Theorem fxpsubm 33233

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 7374	. . . . . . 7 ⊢ (𝑝 = (0g‘𝐺) → (𝑝𝐴𝑥) = ((0g‘𝐺)𝐴𝑥))
3	2	mpteq2dv 5179	. . . . . 6 ⊢ (𝑝 = (0g‘𝐺) → (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥)) = (𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)))
4	1, 3	eqtrid 2783	. . . . 5 ⊢ (𝑝 = (0g‘𝐺) → 𝐹 = (𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)))
5	4	eleq1d 2821	. . . 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 3129	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ 𝐵 𝐹 ∈ (𝑊 MndHom 𝑊))
8		fxpsubm.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
9		gagrp 19267	. . . . . 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 2736	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	11, 12	grpidcl 18941	. . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
14	10, 13	syl 17	. . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝐵)
15	5, 7, 14	rspcdva 3565	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)) ∈ (𝑊 MndHom 𝑊))
16		mhmrcl1 18755	. . 3 ⊢ ((𝑥 ∈ 𝐶 ↦ ((0g‘𝐺)𝐴𝑥)) ∈ (𝑊 MndHom 𝑊) → 𝑊 ∈ Mnd)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → 𝑊 ∈ Mnd)
18		gaset 19268	. . . 4 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐶 ∈ V)
19	8, 18	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
20	19, 8	fxpss 33227	. 2 ⊢ (𝜑 → (𝐶FixPts𝐴) ⊆ 𝐶)
21		oveq2 7375	. . . . . 6 ⊢ (𝑥 = (0g‘𝑊) → (𝑝𝐴𝑥) = (𝑝𝐴(0g‘𝑊)))
22		fxpsubm.c	. . . . . . . . 9 ⊢ 𝐶 = (Base‘𝑊)
23		eqid 2736	. . . . . . . . 9 ⊢ (0g‘𝑊) = (0g‘𝑊)
24	22, 23	mndidcl 18717	. . . . . . . 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 7402	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(0g‘𝑊)) ∈ V)
28	1, 21, 26, 27	fvmptd3 6971	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝐹‘(0g‘𝑊)) = (𝑝𝐴(0g‘𝑊)))
29	23, 23	mhm0 18762	. . . . . 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 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(0g‘𝑊)) = (0g‘𝑊))
32	31	ralrimiva 3129	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ 𝐵 (𝑝𝐴(0g‘𝑊)) = (0g‘𝑊))
33	11, 8, 25	isfxp 33229	. . 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 753	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 MndHom 𝑊))
36	20	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → (𝐶FixPts𝐴) ⊆ 𝐶)
37		simplr 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝑧 ∈ (𝐶FixPts𝐴))
38	36, 37	sseldd 3922	. . . . . . . . 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 3921	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝑦 ∈ 𝐶)
42	41	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑦 ∈ 𝐶)
43		eqid 2736	. . . . . . . . 9 ⊢ (+g‘𝑊) = (+g‘𝑊)
44	22, 43, 43	mhmlin 18761	. . . . . . . 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 1374	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘(𝑧(+g‘𝑊)𝑦)) = ((𝐹‘𝑧)(+g‘𝑊)(𝐹‘𝑦)))
46		oveq2 7375	. . . . . . . 8 ⊢ (𝑥 = (𝑧(+g‘𝑊)𝑦) → (𝑝𝐴𝑥) = (𝑝𝐴(𝑧(+g‘𝑊)𝑦)))
47	17	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → 𝑊 ∈ Mnd)
48	22, 43, 47, 38, 41	mndcld 33082	. . . . . . . . 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 7402	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(𝑧(+g‘𝑊)𝑦)) ∈ V)
51	1, 46, 49, 50	fvmptd3 6971	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘(𝑧(+g‘𝑊)𝑦)) = (𝑝𝐴(𝑧(+g‘𝑊)𝑦)))
52		oveq2 7375	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑝𝐴𝑥) = (𝑝𝐴𝑧))
53		ovexd 7402	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑧) ∈ V)
54	1, 52, 39, 53	fvmptd3 6971	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑧) = (𝑝𝐴𝑧))
55	8	ad2antrr 727	. . . . . . . . . . 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 33230	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑧) = 𝑧)
60	54, 59	eqtrd 2771	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑧) = 𝑧)
61		oveq2 7375	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑝𝐴𝑥) = (𝑝𝐴𝑦))
62		ovexd 7402	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑦) ∈ V)
63	1, 61, 42, 62	fvmptd3 6971	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑦) = (𝑝𝐴𝑦))
64		simplr 769	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → 𝑦 ∈ (𝐶FixPts𝐴))
65	11, 56, 64, 58	fxpgaeq 33230	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴𝑦) = 𝑦)
66	63, 65	eqtrd 2771	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝐹‘𝑦) = 𝑦)
67	60, 66	oveq12d 7385	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → ((𝐹‘𝑧)(+g‘𝑊)(𝐹‘𝑦)) = (𝑧(+g‘𝑊)𝑦))
68	45, 51, 67	3eqtr3d 2779	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) ∧ 𝑝 ∈ 𝐵) → (𝑝𝐴(𝑧(+g‘𝑊)𝑦)) = (𝑧(+g‘𝑊)𝑦))
69	68	ralrimiva 3129	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) ∧ 𝑦 ∈ (𝐶FixPts𝐴)) → ∀𝑝 ∈ 𝐵 (𝑝𝐴(𝑧(+g‘𝑊)𝑦)) = (𝑧(+g‘𝑊)𝑦))
70	11, 55, 48	isfxp 33229	. . . . 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 3129	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐶FixPts𝐴)) → ∀𝑦 ∈ (𝐶FixPts𝐴)(𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴))
73	72	ralrimiva 3129	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (𝐶FixPts𝐴)∀𝑦 ∈ (𝐶FixPts𝐴)(𝑧(+g‘𝑊)𝑦) ∈ (𝐶FixPts𝐴))
74	22, 23, 43	issubm 18771	. . 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 1375	1 ⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubMnd‘𝑊))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ⊆ wss 3889   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402  Mndcmnd 18702   MndHom cmhm 18749  SubMndcsubmnd 18750  Grpcgrp 18909   GrpAct cga 19264  FixPtscfxp 33224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-map 8775  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-ga 19265  df-fxp 33225

This theorem is referenced by:  fxpsubg  33234