(+g‘𝑊)𝑦)) = (𝑧(+g‘𝑊)𝑦))
70	11, 55, 48	isfxp 33133	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → ((𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴) ↔ ∀𝑝 ∈ 𝐵 (𝑝𝐴(𝑧(+g‘𝑊)𝑦)) = (𝑧(+g‘𝑊)𝑦)))
71	69, 70	mpbird 257	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → (𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴))
72	71	ralrimiva 3127	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) → ∀𝑦 ∈ (𝐶FixPts𝐴)(𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴))
73	72	ralrimiva 3127	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (𝐶FixPts𝐴)∀𝑦 ∈ (𝐶FixPts𝐴)(𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴))
74	22, 23, 43	issubm 18736	. . 3 ⊢ (𝑊 ∈ Mnd → ((𝐶FixPts𝐴) ∈ (SubMnd‘𝑊) ↔ ((𝐶FixPts𝐴) ⊆ 𝐶 ∧ (0g‘𝑊) ∈ (𝐶FixPts𝐴) ∧ ∀𝑧 ∈ (𝐶FixPts𝐴)∀𝑦 ∈ (𝐶FixPts𝐴)(𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴))))
75	74	biimpar 477	. 2 ⊢ ((𝑊 ∈ Mnd ∧ ((𝐶FixPts𝐴) ⊆ 𝐶 ∧ (0g‘𝑊) ∈ (𝐶FixPts𝐴) ∧ ∀𝑧 ∈ (𝐶FixPts𝐴)∀𝑦 ∈ (𝐶FixPts𝐴)(𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴))) → (𝐶FixPts𝐴) ∈ (SubMnd‘𝑊))
76	17, 20, 34, 73, 75	syl13anc 1374	1 ⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubMnd‘𝑊))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3046  Vcvv 3455   ⊆ wss 3922   ↦ cmpt 5196  ‘cfv 6519  (class class class)co 7394  Basecbs 17185  +_gcplusg 17226  0_gc0g 17408  Mndcmnd 18667   MndHom cmhm 18714  SubMndcsubmnd 18715  Grpcgrp 18871   GrpAct cga 19227  FixPtscfxp 33128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-fv 6527  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-map 8805  df-0g 17410  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18716  df-submnd 18717  df-grp 18874  df-ga 19228  df-fxp 33129

This theorem is referenced by: (None)