Theorem ressply1evls1 33763

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 2764	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
3	2	subrgss 20624	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝑅 ⊆ (Base‘𝐸))
4		ressply1evls1.1	. . . . . 6 ⊢ 𝐺 = (𝐸 ↾s 𝑅)
5	4, 2	ressbas2 17276	. . . . 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 3973	. . 3 ⊢ (𝜑 → (Base‘𝐺) ⊆ (Base‘𝐸))
9	8	resmptd 6031	. 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 20650	. . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (𝑆 ∈ (SubRing‘𝐺) ↔ (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅)))
17	16	biimpa 480	. . . . . 6 ⊢ ((𝑅 ∈ (SubRing‘𝐸) ∧ 𝑆 ∈ (SubRing‘𝐺)) → (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅))
18	1, 15, 17	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝑆 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅))
19	18	simpld 498	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝐸))
20		ressply1evls1.10	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
21		eqid 2764	. . . 4 ⊢ (.r‘𝐸) = (.r‘𝐸)
22		eqid 2764	. . . 4 ⊢ (.g‘(mulGrp‘𝐸)) = (.g‘(mulGrp‘𝐸))
23		eqid 2764	. . . 4 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
24	10, 2, 11, 12, 13, 14, 19, 20, 21, 22, 23	evls1fpws 22434	. . 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 5968	. 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 2764	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
28		eqid 2764	. . . 4 ⊢ (Poly1‘(𝐺 ↾s 𝑆)) = (Poly1‘(𝐺 ↾s 𝑆))
29		eqid 2764	. . . 4 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆)
30		eqid 2764	. . . 4 ⊢ (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = (Base‘(Poly1‘(𝐺 ↾s 𝑆)))
31	4	subrgcrng 20627	. . . . 5 ⊢ ((𝐸 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝐸)) → 𝐺 ∈ CRing)
32	14, 1, 31	syl2anc 593	. . . 4 ⊢ (𝜑 → 𝐺 ∈ CRing)
33	18	simprd 499	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ 𝑅)
34		ressabs 17286	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (SubRing‘𝐸) ∧ 𝑆 ⊆ 𝑅) → ((𝐸 ↾s 𝑅) ↾s 𝑆) = (𝐸 ↾s 𝑆))
35	1, 33, 34	syl2anc 593	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 ↾s 𝑅) ↾s 𝑆) = (𝐸 ↾s 𝑆))
36	4	oveq1i 7408	. . . . . . . . . 10 ⊢ (𝐺 ↾s 𝑆) = ((𝐸 ↾s 𝑅) ↾s 𝑆)
37	35, 36, 12	3eqtr4g 2824	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾s 𝑆) = 𝐾)
38	37	fveq2d 6873	. . . . . . . 8 ⊢ (𝜑 → (Poly1‘(𝐺 ↾s 𝑆)) = (Poly1‘𝐾))
39	38, 11	eqtr4di 2817	. . . . . . 7 ⊢ (𝜑 → (Poly1‘(𝐺 ↾s 𝑆)) = 𝑃)
40	39	fveq2d 6873	. . . . . 6 ⊢ (𝜑 → (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = (Base‘𝑃))
41	40, 13	eqtr4di 2817	. . . . 5 ⊢ (𝜑 → (Base‘(Poly1‘(𝐺 ↾s 𝑆))) = 𝐵)
42	20, 41	eleqtrrd 2867	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(Poly1‘(𝐺 ↾s 𝑆))))
43		eqid 2764	. . . 4 ⊢ (.r‘𝐺) = (.r‘𝐺)
44		eqid 2764	. . . 4 ⊢ (.g‘(mulGrp‘𝐺)) = (.g‘(mulGrp‘𝐺))
45	26, 27, 28, 29, 30, 32, 15, 42, 43, 44, 23	evls1fpws 22434	. . 3 ⊢ (𝜑 → (𝑄‘𝐹) = (𝑥 ∈ (Base‘𝐺) ↦ (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐺)(𝑘(.g‘(mulGrp‘𝐺))𝑥))))))
46		eqid 2764	. . . . . 6 ⊢ (+g‘𝐸) = (+g‘𝐸)
47	14	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝐸 ∈ CRing)
48		nn0ex 12489	. . . . . . 7 ⊢ ℕ0 ∈ V
49	48	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ℕ0 ∈ V)
50	7	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑅 ⊆ (Base‘𝐸))
51	1	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ (SubRing‘𝐸))
52	33, 7	sstrd 3948	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐸))
53	12, 2	ressbas2 17276	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ (Base‘𝐸) → 𝑆 = (Base‘𝐾))
54	52, 53	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 = (Base‘𝐾))
55	54, 33	eqsstrrd 3973	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐾) ⊆ 𝑅)
56	55	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (Base‘𝐾) ⊆ 𝑅)
57	20	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝐹 ∈ 𝐵)
58		simpr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
59		eqid 2764	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
60	23, 13, 11, 59	coe1fvalcl 22276	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝐾))
61	57, 58, 60	syl2anc 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝐾))
62	56, 61	sseldd 3939	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ 𝑅)
63		eqid 2764	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐸) = (mulGrp‘𝐸)
64	4, 63	mgpress 20198	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝐸)) → ((mulGrp‘𝐸) ↾s 𝑅) = (mulGrp‘𝐺))
65	47, 51, 64	syl2an2r 695	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → ((mulGrp‘𝐸) ↾s 𝑅) = (mulGrp‘𝐺))
66	14	crngringd 20298	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ Ring)
67		eqid 2764	. . . . . . . . . . . . . . . 16 ⊢ (1r‘𝐸) = (1r‘𝐸)
68	67	subrg1cl 20632	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝑅)
69	1, 68	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝑅)
70	4, 2, 67	ress1r 33415	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ Ring ∧ (1r‘𝐸) ∈ 𝑅 ∧ 𝑅 ⊆ (Base‘𝐸)) → (1r‘𝐸) = (1r‘𝐺))
71	66, 69, 7, 70	syl3anc 1392	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐺))
72	71	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (1r‘𝐸) = (1r‘𝐺))
73	63, 67	ringidval 20235	. . . . . . . . . . . 12 ⊢ (1r‘𝐸) = (0g‘(mulGrp‘𝐸))
74		eqid 2764	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐺) = (mulGrp‘𝐺)
75		eqid 2764	. . . . . . . . . . . . 13 ⊢ (1r‘𝐺) = (1r‘𝐺)
76	74, 75	ringidval 20235	. . . . . . . . . . . 12 ⊢ (1r‘𝐺) = (0g‘(mulGrp‘𝐺))
77	72, 73, 76	3eqtr3g 2822	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (0g‘(mulGrp‘𝐸)) = (0g‘(mulGrp‘𝐺)))
78	63, 2	mgpbas 20193	. . . . . . . . . . . . 13 ⊢ (Base‘𝐸) = (Base‘(mulGrp‘𝐸))
79	7, 78	sseqtrdi 3978	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ⊆ (Base‘(mulGrp‘𝐸)))
80	79	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑅 ⊆ (Base‘(mulGrp‘𝐸)))
81	6	eleq2d 2850	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝑅 ↔ 𝑥 ∈ (Base‘𝐺)))
82	81	biimpar 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝑅)
83	82	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑥 ∈ 𝑅)
84	65, 77, 80, 58, 83	ressmulgnn0d 33226	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐺))𝑥) = (𝑘(.g‘(mulGrp‘𝐸))𝑥))
85	74, 27	mgpbas 20193	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘(mulGrp‘𝐺))
86	4	subrgring 20626	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝐺 ∈ Ring)
87	74	ringmgp 20291	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Ring → (mulGrp‘𝐺) ∈ Mnd)
88	1, 86, 87	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝐺) ∈ Mnd)
89	88	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝐺) ∈ Mnd)
90		simplr 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → 𝑥 ∈ (Base‘𝐺))
91	85, 44, 89, 58, 90	mulgnn0cld 19139	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐺))𝑥) ∈ (Base‘𝐺))
92	84, 91	eqeltrrd 2865	. . . . . . . . 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 2867	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝐸))𝑥) ∈ 𝑅)
95	21, 51, 62, 94	subrgmcld 33414	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥)) ∈ 𝑅)
96	95	fmpttd 7098	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝐸)(𝑘(.g‘(mulGrp‘𝐸))𝑥))):ℕ0⟶𝑅)
97		subrgsubg 20629	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝐸) → 𝑅 ∈ (SubGrp‘𝐸))
98		eqid 2764	. . . . . . . . 9 ⊢ (0g‘𝐸) = (0g‘𝐸)
99	98	subg0cl 19178	. . . . . . . 8 ⊢ (𝑅 ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ 𝑅)
100	1, 97, 99	3syl 18	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝑅)
101	100	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (0g‘𝐸) ∈ 𝑅)
102	14	crnggrpd 20299	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Grp)
103	102	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → 𝐸 ∈ Grp)
104		simpr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → 𝑦 ∈ (Base‘𝐸))
105	2, 46, 98, 103, 104	grplidd 19013	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → ((0g‘𝐸)(+g‘𝐸)𝑦) = 𝑦)
106	2, 46, 98, 103, 104	grpridd 19014	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐸)) → (𝑦(+g‘𝐸)(0g‘𝐸)) = 𝑦)
107	105, 106	jca 519	. . . . . 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 18718	. . . . 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 17338	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (SubRing‘𝐸) → (.r‘𝐸) = (.r‘𝐺))
110	1, 109	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (.r‘𝐸) = (.r‘𝐺))
111	110	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑘 ∈ ℕ0) → (.r‘𝐸) = (.r‘𝐺))
112	111	oveqd 7415	. . . . . . . 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 7414	. . . . . . . 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 2801	. . . . . . 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 5195	. . . . . 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 7414	. . . . 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 2802	. . . 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 5195	. . 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 2802	. 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 2810	1 ⊢ (𝜑 → (𝑄‘𝐹) = ((𝑂‘𝐹) ↾ 𝑅))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  Vcvv 3456   ⊆ wss 3906   ↦ cmpt 5183   ↾ cres 5651  ‘cfv 6523  (class class class)co 7398  ℕ₀cn0 12483  Basecbs 17247   ↾_s cress 17268  +_gcplusg 17288  ._rcmulr 17289  0_gc0g 17470   Σ_g cgsu 17471  Mndcmnd 18770  Grpcgrp 18977  ._gcmg 19111  SubGrpcsubg 19164  mulGrpcmgp 20188  1_rcur 20233  Ringcrg 20285  CRingccrg 20286  SubRingcsubrg 20621  Poly₁cpl1 22241  coe₁cco1 22242   evalSub₁ ces1 22378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-of 7662  df-ofr 7663  df-om 7849  df-1st 7972  df-2nd 7973  df-supp 8143  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-er 8680  df-map 8812  df-pm 8813  df-ixp 8882  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-fsupp 9310  df-sup 9390  df-oi 9460  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-z 12571  df-dec 12691  df-uz 12842  df-fz 13515  df-fzo 13662  df-seq 14017  df-hash 14346  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17248  df-ress 17269  df-plusg 17301  df-mulr 17302  df-sca 17304  df-vsca 17305  df-ip 17306  df-tset 17307  df-ple 17308  df-ds 17310  df-hom 17312  df-cco 17313  df-0g 17472  df-gsum 17473  df-prds 17478  df-pws 17480  df-mre 17616  df-mrc 17617  df-acs 17619  df-mgm 18676  df-sgrp 18755  df-mnd 18771  df-mhm 18819  df-submnd 18820  df-grp 18980  df-minusg 18981  df-sbg 18982  df-mulg 19112  df-subg 19167  df-ghm 19256  df-cntz 19359  df-cmn 19824  df-abl 19825  df-mgp 20189  df-rng 20201  df-ur 20234  df-srg 20239  df-ring 20287  df-cring 20288  df-rhm 20523  df-subrng 20598  df-subrg 20622  df-lmod 20931  df-lss 21001  df-lsp 21041  df-assa 21907  df-asp 21908  df-ascl 21909  df-psr 21963  df-mvr 21964  df-mpl 21965  df-opsr 21967  df-evls 22129  df-evl 22130  df-psr1 22244  df-vr1 22245  df-ply1 22246  df-coe1 22247  df-evls1 22380  df-evl1 22381

This theorem is referenced by:  cos9thpiminply  34087