evl1gsumd - Metamath Proof Explorer

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Theorem evl1gsumd 21883

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 3322	. . . . . . 7 ⊢ (𝑛 = ∅ → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝑈))
4	3	anbi2d 629	. . . . . 6 ⊢ (𝑛 = ∅ → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) ↔ (𝜑 ∧ ∀𝑥 ∈ ∅ 𝑀 ∈ 𝑈)))
5		mpteq1 5241	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ ∅ ↦ 𝑀))
6	5	oveq2d 7427	. . . . . . . . 9 ⊢ (𝑛 = ∅ → (𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))
7	6	fveq2d 6895	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀))) = (𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀))))
8	7	fveq1d 6893	. . . . . . 7 ⊢ (𝑛 = ∅ → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = ((𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))‘𝑌))
9		mpteq1 5241	. . . . . . . 8 ⊢ (𝑛 = ∅ → (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)) = (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌)))
10	9	oveq2d 7427	. . . . . . 7 ⊢ (𝑛 = ∅ → (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌))) = (𝑅 Σ_g (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌))))
11	8, 10	eqeq12d 2748	. . . . . 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 344	. . . . 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 3322	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈))
14	13	anbi2d 629	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝑈)))
15		mpteq1 5241	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ 𝑚 ↦ 𝑀))
16	15	oveq2d 7427	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))
17	16	fveq2d 6895	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀))) = (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀))))
18	17	fveq1d 6893	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝑌))
19		mpteq1 5241	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)) = (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌)))
20	19	oveq2d 7427	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌))) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((𝑂‘𝑀)‘𝑌))))
21	18, 20	eqeq12d 2748	. . . . . 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 344	. . . . 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 3322	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈))
24	23	anbi2d 629	. . . . . 6 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) ↔ (𝜑 ∧ ∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝑈)))
25		mpteq1 5241	. . . . . . . . . 10 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀))
26	25	oveq2d 7427	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))
27	26	fveq2d 6895	. . . . . . . 8 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀))) = (𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀))))
28	27	fveq1d 6893	. . . . . . 7 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = ((𝑂‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌))
29		mpteq1 5241	. . . . . . . 8 ⊢ (𝑛 = (𝑚 ∪ {𝑎}) → (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)) = (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂‘𝑀)‘𝑌)))
30	29	oveq2d 7427	. . . . . . 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 2748	. . . . . 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 344	. . . . 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 3322	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈))
34	33	anbi2d 629	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝜑 ∧ ∀𝑥 ∈ 𝑛 𝑀 ∈ 𝑈) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈)))
35		mpteq1 5241	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝑛 ↦ 𝑀) = (𝑥 ∈ 𝑁 ↦ 𝑀))
36	35	oveq2d 7427	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)) = (𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀)))
37	36	fveq2d 6895	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀))) = (𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀))))
38	37	fveq1d 6893	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑛 ↦ 𝑀)))‘𝑌) = ((𝑂‘(𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌))
39		mpteq1 5241	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌)) = (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌)))
40	39	oveq2d 7427	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑅 Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑂‘𝑀)‘𝑌))) = (𝑅 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌))))
41	38, 40	eqeq12d 2748	. . . . . 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 344	. . . . 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 6692	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∅ ↦ 𝑀) = ∅
44	43	oveq2i 7422	. . . . . . . . . . . 12 ⊢ (𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)) = (𝑃 Σ_g ∅)
45		eqid 2732	. . . . . . . . . . . . 13 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
46	45	gsum0 18605	. . . . . . . . . . . 12 ⊢ (𝑃 Σ_g ∅) = (0_g‘𝑃)
47	44, 46	eqtri 2760	. . . . . . . . . . 11 ⊢ (𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)) = (0_g‘𝑃)
48	47	fveq2i 6894	. . . . . . . . . 10 ⊢ (𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀))) = (𝑂‘(0_g‘𝑃))
49		evl1gsumd.r	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ CRing)
50		crngring 20070	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
51	49, 50	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Ring)
52		evl1gsumd.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = (Poly₁‘𝑅)
53		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
54		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
55	52, 53, 54, 45	ply1scl0 21819	. . . . . . . . . . . . 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 2738	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘𝑃) = ((algSc‘𝑃)‘(0_g‘𝑅)))
58	57	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘(0_g‘𝑃)) = (𝑂‘((algSc‘𝑃)‘(0_g‘𝑅))))
59	48, 58	eqtrid 2784	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀))) = (𝑂‘((algSc‘𝑃)‘(0_g‘𝑅))))
60	59	fveq1d 6893	. . . . . . . 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 20063	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
65	51, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Grp)
66	62, 54	grpidcl 18852	. . . . . . . . . . 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 21861	. . . . . . . . 9 ⊢ (𝜑 → (((algSc‘𝑃)‘(0_g‘𝑅)) ∈ 𝑈 ∧ ((𝑂‘((algSc‘𝑃)‘(0_g‘𝑅)))‘𝑌) = (0_g‘𝑅)))
70	69	simprd 496	. . . . . . . 8 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑃)‘(0_g‘𝑅)))‘𝑌) = (0_g‘𝑅))
71	60, 70	eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))‘𝑌) = (0_g‘𝑅))
72		mpt0 6692	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌)) = ∅
73	72	oveq2i 7422	. . . . . . . 8 ⊢ (𝑅 Σ_g (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌))) = (𝑅 Σ_g ∅)
74	54	gsum0 18605	. . . . . . . 8 ⊢ (𝑅 Σ_g ∅) = (0_g‘𝑅)
75	73, 74	eqtri 2760	. . . . . . 7 ⊢ (𝑅 Σ_g (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌))) = (0_g‘𝑅)
76	71, 75	eqtr4di 2790	. . . . . 6 ⊢ (𝜑 → ((𝑂‘(𝑃 Σ_g (𝑥 ∈ ∅ ↦ 𝑀)))‘𝑌) = (𝑅 Σ_g (𝑥 ∈ ∅ ↦ ((𝑂‘𝑀)‘𝑌))))
77	76	adantr 481	. . . . 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 21882	. . . . . . . 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 1121	. . . . . . 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 451	. . . . . 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 451	. . . . . 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 9167	. . . 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 416	. . 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 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∪ cun 3946 ∅c0 4322 {csn 4628 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7411 Fincfn 8941 Basecbs 17146 0_gc0g 17387 Σ_g cgsu 17388 Grpcgrp 18821 Ringcrg 20058 CRingccrg 20059 algSccascl 21413 Poly₁cpl1 21707 eval₁ce1 21840

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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-fz 13487 df-fzo 13630 df-seq 13969 df-hash 14293 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-hom 17223 df-cco 17224 df-0g 17389 df-gsum 17390 df-prds 17395 df-pws 17397 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-mhm 18673 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-ghm 19092 df-cntz 19183 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-srg 20012 df-ring 20060 df-cring 20061 df-rnghom 20255 df-subrg 20321 df-lmod 20477 df-lss 20548 df-lsp 20588 df-assa 21414 df-asp 21415 df-ascl 21416 df-psr 21468 df-mvr 21469 df-mpl 21470 df-opsr 21472 df-evls 21641 df-evl 21642 df-psr1 21710 df-ply1 21712 df-evl1 21842

This theorem is referenced by: evl1gsumaddval 21885

W3C validator