evl1gsumd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > evl1gsumd		Structured version Visualization version GIF version

Theorem evl1gsumd 22097

Description: Polynomial evaluation builder for a finite group sum of polynomials. (Contributed by AV, 17-Sep-2019.)

**Hypotheses**
Ref	Expression
evl1gsumd.q	⊢ 𝑂 = (eval₁‘𝑅)
evl1gsumd.p	⊢ 𝑃 = (Poly₁‘𝑅)
evl1gsumd.b	⊢ 𝐵 = (Base‘𝑅)
evl1gsumd.u	⊢ 𝑈 = (Base‘𝑃)
evl1gsumd.r	⊢ (𝜑 → 𝑅 ∈ CRing)
evl1gsumd.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
evl1gsumd.m	⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈)
evl1gsumd.n	⊢ (𝜑 → 𝑁 ∈ Fin)

**Assertion**
Ref	Expression
evl1gsumd	⊢ (𝜑 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌))))

Distinct variable groups: 𝑥,𝑂 𝑥,𝑈 𝑥,𝑌 𝑥,𝐵 𝑥,𝑁 𝑥,𝑅 𝜑,𝑥 Allowed substitution hints: 𝑃(𝑥) 𝑀(𝑥)

Proof of Theorem evl1gsumd Dummy variables 𝑎 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evl1gsumd.m	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈)
2		evl1gsumd.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ Fin)
3		raleq 3321	. . . . . . 7 ⊢ (𝑛 = ∅ → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝑈))
4	3	anbi2d 628	. . . . . 6 ⊢ (𝑛 = ∅ → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) ↔ (𝜑 ∧ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝑈)))
5		mpteq1 5242	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ ∅ ↦ 𝑀))
6	5	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑛 = ∅ → (𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))
7	6	fveq2d 6896	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀))) = (𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀))))
8	7	fveq1d 6894	. . . . . . 7 ⊢ (𝑛 = ∅ → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = ((𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))‘𝑌))
9		mpteq1 5242	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)) = (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌)))
10	9	oveq2d 7428	. . . . . . 7 ⊢ (𝑛 = ∅ → (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌))) = (𝑅 Σ_g (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌))))
11	8, 10	eqeq12d 2747	. . . . . 6 ⊢ (𝑛 = ∅ → (((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌))) ↔ ((𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌)))))
12	4, 11	imbi12d 343	. . . . 5 ⊢ (𝑛 = ∅ → (((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)))) ↔ ((𝜑 ∧ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌))))))
13		raleq 3321	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈))
14	13	anbi2d 628	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈)))
15		mpteq1 5242	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ 𝑚 ↦ 𝑀))
16	15	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))
17	16	fveq2d 6896	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀))) = (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀))))
18	17	fveq1d 6894	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝑌))
19		mpteq1 5242	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)) = (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌)))
20	19	oveq2d 7428	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌))) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌))))
21	18, 20	eqeq12d 2747	. . . . . 6 ⊢ (𝑛 = 𝑚 → (((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌))) ↔ ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌)))))
22	14, 21	imbi12d 343	. . . . 5 ⊢ (𝑛 = 𝑚 → (((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)))) ↔ ((𝜑 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌))))))
23		raleq 3321	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈))
24	23	anbi2d 628	. . . . . 6 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) ↔ (𝜑 ∧ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈)))
25		mpteq1 5242	. . . . . . . . . 10 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀))
26	25	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))
27	26	fveq2d 6896	. . . . . . . 8 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀))) = (𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀))))
28	27	fveq1d 6894	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = ((𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌))
29		mpteq1 5242	. . . . . . . 8 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)) = (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌)))
30	29	oveq2d 7428	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌))) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌))))
31	28, 30	eqeq12d 2747	. . . . . 6 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌))) ↔ ((𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌)))))
32	24, 31	imbi12d 343	. . . . 5 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)))) ↔ ((𝜑 ∧ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌))))))
33		raleq 3321	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈))
34	33	anbi2d 628	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈)))
35		mpteq1 5242	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ 𝑁 ↦ 𝑀))
36	35	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀)))
37	36	fveq2d 6896	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀))) = (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀))))
38	37	fveq1d 6894	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌))
39		mpteq1 5242	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)) = (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌)))
40	39	oveq2d 7428	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌))) = (𝑅 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌))))
41	38, 40	eqeq12d 2747	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌))) ↔ ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌)))))
42	34, 41	imbi12d 343	. . . . 5 ⊢ (𝑛 = 𝑁 → (((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)))) ↔ ((𝜑 ∧ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌))))))
43		mpt0 6693	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∅ ↦ 𝑀) = ∅
44	43	oveq2i 7423	. . . . . . . . . . . 12 ⊢ (𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)) = (𝑃 Σ_g ∅)
45		eqid 2731	. . . . . . . . . . . . 13 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
46	45	gsum0 18610	. . . . . . . . . . . 12 ⊢ (𝑃 Σ_g ∅) = (0_g‘𝑃)
47	44, 46	eqtri 2759	. . . . . . . . . . 11 ⊢ (𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)) = (0_g‘𝑃)
48	47	fveq2i 6895	. . . . . . . . . 10 ⊢ (𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀))) = (𝑂‘(0_g‘𝑃))
49		evl1gsumd.r	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ CRing)
50		crngring 20140	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
51	49, 50	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Ring)
52		evl1gsumd.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = (Poly₁‘𝑅)
53		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
54		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
55	52, 53, 54, 45	ply1scl0 22033	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘(0_g‘𝑅)) = (0_g‘𝑃))
56	51, 55	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((algSc‘𝑃)‘(0_g‘𝑅)) = (0_g‘𝑃))
57	56	eqcomd 2737	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘𝑃) = ((algSc‘𝑃)‘(0_g‘𝑅)))
58	57	fveq2d 6896	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘(0_g‘𝑃)) = (𝑂‘((algSc‘𝑃)‘(0_g‘𝑅))))
59	48, 58	eqtrid 2783	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀))) = (𝑂‘((algSc‘𝑃)‘(0_g‘𝑅))))
60	59	fveq1d 6894	. . . . . . . 8 ⊢ (𝜑 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))‘𝑌) = ((𝑂‘((algSc‘𝑃)‘(0_g‘𝑅)))‘𝑌))
61		evl1gsumd.q	. . . . . . . . . 10 ⊢ 𝑂 = (eval₁‘𝑅)
62		evl1gsumd.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
63		evl1gsumd.u	. . . . . . . . . 10 ⊢ 𝑈 = (Base‘𝑃)
64		ringgrp 20133	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
65	51, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Grp)
66	62, 54	grpidcl 18887	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Grp → (0_g‘𝑅) ∈ 𝐵)
67	65, 66	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0_g‘𝑅) ∈ 𝐵)
68		evl1gsumd.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
69	61, 52, 62, 53, 63, 49, 67, 68	evl1scad 22075	. . . . . . . . 9 ⊢ (𝜑 → (((algSc‘𝑃)‘(0_g‘𝑅)) ∈ 𝑈 ∧ ((𝑂‘((algSc‘𝑃)‘(0_g‘𝑅)))‘𝑌) = (0_g‘𝑅)))
70	69	simprd 495	. . . . . . . 8 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑃)‘(0_g‘𝑅)))‘𝑌) = (0_g‘𝑅))
71	60, 70	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))‘𝑌) = (0_g‘𝑅))
72		mpt0 6693	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌)) = ∅
73	72	oveq2i 7423	. . . . . . . 8 ⊢ (𝑅 Σ_g (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌))) = (𝑅 Σ_g ∅)
74	54	gsum0 18610	. . . . . . . 8 ⊢ (𝑅 Σ_g ∅) = (0_g‘𝑅)
75	73, 74	eqtri 2759	. . . . . . 7 ⊢ (𝑅 Σ_g (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌))) = (0_g‘𝑅)
76	71, 75	eqtr4di 2789	. . . . . 6 ⊢ (𝜑 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌))))
77	76	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌))))
78	61, 52, 62, 63, 49, 68	evl1gsumdlem 22096	. . . . . . . 8 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → ((∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌))))))
79	78	3expia 1120	. . . . . . 7 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚) → (𝜑 → ((∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌)))))))
80	79	a2d 29	. . . . . 6 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚) → ((𝜑 → (∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌))))) → (𝜑 → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌)))))))
81		impexp 450	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌)))) ↔ (𝜑 → (∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌))))))
82		impexp 450	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌)))) ↔ (𝜑 → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌))))))
83	80, 81, 82	3imtr4g 295	. . . . 5 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚) → (((𝜑 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌)))) → ((𝜑 ∧ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌))))))
84	12, 22, 32, 42, 77, 83	findcard2s 9168	. . . 4 ⊢ (𝑁 ∈ Fin → ((𝜑 ∧ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌)))))
85	84	expd 415	. . 3 ⊢ (𝑁 ∈ Fin → (𝜑 → (∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌))))))
86	2, 85	mpcom 38	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌)))))
87	1, 86	mpd 15	1 ⊢ (𝜑 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∪ cun 3947 ∅c0 4323 {csn 4629 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7412 Fincfn 8942 Basecbs 17149 0_gc0g 17390 Σ_g cgsu 17391 Grpcgrp 18856 Ringcrg 20128 CRingccrg 20129 algSccascl 21627 Poly₁cpl1 21921 eval₁ce1 22054

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-ofr 7674 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-sup 9440 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-fzo 13633 df-seq 13972 df-hash 14296 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-srg 20082 df-ring 20130 df-cring 20131 df-rhm 20364 df-subrng 20435 df-subrg 20460 df-lmod 20617 df-lss 20688 df-lsp 20728 df-assa 21628 df-asp 21629 df-ascl 21630 df-psr 21682 df-mvr 21683 df-mpl 21684 df-opsr 21686 df-evls 21855 df-evl 21856 df-psr1 21924 df-ply1 21926 df-evl1 22056

This theorem is referenced by: evl1gsumaddval 22099

W3C validator