Theorem coe1fzgsumd 21710

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 3307	. . . . . . 7 ⊢ (𝑛 = ∅ → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝐵))
4	3	anbi2d 629	. . . . . 6 ⊢ (𝑛 = ∅ → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝐵)))
5		mpteq1 5203	. . . . . . . . . 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 6851	. . . . . . . 8 ⊢ (𝑛 = ∅ → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀))))
8	7	fveq1d 6849	. . . . . . 7 ⊢ (𝑛 = ∅ → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾))
9		mpteq1 5203	. . . . . . . 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 2747	. . . . . 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 3307	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵))
14	13	anbi2d 629	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵)))
15		mpteq1 5203	. . . . . . . . . 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 6851	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀))))
18	17	fveq1d 6849	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾))
19		mpteq1 5203	. . . . . . . 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 2747	. . . . . 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 3307	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵))
24	23	anbi2d 629	. . . . . 6 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵)))
25		mpteq1 5203	. . . . . . . . . 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 6851	. . . . . . . 8 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀))))
28	27	fveq1d 6849	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾))
29		mpteq1 5203	. . . . . . . 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 2747	. . . . . 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 3307	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵))
34	33	anbi2d 629	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵)))
35		mpteq1 5203	. . . . . . . . . 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 6851	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀))))
38	37	fveq1d 6849	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾))
39		mpteq1 5203	. . . . . . . 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 2747	. . . . . 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 6648	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∅ ↦ 𝑀) = ∅
44	43	oveq2i 7373	. . . . . . . . . . . 12 ⊢ (𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)) = (𝑃 Σg ∅)
45		eqid 2731	. . . . . . . . . . . . 13 ⊢ (0g‘𝑃) = (0g‘𝑃)
46	45	gsum0 18553	. . . . . . . . . . . 12 ⊢ (𝑃 Σg ∅) = (0g‘𝑃)
47	44, 46	eqtri 2759	. . . . . . . . . . 11 ⊢ (𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)) = (0g‘𝑃)
48	47	fveq2i 6850	. . . . . . . . . 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 6849	. . . . . . . 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 2731	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
54	52, 45, 53	coe1z 21671	. . . . . . . . . 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 6849	. . . . . . . 8 ⊢ (𝜑 → ((coe1‘(0g‘𝑃))‘𝐾) = ((ℕ0 × {(0g‘𝑅)})‘𝐾))
57		fvex 6860	. . . . . . . . 9 ⊢ (0g‘𝑅) ∈ V
58		coe1fzgsumd.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
59		fvconst2g 7156	. . . . . . . . 9 ⊢ (((0g‘𝑅) ∈ V ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝐾) = (0g‘𝑅))
60	57, 58, 59	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → ((ℕ0 × {(0g‘𝑅)})‘𝐾) = (0g‘𝑅))
61	50, 56, 60	3eqtrd 2775	. . . . . . 7 ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾) = (0g‘𝑅))
62		mpt0 6648	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾)) = ∅
63	62	oveq2i 7373	. . . . . . . 8 ⊢ (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾))) = (𝑅 Σg ∅)
64	53	gsum0 18553	. . . . . . . 8 ⊢ (𝑅 Σg ∅) = (0g‘𝑅)
65	63, 64	eqtri 2759	. . . . . . 7 ⊢ (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾))) = (0g‘𝑅)
66	61, 65	eqtr4di 2789	. . . . . 6 ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾))))
67	66	adantr 481	. . . . 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 21709	. . . . . . . 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 1121	. . . . . . 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 451	. . . . . 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 451	. . . . . 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 295	. . . . 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 9116	. . . 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 416	. . 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  Vcvv 3446   ∪ cun 3911  ∅c0 4287  {csn 4591   ↦ cmpt 5193   × cxp 5636  ‘cfv 6501  (class class class)co 7362  Fincfn 8890  ℕ₀cn0 12422  Basecbs 17094  0_gc0g 17335   Σ_g cgsu 17336  Ringcrg 19978  Poly₁cpl1 21585  coe₁cco1 21586

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-ofr 7623  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-pm 8775  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9313  df-sup 9387  df-oi 9455  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12423  df-z 12509  df-dec 12628  df-uz 12773  df-fz 13435  df-fzo 13578  df-seq 13917  df-hash 14241  df-struct 17030  df-sets 17047  df-slot 17065  df-ndx 17077  df-base 17095  df-ress 17124  df-plusg 17160  df-mulr 17161  df-sca 17163  df-vsca 17164  df-ip 17165  df-tset 17166  df-ple 17167  df-ds 17169  df-hom 17171  df-cco 17172  df-0g 17337  df-gsum 17338  df-prds 17343  df-pws 17345  df-mre 17480  df-mrc 17481  df-acs 17483  df-mgm 18511  df-sgrp 18560  df-mnd 18571  df-mhm 18615  df-submnd 18616  df-grp 18765  df-minusg 18766  df-mulg 18887  df-subg 18939  df-ghm 19020  df-cntz 19111  df-cmn 19578  df-abl 19579  df-mgp 19911  df-ur 19928  df-ring 19980  df-subrg 20268  df-psr 21348  df-mpl 21350  df-opsr 21352  df-psr1 21588  df-ply1 21590  df-coe1 21591

This theorem is referenced by:  gsummoncoe1  21712  cpmatmcllem  22104  decpmatmullem  22157  mp2pm2mplem4  22195