Theorem ressply1evls1 33524

Description: Subring evaluation of a univariate polynomial is the same as the subring evaluation in the bigger ring. (Contributed by Thierry Arnoux, 14-Nov-2025.)

Hypotheses

Ref	Expression
ressply1evls1.1	⊢ 𝐺 = (𝐸 ↾s 𝑅)
ressply1evls1.2	⊢ 𝑂 = (𝐸 evalSub1 𝑆)
ressply1evls1.3	⊢ 𝑄 = (𝐺 evalSub1 𝑆)
ressply1evls1.4	⊢ 𝑃 = (Poly1‘𝐾)
ressply1evls1.5	⊢ 𝐾 = (𝐸 ↾s 𝑆)
ressply1evls1.6	⊢ 𝐵 = (Base‘𝑃)
ressply1evls1.7	⊢ (𝜑 → 𝐸 ∈ CRing)
ressply1evls1.8	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝐸))
ressply1evls1.9	⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝐺))
ressply1evls1.10	⊢ (𝜑 → 𝐹 ∈ 𝐵)

Assertion

Ref	Expression
ressply1evls1	⊢ (𝜑 → (𝑄‘𝐹) = ((𝑂‘𝐹) ↾ 𝑅))

Proof of Theorem ressply1evls1

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

Step	Hyp	Ref	Expression
1		ressply1evls1.8	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝐸))
2		eqid 2735	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
3	2	subrgss 20530	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝑅 ⊆ (Base‘𝐸))
4		ressply1evls1.1	. . . . . 6 ⊢ 𝐺 = (𝐸 ↾s 𝑅)
5	4, 2	ressbas2 17257	. . . . 5 ⊢ (𝑅 ⊆ (Base‘𝐸) → 𝑅 = (Base‘𝐺))
6	1, 3, 5	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 = (Base‘𝐺))
7	1, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ⊆ (Base‘𝐸))
8	6, 7	eqsstrrd 3994	. . 3 ⊢ (𝜑 → (Base‘𝐺) ⊆ (Base‘𝐸))
9	8	resmptd 6027	. 2 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐸) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))) ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))))
10		ressply1evls1.2	. . . 4 ⊢ 𝑂 = (𝐸 evalSub1 𝑆)
11		ressply1evls1.4	. . . 4 ⊢ 𝑃 = (Poly1‘𝐾)
12		ressply1evls1.5	. . . 4 ⊢ 𝐾 = (𝐸 ↾s 𝑆)
13		ressply1evls1.6	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
14		ressply1evls1.7	. . . 4 ⊢ (𝜑 → 𝐸 ∈ CRing)
15		ressply1evls1.9	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝐺))
16	4	subsubrg 20556	. . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (𝑆 ∈ (SubRing‘𝐺) ↔ (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅)))
17	16	biimpa 476	. . . . . 6 ⊢ ((𝑅 ∈ (SubRing‘𝐸) ∧ 𝑆 ∈ (SubRing‘𝐺)) → (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅))
18	1, 15, 17	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅))
19	18	simpld 494	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝐸))
20		ressply1evls1.10	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
21		eqid 2735	. . . 4 ⊢ (.r‘𝐸) = (.r‘𝐸)
22		eqid 2735	. . . 4 ⊢ (.g‘(mulGrp‘𝐸)) = (.g‘(mulGrp‘𝐸))
23		eqid 2735	. . . 4 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
24	10, 2, 11, 12, 13, 14, 19, 20, 21, 22, 23	evls1fpws 22305	. . 3 ⊢ (𝜑 → (𝑂‘𝐹) = (𝑥 ∈ (Base‘𝐸) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))))
25	24, 6	reseq12d 5967	. 2 ⊢ (𝜑 → ((𝑂‘𝐹) ↾ 𝑅) = ((𝑥 ∈ (Base‘𝐸) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))) ↾ (Base‘𝐺)))
26		ressply1evls1.3	. . . 4 ⊢ 𝑄 = (𝐺 evalSub1 𝑆)
27		eqid 2735	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
28		eqid 2735	. . . 4 ⊢ (Poly1‘(𝐺 ↾s 𝑆)) = (Poly1‘(𝐺 ↾s 𝑆))
29		eqid 2735	. . . 4 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆)
30		eqid 2735	. . . 4 ⊢ (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = (Base‘(Poly1‘(𝐺 ↾s 𝑆)))
31	4	subrgcrng 20533	. . . . 5 ⊢ ((𝐸 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝐸)) → 𝐺 ∈ CRing)
32	14, 1, 31	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐺 ∈ CRing)
33	18	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ 𝑅)
34		ressabs 17267	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅) → ((𝐸 ↾s 𝑅) ↾s 𝑆) = (𝐸 ↾s 𝑆))
35	1, 33, 34	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 ↾s 𝑅) ↾s 𝑆) = (𝐸 ↾s 𝑆))
36	4	oveq1i 7413	. . . . . . . . . 10 ⊢ (𝐺 ↾s 𝑆) = ((𝐸 ↾s 𝑅) ↾s 𝑆)
37	35, 36, 12	3eqtr4g 2795	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾s 𝑆) = 𝐾)
38	37	fveq2d 6879	. . . . . . . 8 ⊢ (𝜑 → (Poly1‘(𝐺 ↾s 𝑆)) = (Poly1‘𝐾))
39	38, 11	eqtr4di 2788	. . . . . . 7 ⊢ (𝜑 → (Poly1‘(𝐺 ↾s 𝑆)) = 𝑃)
40	39	fveq2d 6879	. . . . . 6 ⊢ (𝜑 → (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = (Base‘𝑃))
41	40, 13	eqtr4di 2788	. . . . 5 ⊢ (𝜑 → (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = 𝐵)
42	20, 41	eleqtrrd 2837	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(Poly1‘(𝐺 ↾s 𝑆))))
43		eqid 2735	. . . 4 ⊢ (.r‘𝐺) = (.r‘𝐺)
44		eqid 2735	. . . 4 ⊢ (.g‘(mulGrp‘𝐺)) = (.g‘(mulGrp‘𝐺))
45	26, 27, 28, 29, 30, 32, 15, 42, 43, 44, 23	evls1fpws 22305	. . 3 ⊢ (𝜑 → (𝑄‘𝐹) = (𝑥 ∈ (Base‘𝐺) ↦ (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥))))))
46		eqid 2735	. . . . . 6 ⊢ (+g‘𝐸) = (+g‘𝐸)
47	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐸 ∈ CRing)
48		nn0ex 12505	. . . . . . 7 ⊢ ℕ0 ∈ V
49	48	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ℕ0 ∈ V)
50	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑅 ⊆ (Base‘𝐸))
51	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ (SubRing‘𝐸))
52	33, 7	sstrd 3969	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐸))
53	12, 2	ressbas2 17257	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ (Base‘𝐸) → 𝑆 = (Base‘𝐾))
54	52, 53	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 = (Base‘𝐾))
55	54, 33	eqsstrrd 3994	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐾) ⊆ 𝑅)
56	55	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (Base‘𝐾) ⊆ 𝑅)
57	20	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝐹 ∈ 𝐵)
58		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
59		eqid 2735	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
60	23, 13, 11, 59	coe1fvalcl 22146	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝐾))
61	57, 58, 60	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝐾))
62	56, 61	sseldd 3959	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ 𝑅)
63		eqid 2735	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐸) = (mulGrp‘𝐸)
64	4, 63	mgpress 20108	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝐸)) → ((mulGrp‘𝐸) ↾s 𝑅) = (mulGrp‘𝐺))
65	47, 51, 64	syl2an2r 685	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((mulGrp‘𝐸) ↾s 𝑅) = (mulGrp‘𝐺))
66	14	crngringd 20204	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Ring)
67		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (1r‘𝐸) = (1r‘𝐸)
68	67	subrg1cl 20538	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝑅)
69	1, 68	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝑅)
70	4, 2, 67	ress1r 33175	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ Ring ∧ (1r‘𝐸) ∈ 𝑅 ∧ 𝑅 ⊆ (Base‘𝐸)) → (1r‘𝐸) = (1r‘𝐺))
71	66, 69, 7, 70	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐺))
72	71	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (1r‘𝐸) = (1r‘𝐺))
73	63, 67	ringidval 20141	. . . . . . . . . . . 12 ⊢ (1r‘𝐸) = (0g‘(mulGrp‘𝐸))
74		eqid 2735	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐺) = (mulGrp‘𝐺)
75		eqid 2735	. . . . . . . . . . . . 13 ⊢ (1r‘𝐺) = (1r‘𝐺)
76	74, 75	ringidval 20141	. . . . . . . . . . . 12 ⊢ (1r‘𝐺) = (0g‘(mulGrp‘𝐺))
77	72, 73, 76	3eqtr3g 2793	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (0g‘(mulGrp‘𝐸)) = (0g‘(mulGrp‘𝐺)))
78	63, 2	mgpbas 20103	. . . . . . . . . . . . 13 ⊢ (Base‘𝐸) = (Base‘(mulGrp‘𝐸))
79	7, 78	sseqtrdi 3999	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ⊆ (Base‘(mulGrp‘𝐸)))
80	79	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ⊆ (Base‘(mulGrp‘𝐸)))
81	6	eleq2d 2820	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝑅 ↔ 𝑥 ∈ (Base‘𝐺)))
82	81	biimpar 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝑅)
83	82	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑥 ∈ 𝑅)
84	65, 77, 80, 58, 83	ressmulgnn0d 32985	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐺))𝑥) = (𝑘(.g‘(mulGrp‘𝐸))𝑥))
85	74, 27	mgpbas 20103	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘(mulGrp‘𝐺))
86	4	subrgring 20532	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝐺 ∈ Ring)
87	74	ringmgp 20197	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Ring → (mulGrp‘𝐺) ∈ Mnd)
88	1, 86, 87	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝐺) ∈ Mnd)
89	88	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝐺) ∈ Mnd)
90		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑥 ∈ (Base‘𝐺))
91	85, 44, 89, 58, 90	mulgnn0cld 19076	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐺))𝑥) ∈ (Base‘𝐺))
92	84, 91	eqeltrrd 2835	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐸))𝑥) ∈ (Base‘𝐺))
93	51, 3, 5	3syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑅 = (Base‘𝐺))
94	92, 93	eleqtrrd 2837	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐸))𝑥) ∈ 𝑅)
95	21, 51, 62, 94	subrgmcld 33174	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)) ∈ 𝑅)
96	95	fmpttd 7104	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))):ℕ0⟶𝑅)
97		subrgsubg 20535	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝑅 ∈ (SubGrp‘𝐸))
98		eqid 2735	. . . . . . . . 9 ⊢ (0g‘𝐸) = (0g‘𝐸)
99	98	subg0cl 19115	. . . . . . . 8 ⊢ (𝑅 ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ 𝑅)
100	1, 97, 99	3syl 18	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝑅)
101	100	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (0g‘𝐸) ∈ 𝑅)
102	14	crnggrpd 20205	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Grp)
103	102	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → 𝐸 ∈ Grp)
104		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → 𝑦 ∈ (Base‘𝐸))
105	2, 46, 98, 103, 104	grplidd 18950	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → ((0g‘𝐸)(+g‘𝐸)𝑦) = 𝑦)
106	2, 46, 98, 103, 104	grpridd 18951	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → (𝑦(+g‘𝐸)(0g‘𝐸)) = 𝑦)
107	105, 106	jca 511	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → (((0g‘𝐸)(+g‘𝐸)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐸)(0g‘𝐸)) = 𝑦))
108	2, 46, 4, 47, 49, 50, 96, 101, 107	gsumress 18658	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))) = (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))))
109	4, 21	ressmulr 17319	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (.r‘𝐸) = (.r‘𝐺))
110	1, 109	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (.r‘𝐸) = (.r‘𝐺))
111	110	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (.r‘𝐸) = (.r‘𝐺))
112	111	oveqd 7420	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐺))𝑥)) = (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥)))
113	84	oveq2d 7419	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐺))𝑥)) = (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))
114	112, 113	eqtr3d 2772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥)) = (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))
115	114	mpteq2dva 5214	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥))) = (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))
116	115	oveq2d 7419	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥)))) = (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))))
117	108, 116	eqtr4d 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)))) = (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥)))))
118	117	mpteq2dva 5214	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐺) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))) = (𝑥 ∈ (Base‘𝐺) ↦ (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥))))))
119	45, 118	eqtr4d 2773	. 2 ⊢ (𝜑 → (𝑄‘𝐹) = (𝑥 ∈ (Base‘𝐺) ↦ (𝐸 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))))))
120	9, 25, 119	3eqtr4rd 2781	1 ⊢ (𝜑 → (𝑄‘𝐹) = ((𝑂‘𝐹) ↾ 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ⊆ wss 3926   ↦ cmpt 5201   ↾ cres 5656  ‘cfv 6530  (class class class)co 7403  ℕ₀cn0 12499  Basecbs 17226   ↾_s cress 17249  +_gcplusg 17269  ._rcmulr 17270  0_gc0g 17451   Σ_g cgsu 17452  Mndcmnd 18710  Grpcgrp 18914  ._gcmg 19048  SubGrpcsubg 19101  mulGrpcmgp 20098  1_rcur 20139  Ringcrg 20191  CRingccrg 20192  SubRingcsubrg 20527  Poly₁cpl1 22110  coe₁cco1 22111   evalSub₁ ces1 22249

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-isom 6539  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-of 7669  df-ofr 7670  df-om 7860  df-1st 7986  df-2nd 7987  df-supp 8158  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-er 8717  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9372  df-sup 9452  df-oi 9522  df-card 9951  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12851  df-fz 13523  df-fzo 13670  df-seq 14018  df-hash 14347  df-struct 17164  df-sets 17181  df-slot 17199  df-ndx 17211  df-base 17227  df-ress 17250  df-plusg 17282  df-mulr 17283  df-sca 17285  df-vsca 17286  df-ip 17287  df-tset 17288  df-ple 17289  df-ds 17291  df-hom 17293  df-cco 17294  df-0g 17453  df-gsum 17454  df-prds 17459  df-pws 17461  df-mre 17596  df-mrc 17597  df-acs 17599  df-mgm 18616  df-sgrp 18695  df-mnd 18711  df-mhm 18759  df-submnd 18760  df-grp 18917  df-minusg 18918  df-sbg 18919  df-mulg 19049  df-subg 19104  df-ghm 19194  df-cntz 19298  df-cmn 19761  df-abl 19762  df-mgp 20099  df-rng 20111  df-ur 20140  df-srg 20145  df-ring 20193  df-cring 20194  df-rhm 20430  df-subrng 20504  df-subrg 20528  df-lmod 20817  df-lss 20887  df-lsp 20927  df-assa 21811  df-asp 21812  df-ascl 21813  df-psr 21867  df-mvr 21868  df-mpl 21869  df-opsr 21871  df-evls 22030  df-evl 22031  df-psr1 22113  df-vr1 22114  df-ply1 22115  df-coe1 22116  df-evls1 22251  df-evl1 22252

This theorem is referenced by:  cos9thpiminply  33768