Theorem coe1fzgsumd 22283

Description: Value of an evaluated coefficient in a finite group sum of polynomials. (Contributed by AV, 8-Oct-2019.)

Hypotheses

Ref	Expression
coe1fzgsumd.p	⊢ 𝑃 = (Poly1‘𝑅)
coe1fzgsumd.b	⊢ 𝐵 = (Base‘𝑃)
coe1fzgsumd.r	⊢ (𝜑 → 𝑅 ∈ Ring)
coe1fzgsumd.k	⊢ (𝜑 → 𝐾 ∈ ℕ0)
coe1fzgsumd.m	⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵)
coe1fzgsumd.n	⊢ (𝜑 → 𝑁 ∈ Fin)

Assertion

Ref	Expression
coe1fzgsumd	⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾))))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐾   𝑥,𝑁

Allowed substitution hints:   𝜑(𝑥)   𝑃(𝑥)   𝑅(𝑥)   𝑀(𝑥)

Proof of Theorem coe1fzgsumd

Dummy variables 𝑎 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coe1fzgsumd.m	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵)
2		coe1fzgsumd.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ Fin)
3		raleq 3293	. . . . . . 7 ⊢ (𝑛 = ∅ → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝐵))
4	3	anbi2d 631	. . . . . 6 ⊢ (𝑛 = ∅ → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝐵)))
5		mpteq1 5175	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ ∅ ↦ 𝑀))
6	5	oveq2d 7378	. . . . . . . . 9 ⊢ (𝑛 = ∅ → (𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))
7	6	fveq2d 6840	. . . . . . . 8 ⊢ (𝑛 = ∅ → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀))))
8	7	fveq1d 6838	. . . . . . 7 ⊢ (𝑛 = ∅ → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾))
9		mpteq1 5175	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)) = (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾)))
10	9	oveq2d 7378	. . . . . . 7 ⊢ (𝑛 = ∅ → (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾))) = (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾))))
11	8, 10	eqeq12d 2753	. . . . . 6 ⊢ (𝑛 = ∅ → (((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾))) ↔ ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾)))))
12	4, 11	imbi12d 344	. . . . 5 ⊢ (𝑛 = ∅ → (((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)))) ↔ ((𝜑 ∧ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾))))))
13		raleq 3293	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵))
14	13	anbi2d 631	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵)))
15		mpteq1 5175	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ 𝑚 ↦ 𝑀))
16	15	oveq2d 7378	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))
17	16	fveq2d 6840	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀))))
18	17	fveq1d 6838	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾))
19		mpteq1 5175	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)) = (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾)))
20	19	oveq2d 7378	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾))) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾))))
21	18, 20	eqeq12d 2753	. . . . . 6 ⊢ (𝑛 = 𝑚 → (((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾))) ↔ ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾)))))
22	14, 21	imbi12d 344	. . . . 5 ⊢ (𝑛 = 𝑚 → (((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)))) ↔ ((𝜑 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾))))))
23		raleq 3293	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵))
24	23	anbi2d 631	. . . . . 6 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵)))
25		mpteq1 5175	. . . . . . . . . 10 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀))
26	25	oveq2d 7378	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))
27	26	fveq2d 6840	. . . . . . . 8 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀))))
28	27	fveq1d 6838	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾))
29		mpteq1 5175	. . . . . . . 8 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)) = (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾)))
30	29	oveq2d 7378	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾))) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾))))
31	28, 30	eqeq12d 2753	. . . . . 6 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾))) ↔ ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾)))))
32	24, 31	imbi12d 344	. . . . 5 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)))) ↔ ((𝜑 ∧ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾))))))
33		raleq 3293	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵))
34	33	anbi2d 631	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵)))
35		mpteq1 5175	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ 𝑁 ↦ 𝑀))
36	35	oveq2d 7378	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))
37	36	fveq2d 6840	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀))))
38	37	fveq1d 6838	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾))
39		mpteq1 5175	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)) = (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾)))
40	39	oveq2d 7378	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾))) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾))))
41	38, 40	eqeq12d 2753	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾))) ↔ ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾)))))
42	34, 41	imbi12d 344	. . . . 5 ⊢ (𝑛 = 𝑁 → (((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)))) ↔ ((𝜑 ∧ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾))))))
43		mpt0 6636	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∅ ↦ 𝑀) = ∅
44	43	oveq2i 7373	. . . . . . . . . . . 12 ⊢ (𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)) = (𝑃 Σg ∅)
45		eqid 2737	. . . . . . . . . . . . 13 ⊢ (0g‘𝑃) = (0g‘𝑃)
46	45	gsum0 18647	. . . . . . . . . . . 12 ⊢ (𝑃 Σg ∅) = (0g‘𝑃)
47	44, 46	eqtri 2760	. . . . . . . . . . 11 ⊢ (𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)) = (0g‘𝑃)
48	47	fveq2i 6839	. . . . . . . . . 10 ⊢ (coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀))) = (coe1‘(0g‘𝑃))
49	48	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀))) = (coe1‘(0g‘𝑃)))
50	49	fveq1d 6838	. . . . . . . 8 ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾) = ((coe1‘(0g‘𝑃))‘𝐾))
51		coe1fzgsumd.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
52		coe1fzgsumd.p	. . . . . . . . . . 11 ⊢ 𝑃 = (Poly1‘𝑅)
53		eqid 2737	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
54	52, 45, 53	coe1z 22242	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
55	51, 54	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
56	55	fveq1d 6838	. . . . . . . 8 ⊢ (𝜑 → ((coe1‘(0g‘𝑃))‘𝐾) = ((ℕ0 × {(0g‘𝑅)})‘𝐾))
57		fvex 6849	. . . . . . . . 9 ⊢ (0g‘𝑅) ∈ V
58		coe1fzgsumd.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
59		fvconst2g 7152	. . . . . . . . 9 ⊢ (((0g‘𝑅) ∈ V ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝐾) = (0g‘𝑅))
60	57, 58, 59	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → ((ℕ0 × {(0g‘𝑅)})‘𝐾) = (0g‘𝑅))
61	50, 56, 60	3eqtrd 2776	. . . . . . 7 ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾) = (0g‘𝑅))
62		mpt0 6636	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾)) = ∅
63	62	oveq2i 7373	. . . . . . . 8 ⊢ (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾))) = (𝑅 Σg ∅)
64	53	gsum0 18647	. . . . . . . 8 ⊢ (𝑅 Σg ∅) = (0g‘𝑅)
65	63, 64	eqtri 2760	. . . . . . 7 ⊢ (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾))) = (0g‘𝑅)
66	61, 65	eqtr4di 2790	. . . . . 6 ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾))))
67	66	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾))))
68		coe1fzgsumd.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑃)
69	52, 68, 51, 58	coe1fzgsumdlem 22282	. . . . . . . 8 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → ((∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾))))))
70	69	3expia 1122	. . . . . . 7 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚) → (𝜑 → ((∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾)))))))
71	70	a2d 29	. . . . . 6 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚) → ((𝜑 → (∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾))))) → (𝜑 → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾)))))))
72		impexp 450	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾)))) ↔ (𝜑 → (∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾))))))
73		impexp 450	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾)))) ↔ (𝜑 → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾))))))
74	71, 72, 73	3imtr4g 296	. . . . 5 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚) → (((𝜑 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾)))) → ((𝜑 ∧ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾))))))
75	12, 22, 32, 42, 67, 74	findcard2s 9095	. . . 4 ⊢ (𝑁 ∈ Fin → ((𝜑 ∧ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾)))))
76	75	expd 415	. . 3 ⊢ (𝑁 ∈ Fin → (𝜑 → (∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾))))))
77	2, 76	mpcom 38	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾)))))
78	1, 77	mpd 15	1 ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ∪ cun 3888  ∅c0 4274  {csn 4568   ↦ cmpt 5167   × cxp 5624  ‘cfv 6494  (class class class)co 7362  Fincfn 8888  ℕ₀cn0 12432  Basecbs 17174  0_gc0g 17397   Σ_g cgsu 17398  Ringcrg 20209  Poly₁cpl1 22154  coe₁cco1 22155

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7626  df-ofr 7627  df-om 7813  df-1st 7937  df-2nd 7938  df-supp 8106  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-er 8638  df-map 8770  df-pm 8771  df-ixp 8841  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-fsupp 9270  df-sup 9350  df-oi 9420  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-fzo 13604  df-seq 13959  df-hash 14288  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-sca 17231  df-vsca 17232  df-ip 17233  df-tset 17234  df-ple 17235  df-ds 17237  df-hom 17239  df-cco 17240  df-0g 17399  df-gsum 17400  df-prds 17405  df-pws 17407  df-mre 17543  df-mrc 17544  df-acs 17546  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18746  df-submnd 18747  df-grp 18907  df-minusg 18908  df-mulg 19039  df-subg 19094  df-ghm 19183  df-cntz 19287  df-cmn 19752  df-abl 19753  df-mgp 20117  df-rng 20129  df-ur 20158  df-ring 20211  df-subrng 20518  df-subrg 20542  df-psr 21903  df-mpl 21905  df-opsr 21907  df-psr1 22157  df-ply1 22159  df-coe1 22160

This theorem is referenced by:  gsummoncoe1  22287  cpmatmcllem  22697  decpmatmullem  22750  mp2pm2mplem4  22788