Theorem coe1fzgsumd 22219

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 3289	. . . . . . 7 ⊢ (𝑛 = ∅ → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝐵))
4	3	anbi2d 630	. . . . . 6 ⊢ (𝑛 = ∅ → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝐵)))
5		mpteq1 5178	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ ∅ ↦ 𝑀))
6	5	oveq2d 7362	. . . . . . . . 9 ⊢ (𝑛 = ∅ → (𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))
7	6	fveq2d 6826	. . . . . . . 8 ⊢ (𝑛 = ∅ → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀))))
8	7	fveq1d 6824	. . . . . . 7 ⊢ (𝑛 = ∅ → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾))
9		mpteq1 5178	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)) = (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾)))
10	9	oveq2d 7362	. . . . . . 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 3289	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵))
14	13	anbi2d 630	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵)))
15		mpteq1 5178	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ 𝑚 ↦ 𝑀))
16	15	oveq2d 7362	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))
17	16	fveq2d 6826	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀))))
18	17	fveq1d 6824	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾))
19		mpteq1 5178	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)) = (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾)))
20	19	oveq2d 7362	. . . . . . 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 3289	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵))
24	23	anbi2d 630	. . . . . 6 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵)))
25		mpteq1 5178	. . . . . . . . . 10 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀))
26	25	oveq2d 7362	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))
27	26	fveq2d 6826	. . . . . . . 8 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀))))
28	27	fveq1d 6824	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾))
29		mpteq1 5178	. . . . . . . 8 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)) = (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾)))
30	29	oveq2d 7362	. . . . . . 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 3289	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵))
34	33	anbi2d 630	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵)))
35		mpteq1 5178	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ 𝑁 ↦ 𝑀))
36	35	oveq2d 7362	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))
37	36	fveq2d 6826	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀))) = (coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀))))
38	37	fveq1d 6824	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝐾) = ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾))
39		mpteq1 5178	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝑛 ↦ ((coe1‘𝑀)‘𝐾)) = (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾)))
40	39	oveq2d 7362	. . . . . . 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 6623	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∅ ↦ 𝑀) = ∅
44	43	oveq2i 7357	. . . . . . . . . . . 12 ⊢ (𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)) = (𝑃 Σg ∅)
45		eqid 2731	. . . . . . . . . . . . 13 ⊢ (0g‘𝑃) = (0g‘𝑃)
46	45	gsum0 18592	. . . . . . . . . . . 12 ⊢ (𝑃 Σg ∅) = (0g‘𝑃)
47	44, 46	eqtri 2754	. . . . . . . . . . 11 ⊢ (𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)) = (0g‘𝑃)
48	47	fveq2i 6825	. . . . . . . . . 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 6824	. . . . . . . 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 22177	. . . . . . . . . 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 6824	. . . . . . . 8 ⊢ (𝜑 → ((coe1‘(0g‘𝑃))‘𝐾) = ((ℕ0 × {(0g‘𝑅)})‘𝐾))
57		fvex 6835	. . . . . . . . 9 ⊢ (0g‘𝑅) ∈ V
58		coe1fzgsumd.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
59		fvconst2g 7136	. . . . . . . . 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 2770	. . . . . . 7 ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑥 ∈ ∅ ↦ 𝑀)))‘𝐾) = (0g‘𝑅))
62		mpt0 6623	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾)) = ∅
63	62	oveq2i 7357	. . . . . . . 8 ⊢ (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾))) = (𝑅 Σg ∅)
64	53	gsum0 18592	. . . . . . . 8 ⊢ (𝑅 Σg ∅) = (0g‘𝑅)
65	63, 64	eqtri 2754	. . . . . . 7 ⊢ (𝑅 Σg (𝑥 ∈ ∅ ↦ ((coe1‘𝑀)‘𝐾))) = (0g‘𝑅)
66	61, 65	eqtr4di 2784	. . . . . 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 22218	. . . . . . . 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 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 9075	. . . 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 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∪ cun 3895  ∅c0 4280  {csn 4573   ↦ cmpt 5170   × cxp 5612  ‘cfv 6481  (class class class)co 7346  Fincfn 8869  ℕ₀cn0 12381  Basecbs 17120  0_gc0g 17343   Σ_g cgsu 17344  Ringcrg 20151  Poly₁cpl1 22089  coe₁cco1 22090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-ofr 7611  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-sup 9326  df-oi 9396  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-fzo 13555  df-seq 13909  df-hash 14238  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-hom 17185  df-cco 17186  df-0g 17345  df-gsum 17346  df-prds 17351  df-pws 17353  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-mhm 18691  df-submnd 18692  df-grp 18849  df-minusg 18850  df-mulg 18981  df-subg 19036  df-ghm 19125  df-cntz 19229  df-cmn 19694  df-abl 19695  df-mgp 20059  df-rng 20071  df-ur 20100  df-ring 20153  df-subrng 20461  df-subrg 20485  df-psr 21846  df-mpl 21848  df-opsr 21850  df-psr1 22092  df-ply1 22094  df-coe1 22095

This theorem is referenced by:  gsummoncoe1  22223  cpmatmcllem  22633  decpmatmullem  22686  mp2pm2mplem4  22724