Theorem fxpsubm 33523

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)
fxpsubm.c	⊢ 𝐶 = (Base‘𝑊)
fxpsubm.f	⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥))
fxpsubm.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 7430	. . . . . . 7 ⊢ (𝑝 = (0g‘𝐺) → (𝑝𝐴𝑥) = ((0g‘𝐺)𝐴𝑥))
3	2	mpteq2dv 5210	. . . . . 6 ⊢ (𝑝 = (0g‘𝐺) → (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥)) = (𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)))
4	1, 3	eqtrid 2813	. . . . 5 ⊢ (𝑝 = (0g‘𝐺) → 𝐹 = (𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)))
5	4	eleq1d 2851	. . . 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 3160	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝐵 𝐹 ∈ (𝑊 MndHom 𝑊))
8		fxpsubm.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
9		gagrp 19393	. . . . . 6 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐺 ∈ Grp)
10	8, 9	syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
11		fxpsubm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
12		eqid 2766	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	11, 12	grpidcl 19063	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
14	10, 13	syl 18	. . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝐵)
15	5, 7, 14	rspcdva 3585	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)) ∈ (𝑊 MndHom 𝑊))
16		mhmrcl1 18876	. . 3 ⊢ ((𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)) ∈ (𝑊 MndHom 𝑊) → 𝑊 ∈ Mnd)
17	15, 16	syl 18	. 2 ⊢ (𝜑 → 𝑊 ∈ Mnd)
18		gaset 19394	. . . 4 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐶 ∈ V)
19	8, 18	syl 18	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
20	19, 8	fxpss 33517	. 2 ⊢ (𝜑 → (𝐶FixPts𝐴) ⊆ 𝐶)
21		oveq2 7431	. . . . . 6 ⊢ (𝑥 = (0g‘𝑊) → (𝑝𝐴𝑥) = (𝑝𝐴(0g‘𝑊)))
22		fxpsubm.c	. . . . . . . . 9 ⊢ 𝐶 = (Base‘𝑊)
23		eqid 2766	. . . . . . . . 9 ⊢ (0g‘𝑊) = (0g‘𝑊)
24	22, 23	mndidcl 18836	. . . . . . . 8 ⊢ (𝑊 ∈ Mnd → (0g‘𝑊) ∈ 𝐶)
25	17, 24	syl 18	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑊) ∈ 𝐶)
26	25	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (0g‘𝑊) ∈ 𝐶)
27		ovexd 7458	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(0g‘𝑊)) ∈ V)
28	1, 21, 26, 27	fvmptd3 7020	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝐹‘(0g‘𝑊)) = (𝑝𝐴(0g‘𝑊)))
29	23, 23	mhm0 18883	. . . . . 6 ⊢ (𝐹 ∈ (𝑊 MndHom 𝑊) → (𝐹‘(0g‘𝑊)) = (0g‘𝑊))
30	6, 29	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝐹‘(0g‘𝑊)) = (0g‘𝑊))
31	28, 30	eqtr3d 2803	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(0g‘𝑊)) = (0g‘𝑊))
32	31	ralrimiva 3160	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ 𝐵 (𝑝𝐴(0g‘𝑊)) = (0g‘𝑊))
33	11, 8, 25	isfxp 33519	. . 3 ⊢ (𝜑 → ((0g‘𝑊) ∈ (𝐶FixPts𝐴) ↔ ∀𝑝 ∈ 𝐵 (𝑝𝐴(0g‘𝑊)) = (0g‘𝑊)))
34	32, 33	mpbird 260	. 2 ⊢ (𝜑 → (0g‘𝑊) ∈ (𝐶FixPts𝐴))
35	6	ad4ant14 765	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 MndHom 𝑊))
36	20	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → (𝐶FixPts𝐴) ⊆ 𝐶)
37		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝑧 ∈ (𝐶FixPts𝐴))
38	36, 37	sseldd 3941	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝑧 ∈ 𝐶)
39	38	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑧 ∈ 𝐶)
40	20	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) → (𝐶FixPts𝐴) ⊆ 𝐶)
41	40	sselda 3940	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝑦 ∈ 𝐶)
42	41	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑦 ∈ 𝐶)
43		eqid 2766	. . . . . . . . 9 ⊢ (+g‘𝑊) = (+g‘𝑊)
44	22, 43, 43	mhmlin 18882	. . . . . . . 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 1398	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘(𝑧(+g‘𝑊)𝑦)) = ((𝐹‘𝑧)(+g‘𝑊)(𝐹‘𝑦)))
46		oveq2 7431	. . . . . . . 8 ⊢ (𝑥 = (𝑧(+g‘𝑊)𝑦) → (𝑝𝐴𝑥) = (𝑝𝐴(𝑧(+g‘𝑊)𝑦)))
47	17	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝑊 ∈ Mnd)
48	22, 43, 47, 38, 41	mndcld 33373	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → (𝑧(+g‘𝑊)𝑦) ∈ 𝐶)
49	48	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑧(+g‘𝑊)𝑦) ∈ 𝐶)
50		ovexd 7458	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(𝑧(+g‘𝑊)𝑦)) ∈ V)
51	1, 46, 49, 50	fvmptd3 7020	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘(𝑧(+g‘𝑊)𝑦)) = (𝑝𝐴(𝑧(+g‘𝑊)𝑦)))
52		oveq2 7431	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑝𝐴𝑥) = (𝑝𝐴𝑧))
53		ovexd 7458	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑧) ∈ V)
54	1, 52, 39, 53	fvmptd3 7020	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑧) = (𝑝𝐴𝑧))
55	8	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝐴 ∈ (𝐺 GrpAct 𝐶))
56	55	adantr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝐴 ∈ (𝐺 GrpAct 𝐶))
57	37	adantr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑧 ∈ (𝐶FixPts𝐴))
58		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑝 ∈ 𝐵)
59	11, 56, 57, 58	fxpgaeq 33520	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑧) = 𝑧)
60	54, 59	eqtrd 2801	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑧) = 𝑧)
61		oveq2 7431	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑝𝐴𝑥) = (𝑝𝐴𝑦))
62		ovexd 7458	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑦) ∈ V)
63	1, 61, 42, 62	fvmptd3 7020	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑦) = (𝑝𝐴𝑦))
64		simplr 781	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑦 ∈ (𝐶FixPts𝐴))
65	11, 56, 64, 58	fxpgaeq 33520	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑦) = 𝑦)
66	63, 65	eqtrd 2801	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑦) = 𝑦)
67	60, 66	oveq12d 7441	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → ((𝐹‘𝑧)(+g‘𝑊)(𝐹‘𝑦)) = (𝑧(+g‘𝑊)𝑦))
68	45, 51, 67	3eqtr3d 2809	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(𝑧(+g‘𝑊)𝑦)) = (𝑧(+g‘𝑊)𝑦))
69	68	ralrimiva 3160	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → ∀𝑝 ∈ 𝐵 (𝑝𝐴(𝑧(+g‘𝑊)𝑦)) = (𝑧(+g‘𝑊)𝑦))
70	11, 55, 48	isfxp 33519	. . . . 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 260	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → (𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴))
72	71	ralrimiva 3160	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) → ∀𝑦 ∈ (𝐶FixPts𝐴)(𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴))
73	72	ralrimiva 3160	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (𝐶FixPts𝐴)∀𝑦 ∈ (𝐶FixPts𝐴)(𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴))
74	22, 23, 43	issubm 18892	. . 3 ⊢ (𝑊 ∈ Mnd → ((𝐶FixPts𝐴) ∈ (SubMnd‘𝑊) ↔ ((𝐶FixPts𝐴) ⊆ 𝐶 ∧ (0g‘𝑊) ∈ (𝐶FixPts𝐴) ∧ ∀𝑧 ∈ (𝐶FixPts𝐴)∀𝑦 ∈ (𝐶FixPts𝐴)(𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴))))
75	74	biimpar 483	. 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 1399	1 ⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubMnd‘𝑊))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  Vcvv 3458   ⊆ wss 3908   ↦ cmpt 5197  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  0_gc0g 17517  Mndcmnd 18821   MndHom cmhm 18870  SubMndcsubmnd 18871  Grpcgrp 19031   GrpAct cga 19390  FixPtscfxp 33514

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-map 8835  df-0g 17519  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-mhm 18872  df-submnd 18873  df-grp 19034  df-ga 19391  df-fxp 33515

This theorem is used by:  fxpsubg  33524