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Theorem coe1mul3 25616

Description: The coefficient vector of multiplication in the univariate polynomial ring, at indices high enough that at most one component can be active in the sum. (Contributed by Stefan O'Rear, 25-Mar-2015.)

**Hypotheses**
Ref	Expression
coe1mul3.s	⊢ 𝑌 = (Poly₁‘𝑅)
coe1mul3.t	⊢ ∙ = (._r‘𝑌)
coe1mul3.u	⊢ · = (._r‘𝑅)
coe1mul3.b	⊢ 𝐵 = (Base‘𝑌)
coe1mul3.d	⊢ 𝐷 = ( deg₁ ‘𝑅)
coe1mul3.r	⊢ (𝜑 → 𝑅 ∈ Ring)
coe1mul3.f1	⊢ (𝜑 → 𝐹 ∈ 𝐵)
coe1mul3.f2	⊢ (𝜑 → 𝐼 ∈ ℕ₀)
coe1mul3.f3	⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐼)
coe1mul3.g1	⊢ (𝜑 → 𝐺 ∈ 𝐵)
coe1mul3.g2	⊢ (𝜑 → 𝐽 ∈ ℕ₀)
coe1mul3.g3	⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐽)

**Assertion**
Ref	Expression
coe1mul3	⊢ (𝜑 → ((coe₁‘(𝐹 ∙ 𝐺))‘(𝐼 + 𝐽)) = (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘𝐽)))

Proof of Theorem coe1mul3 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coe1mul3.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		coe1mul3.f1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
3		coe1mul3.g1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
4		coe1mul3.s	. . . . 5 ⊢ 𝑌 = (Poly₁‘𝑅)
5		coe1mul3.t	. . . . 5 ⊢ ∙ = (._r‘𝑌)
6		coe1mul3.u	. . . . 5 ⊢ · = (._r‘𝑅)
7		coe1mul3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑌)
8	4, 5, 6, 7	coe1mul 21791	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe₁‘(𝐹 ∙ 𝐺)) = (𝑥 ∈ ℕ₀ ↦ (𝑅 Σ_g (𝑦 ∈ (0...𝑥) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦)))))))
9	1, 2, 3, 8	syl3anc 1371	. . 3 ⊢ (𝜑 → (coe₁‘(𝐹 ∙ 𝐺)) = (𝑥 ∈ ℕ₀ ↦ (𝑅 Σ_g (𝑦 ∈ (0...𝑥) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦)))))))
10	9	fveq1d 6893	. 2 ⊢ (𝜑 → ((coe₁‘(𝐹 ∙ 𝐺))‘(𝐼 + 𝐽)) = ((𝑥 ∈ ℕ₀ ↦ (𝑅 Σ_g (𝑦 ∈ (0...𝑥) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦))))))‘(𝐼 + 𝐽)))
11		coe1mul3.f2	. . . 4 ⊢ (𝜑 → 𝐼 ∈ ℕ₀)
12		coe1mul3.g2	. . . 4 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
13	11, 12	nn0addcld 12535	. . 3 ⊢ (𝜑 → (𝐼 + 𝐽) ∈ ℕ₀)
14		oveq2 7416	. . . . . 6 ⊢ (𝑥 = (𝐼 + 𝐽) → (0...𝑥) = (0...(𝐼 + 𝐽)))
15		fvoveq1 7431	. . . . . . 7 ⊢ (𝑥 = (𝐼 + 𝐽) → ((coe₁‘𝐺)‘(𝑥 − 𝑦)) = ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)))
16	15	oveq2d 7424	. . . . . 6 ⊢ (𝑥 = (𝐼 + 𝐽) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦))) = (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))
17	14, 16	mpteq12dv 5239	. . . . 5 ⊢ (𝑥 = (𝐼 + 𝐽) → (𝑦 ∈ (0...𝑥) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦)))) = (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)))))
18	17	oveq2d 7424	. . . 4 ⊢ (𝑥 = (𝐼 + 𝐽) → (𝑅 Σ_g (𝑦 ∈ (0...𝑥) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦))))) = (𝑅 Σ_g (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))))
19		eqid 2732	. . . 4 ⊢ (𝑥 ∈ ℕ₀ ↦ (𝑅 Σ_g (𝑦 ∈ (0...𝑥) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦)))))) = (𝑥 ∈ ℕ₀ ↦ (𝑅 Σ_g (𝑦 ∈ (0...𝑥) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦))))))
20		ovex 7441	. . . 4 ⊢ (𝑅 Σ_g (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))) ∈ V
21	18, 19, 20	fvmpt 6998	. . 3 ⊢ ((𝐼 + 𝐽) ∈ ℕ₀ → ((𝑥 ∈ ℕ₀ ↦ (𝑅 Σ_g (𝑦 ∈ (0...𝑥) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦))))))‘(𝐼 + 𝐽)) = (𝑅 Σ_g (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))))
22	13, 21	syl 17	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℕ₀ ↦ (𝑅 Σ_g (𝑦 ∈ (0...𝑥) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦))))))‘(𝐼 + 𝐽)) = (𝑅 Σ_g (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))))
23		eqid 2732	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
24		eqid 2732	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
25		ringmnd 20065	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
26	1, 25	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
27		ovexd 7443	. . . 4 ⊢ (𝜑 → (0...(𝐼 + 𝐽)) ∈ V)
28	11	nn0red 12532	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℝ)
29		nn0addge1 12517	. . . . . 6 ⊢ ((𝐼 ∈ ℝ ∧ 𝐽 ∈ ℕ₀) → 𝐼 ≤ (𝐼 + 𝐽))
30	28, 12, 29	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐼 ≤ (𝐼 + 𝐽))
31		fznn0 13592	. . . . . 6 ⊢ ((𝐼 + 𝐽) ∈ ℕ₀ → (𝐼 ∈ (0...(𝐼 + 𝐽)) ↔ (𝐼 ∈ ℕ₀ ∧ 𝐼 ≤ (𝐼 + 𝐽))))
32	13, 31	syl 17	. . . . 5 ⊢ (𝜑 → (𝐼 ∈ (0...(𝐼 + 𝐽)) ↔ (𝐼 ∈ ℕ₀ ∧ 𝐼 ≤ (𝐼 + 𝐽))))
33	11, 30, 32	mpbir2and 711	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (0...(𝐼 + 𝐽)))
34	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝑅 ∈ Ring)
35		eqid 2732	. . . . . . . . 9 ⊢ (coe₁‘𝐹) = (coe₁‘𝐹)
36	35, 7, 4, 23	coe1f 21734	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → (coe₁‘𝐹):ℕ₀⟶(Base‘𝑅))
37	2, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (coe₁‘𝐹):ℕ₀⟶(Base‘𝑅))
38		elfznn0 13593	. . . . . . 7 ⊢ (𝑦 ∈ (0...(𝐼 + 𝐽)) → 𝑦 ∈ ℕ₀)
39		ffvelcdm 7083	. . . . . . 7 ⊢ (((coe₁‘𝐹):ℕ₀⟶(Base‘𝑅) ∧ 𝑦 ∈ ℕ₀) → ((coe₁‘𝐹)‘𝑦) ∈ (Base‘𝑅))
40	37, 38, 39	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((coe₁‘𝐹)‘𝑦) ∈ (Base‘𝑅))
41		eqid 2732	. . . . . . . . 9 ⊢ (coe₁‘𝐺) = (coe₁‘𝐺)
42	41, 7, 4, 23	coe1f 21734	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐵 → (coe₁‘𝐺):ℕ₀⟶(Base‘𝑅))
43	3, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → (coe₁‘𝐺):ℕ₀⟶(Base‘𝑅))
44		fznn0sub 13532	. . . . . . 7 ⊢ (𝑦 ∈ (0...(𝐼 + 𝐽)) → ((𝐼 + 𝐽) − 𝑦) ∈ ℕ₀)
45		ffvelcdm 7083	. . . . . . 7 ⊢ (((coe₁‘𝐺):ℕ₀⟶(Base‘𝑅) ∧ ((𝐼 + 𝐽) − 𝑦) ∈ ℕ₀) → ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)) ∈ (Base‘𝑅))
46	43, 44, 45	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)) ∈ (Base‘𝑅))
47	23, 6	ringcl 20072	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((coe₁‘𝐹)‘𝑦) ∈ (Base‘𝑅) ∧ ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)) ∈ (Base‘𝑅)) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) ∈ (Base‘𝑅))
48	34, 40, 46, 47	syl3anc 1371	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) ∈ (Base‘𝑅))
49	48	fmpttd 7114	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)))):(0...(𝐼 + 𝐽))⟶(Base‘𝑅))
50		eldifsn 4790	. . . . . 6 ⊢ (𝑦 ∈ ((0...(𝐼 + 𝐽)) ∖ {𝐼}) ↔ (𝑦 ∈ (0...(𝐼 + 𝐽)) ∧ 𝑦 ≠ 𝐼))
51	38	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝑦 ∈ ℕ₀)
52	51	nn0red 12532	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝑦 ∈ ℝ)
53	28	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝐼 ∈ ℝ)
54	52, 53	lttri2d 11352	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (𝑦 ≠ 𝐼 ↔ (𝑦 < 𝐼 ∨ 𝐼 < 𝑦)))
55	3	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → 𝐺 ∈ 𝐵)
56	44	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((𝐼 + 𝐽) − 𝑦) ∈ ℕ₀)
57	56	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → ((𝐼 + 𝐽) − 𝑦) ∈ ℕ₀)
58		coe1mul3.d	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 = ( deg₁ ‘𝑅)
59	58, 4, 7	deg1xrcl 25599	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ^*)
60	3, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ^*)
61	60	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → (𝐷‘𝐺) ∈ ℝ^*)
62	12	nn0red 12532	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽 ∈ ℝ)
63	62	rexrd 11263	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ ℝ^*)
64	63	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → 𝐽 ∈ ℝ^*)
65	13	nn0red 12532	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐼 + 𝐽) ∈ ℝ)
66	65	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (𝐼 + 𝐽) ∈ ℝ)
67	66, 52	resubcld 11641	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((𝐼 + 𝐽) − 𝑦) ∈ ℝ)
68	67	rexrd 11263	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((𝐼 + 𝐽) − 𝑦) ∈ ℝ^*)
69	68	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → ((𝐼 + 𝐽) − 𝑦) ∈ ℝ^*)
70		coe1mul3.g3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐽)
71	70	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → (𝐷‘𝐺) ≤ 𝐽)
72	62	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝐽 ∈ ℝ)
73	52, 53, 72	ltadd1d 11806	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (𝑦 < 𝐼 ↔ (𝑦 + 𝐽) < (𝐼 + 𝐽)))
74	52, 72, 66	ltaddsub2d 11814	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((𝑦 + 𝐽) < (𝐼 + 𝐽) ↔ 𝐽 < ((𝐼 + 𝐽) − 𝑦)))
75	73, 74	bitrd 278	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (𝑦 < 𝐼 ↔ 𝐽 < ((𝐼 + 𝐽) − 𝑦)))
76	75	biimpa 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → 𝐽 < ((𝐼 + 𝐽) − 𝑦))
77	61, 64, 69, 71, 76	xrlelttrd 13138	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → (𝐷‘𝐺) < ((𝐼 + 𝐽) − 𝑦))
78	58, 4, 7, 24, 41	deg1lt 25614	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝐵 ∧ ((𝐼 + 𝐽) − 𝑦) ∈ ℕ₀ ∧ (𝐷‘𝐺) < ((𝐼 + 𝐽) − 𝑦)) → ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)) = (0_g‘𝑅))
79	55, 57, 77, 78	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)) = (0_g‘𝑅))
80	79	oveq2d 7424	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (((coe₁‘𝐹)‘𝑦) · (0_g‘𝑅)))
81	23, 6, 24	ringrz 20107	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ ((coe₁‘𝐹)‘𝑦) ∈ (Base‘𝑅)) → (((coe₁‘𝐹)‘𝑦) · (0_g‘𝑅)) = (0_g‘𝑅))
82	34, 40, 81	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (((coe₁‘𝐹)‘𝑦) · (0_g‘𝑅)) = (0_g‘𝑅))
83	82	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → (((coe₁‘𝐹)‘𝑦) · (0_g‘𝑅)) = (0_g‘𝑅))
84	80, 83	eqtrd 2772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
85	2	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → 𝐹 ∈ 𝐵)
86	51	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → 𝑦 ∈ ℕ₀)
87	58, 4, 7	deg1xrcl 25599	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ^*)
88	2, 87	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ^*)
89	88	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → (𝐷‘𝐹) ∈ ℝ^*)
90	28	rexrd 11263	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ ℝ^*)
91	90	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → 𝐼 ∈ ℝ^*)
92	52	rexrd 11263	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝑦 ∈ ℝ^*)
93	92	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → 𝑦 ∈ ℝ^*)
94		coe1mul3.f3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐼)
95	94	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → (𝐷‘𝐹) ≤ 𝐼)
96		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → 𝐼 < 𝑦)
97	89, 91, 93, 95, 96	xrlelttrd 13138	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → (𝐷‘𝐹) < 𝑦)
98	58, 4, 7, 24, 35	deg1lt 25614	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑦 ∈ ℕ₀ ∧ (𝐷‘𝐹) < 𝑦) → ((coe₁‘𝐹)‘𝑦) = (0_g‘𝑅))
99	85, 86, 97, 98	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → ((coe₁‘𝐹)‘𝑦) = (0_g‘𝑅))
100	99	oveq1d 7423	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = ((0_g‘𝑅) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))
101	23, 6, 24	ringlz 20106	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)) ∈ (Base‘𝑅)) → ((0_g‘𝑅) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
102	34, 46, 101	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((0_g‘𝑅) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
103	102	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → ((0_g‘𝑅) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
104	100, 103	eqtrd 2772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
105	84, 104	jaodan 956	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ (𝑦 < 𝐼 ∨ 𝐼 < 𝑦)) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
106	105	ex 413	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((𝑦 < 𝐼 ∨ 𝐼 < 𝑦) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅)))
107	54, 106	sylbid 239	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (𝑦 ≠ 𝐼 → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅)))
108	107	impr 455	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (0...(𝐼 + 𝐽)) ∧ 𝑦 ≠ 𝐼)) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
109	50, 108	sylan2b 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ((0...(𝐼 + 𝐽)) ∖ {𝐼})) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
110	109, 27	suppss2 8184	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)))) supp (0_g‘𝑅)) ⊆ {𝐼})
111	23, 24, 26, 27, 33, 49, 110	gsumpt 19829	. . 3 ⊢ (𝜑 → (𝑅 Σ_g (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))) = ((𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))‘𝐼))
112		fveq2 6891	. . . . . 6 ⊢ (𝑦 = 𝐼 → ((coe₁‘𝐹)‘𝑦) = ((coe₁‘𝐹)‘𝐼))
113		oveq2 7416	. . . . . . 7 ⊢ (𝑦 = 𝐼 → ((𝐼 + 𝐽) − 𝑦) = ((𝐼 + 𝐽) − 𝐼))
114	113	fveq2d 6895	. . . . . 6 ⊢ (𝑦 = 𝐼 → ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)) = ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼)))
115	112, 114	oveq12d 7426	. . . . 5 ⊢ (𝑦 = 𝐼 → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼))))
116		eqid 2732	. . . . 5 ⊢ (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)))) = (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))
117		ovex 7441	. . . . 5 ⊢ (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼))) ∈ V
118	115, 116, 117	fvmpt 6998	. . . 4 ⊢ (𝐼 ∈ (0...(𝐼 + 𝐽)) → ((𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))‘𝐼) = (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼))))
119	33, 118	syl 17	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))‘𝐼) = (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼))))
120	11	nn0cnd 12533	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℂ)
121	12	nn0cnd 12533	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℂ)
122	120, 121	pncan2d 11572	. . . . 5 ⊢ (𝜑 → ((𝐼 + 𝐽) − 𝐼) = 𝐽)
123	122	fveq2d 6895	. . . 4 ⊢ (𝜑 → ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼)) = ((coe₁‘𝐺)‘𝐽))
124	123	oveq2d 7424	. . 3 ⊢ (𝜑 → (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼))) = (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘𝐽)))
125	111, 119, 124	3eqtrd 2776	. 2 ⊢ (𝜑 → (𝑅 Σ_g (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))) = (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘𝐽)))
126	10, 22, 125	3eqtrd 2776	1 ⊢ (𝜑 → ((coe₁‘(𝐹 ∙ 𝐺))‘(𝐼 + 𝐽)) = (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ∖ cdif 3945 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ℝcr 11108 0cc0 11109 + caddc 11112 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 − cmin 11443 ℕ₀cn0 12471 ...cfz 13483 Basecbs 17143 ._rcmulr 17197 0_gc0g 17384 Σ_g cgsu 17385 Mndcmnd 18624 Ringcrg 20055 Poly₁cpl1 21700 coe₁cco1 21701 deg₁ cdg1 25568

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 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-ofr 7670 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-mulg 18950 df-ghm 19089 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-cnfld 20944 df-psr 21461 df-mpl 21463 df-opsr 21465 df-psr1 21703 df-ply1 21705 df-coe1 21706 df-mdeg 25569 df-deg1 25570

This theorem is referenced by: coe1mul4 25617

W3C validator