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Theorem coe1mul3 25852

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 22012	. . . 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 1369	. . 3 ⊢ (𝜑 → (coe₁‘(𝐹 ∙ 𝐺)) = (𝑥 ∈ ℕ₀ ↦ (𝑅 Σ_g (𝑦 ∈ (0...𝑥) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦)))))))
10	9	fveq1d 6892	. 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 12540	. . 3 ⊢ (𝜑 → (𝐼 + 𝐽) ∈ ℕ₀)
14		oveq2 7419	. . . . . 6 ⊢ (𝑥 = (𝐼 + 𝐽) → (0...𝑥) = (0...(𝐼 + 𝐽)))
15		fvoveq1 7434	. . . . . . 7 ⊢ (𝑥 = (𝐼 + 𝐽) → ((coe₁‘𝐺)‘(𝑥 − 𝑦)) = ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)))
16	15	oveq2d 7427	. . . . . 6 ⊢ (𝑥 = (𝐼 + 𝐽) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘(𝑥 − 𝑦))) = (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))
17	14, 16	mpteq12dv 5238	. . . . 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 7427	. . . 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 2730	. . . 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 7444	. . . 4 ⊢ (𝑅 Σ_g (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))))) ∈ V
21	18, 19, 20	fvmpt 6997	. . 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 2730	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
24		eqid 2730	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
25		ringmnd 20137	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
26	1, 25	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Mnd)
27		ovexd 7446	. . . 4 ⊢ (𝜑 → (0...(𝐼 + 𝐽)) ∈ V)
28	11	nn0red 12537	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℝ)
29		nn0addge1 12522	. . . . . 6 ⊢ ((𝐼 ∈ ℝ ∧ 𝐽 ∈ ℕ₀) → 𝐼 ≤ (𝐼 + 𝐽))
30	28, 12, 29	syl2anc 582	. . . . 5 ⊢ (𝜑 → 𝐼 ≤ (𝐼 + 𝐽))
31		fznn0 13597	. . . . . 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 709	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (0...(𝐼 + 𝐽)))
34	1	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝑅 ∈ Ring)
35		eqid 2730	. . . . . . . . 9 ⊢ (coe₁‘𝐹) = (coe₁‘𝐹)
36	35, 7, 4, 23	coe1f 21954	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐵 → (coe₁‘𝐹):ℕ₀⟶(Base‘𝑅))
37	2, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (coe₁‘𝐹):ℕ₀⟶(Base‘𝑅))
38		elfznn0 13598	. . . . . . 7 ⊢ (𝑦 ∈ (0...(𝐼 + 𝐽)) → 𝑦 ∈ ℕ₀)
39		ffvelcdm 7082	. . . . . . 7 ⊢ (((coe₁‘𝐹):ℕ₀⟶(Base‘𝑅) ∧ 𝑦 ∈ ℕ₀) → ((coe₁‘𝐹)‘𝑦) ∈ (Base‘𝑅))
40	37, 38, 39	syl2an 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((coe₁‘𝐹)‘𝑦) ∈ (Base‘𝑅))
41		eqid 2730	. . . . . . . . 9 ⊢ (coe₁‘𝐺) = (coe₁‘𝐺)
42	41, 7, 4, 23	coe1f 21954	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐵 → (coe₁‘𝐺):ℕ₀⟶(Base‘𝑅))
43	3, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → (coe₁‘𝐺):ℕ₀⟶(Base‘𝑅))
44		fznn0sub 13537	. . . . . . 7 ⊢ (𝑦 ∈ (0...(𝐼 + 𝐽)) → ((𝐼 + 𝐽) − 𝑦) ∈ ℕ₀)
45		ffvelcdm 7082	. . . . . . 7 ⊢ (((coe₁‘𝐺):ℕ₀⟶(Base‘𝑅) ∧ ((𝐼 + 𝐽) − 𝑦) ∈ ℕ₀) → ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)) ∈ (Base‘𝑅))
46	43, 44, 45	syl2an 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)) ∈ (Base‘𝑅))
47	23, 6	ringcl 20144	. . . . . 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 1369	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) ∈ (Base‘𝑅))
49	48	fmpttd 7115	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (0...(𝐼 + 𝐽)) ↦ (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)))):(0...(𝐼 + 𝐽))⟶(Base‘𝑅))
50		eldifsn 4789	. . . . . 6 ⊢ (𝑦 ∈ ((0...(𝐼 + 𝐽)) ∖ {𝐼}) ↔ (𝑦 ∈ (0...(𝐼 + 𝐽)) ∧ 𝑦 ≠ 𝐼))
51	38	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝑦 ∈ ℕ₀)
52	51	nn0red 12537	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝑦 ∈ ℝ)
53	28	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝐼 ∈ ℝ)
54	52, 53	lttri2d 11357	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (𝑦 ≠ 𝐼 ↔ (𝑦 < 𝐼 ∨ 𝐼 < 𝑦)))
55	3	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → 𝐺 ∈ 𝐵)
56	44	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((𝐼 + 𝐽) − 𝑦) ∈ ℕ₀)
57	56	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → ((𝐼 + 𝐽) − 𝑦) ∈ ℕ₀)
58		coe1mul3.d	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 = ( deg₁ ‘𝑅)
59	58, 4, 7	deg1xrcl 25835	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ^*)
60	3, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ^*)
61	60	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → (𝐷‘𝐺) ∈ ℝ^*)
62	12	nn0red 12537	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽 ∈ ℝ)
63	62	rexrd 11268	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ ℝ^*)
64	63	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → 𝐽 ∈ ℝ^*)
65	13	nn0red 12537	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐼 + 𝐽) ∈ ℝ)
66	65	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (𝐼 + 𝐽) ∈ ℝ)
67	66, 52	resubcld 11646	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((𝐼 + 𝐽) − 𝑦) ∈ ℝ)
68	67	rexrd 11268	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((𝐼 + 𝐽) − 𝑦) ∈ ℝ^*)
69	68	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → ((𝐼 + 𝐽) − 𝑦) ∈ ℝ^*)
70		coe1mul3.g3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐽)
71	70	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → (𝐷‘𝐺) ≤ 𝐽)
72	62	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝐽 ∈ ℝ)
73	52, 53, 72	ltadd1d 11811	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (𝑦 < 𝐼 ↔ (𝑦 + 𝐽) < (𝐼 + 𝐽)))
74	52, 72, 66	ltaddsub2d 11819	. . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → 𝐽 < ((𝐼 + 𝐽) − 𝑦))
77	61, 64, 69, 71, 76	xrlelttrd 13143	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → (𝐷‘𝐺) < ((𝐼 + 𝐽) − 𝑦))
78	58, 4, 7, 24, 41	deg1lt 25850	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)) = (0_g‘𝑅))
80	79	oveq2d 7427	. . . . . . . . . . 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 20182	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ ((coe₁‘𝐹)‘𝑦) ∈ (Base‘𝑅)) → (((coe₁‘𝐹)‘𝑦) · (0_g‘𝑅)) = (0_g‘𝑅))
82	34, 40, 81	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → (((coe₁‘𝐹)‘𝑦) · (0_g‘𝑅)) = (0_g‘𝑅))
83	82	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → (((coe₁‘𝐹)‘𝑦) · (0_g‘𝑅)) = (0_g‘𝑅))
84	80, 83	eqtrd 2770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝑦 < 𝐼) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
85	2	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → 𝐹 ∈ 𝐵)
86	51	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → 𝑦 ∈ ℕ₀)
87	58, 4, 7	deg1xrcl 25835	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ^*)
88	2, 87	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ^*)
89	88	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → (𝐷‘𝐹) ∈ ℝ^*)
90	28	rexrd 11268	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ ℝ^*)
91	90	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → 𝐼 ∈ ℝ^*)
92	52	rexrd 11268	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → 𝑦 ∈ ℝ^*)
93	92	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → 𝑦 ∈ ℝ^*)
94		coe1mul3.f3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐼)
95	94	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → (𝐷‘𝐹) ≤ 𝐼)
96		simpr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → 𝐼 < 𝑦)
97	89, 91, 93, 95, 96	xrlelttrd 13143	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → (𝐷‘𝐹) < 𝑦)
98	58, 4, 7, 24, 35	deg1lt 25850	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑦 ∈ ℕ₀ ∧ (𝐷‘𝐹) < 𝑦) → ((coe₁‘𝐹)‘𝑦) = (0_g‘𝑅))
99	85, 86, 97, 98	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → ((coe₁‘𝐹)‘𝑦) = (0_g‘𝑅))
100	99	oveq1d 7426	. . . . . . . . . . 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 20181	. . . . . . . . . . . . 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 582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) → ((0_g‘𝑅) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
103	102	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → ((0_g‘𝑅) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
104	100, 103	eqtrd 2770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ 𝐼 < 𝑦) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
105	84, 104	jaodan 954	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝐼 + 𝐽))) ∧ (𝑦 < 𝐼 ∨ 𝐼 < 𝑦)) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
106	105	ex 411	. . . . . . . 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 453	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (0...(𝐼 + 𝐽)) ∧ 𝑦 ≠ 𝐼)) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
109	50, 108	sylan2b 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ((0...(𝐼 + 𝐽)) ∖ {𝐼})) → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (0_g‘𝑅))
110	109, 27	suppss2 8187	. . . 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 19871	. . 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 6890	. . . . . 6 ⊢ (𝑦 = 𝐼 → ((coe₁‘𝐹)‘𝑦) = ((coe₁‘𝐹)‘𝐼))
113		oveq2 7419	. . . . . . 7 ⊢ (𝑦 = 𝐼 → ((𝐼 + 𝐽) − 𝑦) = ((𝐼 + 𝐽) − 𝐼))
114	113	fveq2d 6894	. . . . . 6 ⊢ (𝑦 = 𝐼 → ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦)) = ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼)))
115	112, 114	oveq12d 7429	. . . . 5 ⊢ (𝑦 = 𝐼 → (((coe₁‘𝐹)‘𝑦) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝑦))) = (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼))))
116		eqid 2730	. . . . 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 7444	. . . . 5 ⊢ (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼))) ∈ V
118	115, 116, 117	fvmpt 6997	. . . 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 12538	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℂ)
121	12	nn0cnd 12538	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℂ)
122	120, 121	pncan2d 11577	. . . . 5 ⊢ (𝜑 → ((𝐼 + 𝐽) − 𝐼) = 𝐽)
123	122	fveq2d 6894	. . . 4 ⊢ (𝜑 → ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼)) = ((coe₁‘𝐺)‘𝐽))
124	123	oveq2d 7427	. . 3 ⊢ (𝜑 → (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘((𝐼 + 𝐽) − 𝐼))) = (((coe₁‘𝐹)‘𝐼) · ((coe₁‘𝐺)‘𝐽)))
125	111, 119, 124	3eqtrd 2774	. 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 2774	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 394 ∨ wo 843 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 Vcvv 3472 ∖ cdif 3944 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ℝcr 11111 0cc0 11112 + caddc 11115 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 − cmin 11448 ℕ₀cn0 12476 ...cfz 13488 Basecbs 17148 ._rcmulr 17202 0_gc0g 17389 Σ_g cgsu 17390 Mndcmnd 18659 Ringcrg 20127 Poly₁cpl1 21920 coe₁cco1 21921 deg₁ cdg1 25804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 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 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 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 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-mulg 18987 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-cnfld 21145 df-psr 21681 df-mpl 21683 df-opsr 21685 df-psr1 21923 df-ply1 21925 df-coe1 21926 df-mdeg 25805 df-deg1 25806

This theorem is referenced by: coe1mul4 25853

W3C validator