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Theorem cpmadugsumlemB 22763

Description: Lemma B for cpmadugsum 22767. (Contributed by AV, 2-Nov-2019.)

**Hypotheses**
Ref	Expression
cpmadugsum.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cpmadugsum.b	⊢ 𝐵 = (Base‘𝐴)
cpmadugsum.p	⊢ 𝑃 = (Poly₁‘𝑅)
cpmadugsum.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cpmadugsum.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cpmadugsum.x	⊢ 𝑋 = (var₁‘𝑅)
cpmadugsum.e	⊢ ↑ = (._g‘(mulGrp‘𝑃))
cpmadugsum.m	⊢ · = ( ·_𝑠 ‘𝑌)
cpmadugsum.r	⊢ × = (._r‘𝑌)
cpmadugsum.1	⊢ 1 = (1_r‘𝑌)

**Assertion**
Ref	Expression
cpmadugsumlemB	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))

Distinct variable groups: 𝐵,𝑖 𝑖,𝑀 𝑖,𝑁 𝑅,𝑖 𝑖,𝑋 𝑖,𝑌 × ,𝑖 · ,𝑖 1 ,𝑖 𝑖,𝑏 𝑖,𝑠 Allowed substitution hints: 𝐴(𝑖,𝑠,𝑏) 𝐵(𝑠,𝑏) 𝑃(𝑖,𝑠,𝑏) 𝑅(𝑠,𝑏) 𝑇(𝑖,𝑠,𝑏) · (𝑠,𝑏) × (𝑠,𝑏) 1 (𝑠,𝑏) ↑ (𝑖,𝑠,𝑏) 𝑀(𝑠,𝑏) 𝑁(𝑠,𝑏) 𝑋(𝑠,𝑏) 𝑌(𝑠,𝑏)

**Proof of Theorem cpmadugsumlemB**
Step	Hyp	Ref	Expression
1		crngring 20176	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		cpmadugsum.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 22153	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
5	4	3ad2ant2 1132	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
6		eqid 2727	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
7	6	ringmgp 20170	. . . . . . . . . 10 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
8	5, 7	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd)
9	8	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (mulGrp‘𝑃) ∈ Mnd)
10		elfznn0 13618	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ₀)
11	10	adantl 481	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑖 ∈ ℕ₀)
12		1nn0 12510	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
13	12	a1i 11	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 1 ∈ ℕ₀)
14	1	3ad2ant2 1132	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
15		cpmadugsum.x	. . . . . . . . . . 11 ⊢ 𝑋 = (var₁‘𝑅)
16		eqid 2727	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
17	15, 2, 16	vr1cl 22123	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
18	14, 17	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃))
19	18	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑋 ∈ (Base‘𝑃))
20	6, 16	mgpbas 20071	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
21		cpmadugsum.e	. . . . . . . . 9 ⊢ ↑ = (._g‘(mulGrp‘𝑃))
22		eqid 2727	. . . . . . . . . 10 ⊢ (._r‘𝑃) = (._r‘𝑃)
23	6, 22	mgpplusg 20069	. . . . . . . . 9 ⊢ (._r‘𝑃) = (+_g‘(mulGrp‘𝑃))
24	20, 21, 23	mulgnn0dir 19050	. . . . . . . 8 ⊢ (((mulGrp‘𝑃) ∈ Mnd ∧ (𝑖 ∈ ℕ₀ ∧ 1 ∈ ℕ₀ ∧ 𝑋 ∈ (Base‘𝑃))) → ((𝑖 + 1) ↑ 𝑋) = ((𝑖 ↑ 𝑋)(._r‘𝑃)(1 ↑ 𝑋)))
25	9, 11, 13, 19, 24	syl13anc 1370	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 + 1) ↑ 𝑋) = ((𝑖 ↑ 𝑋)(._r‘𝑃)(1 ↑ 𝑋)))
26	2	ply1crng 22104	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
27	26	anim2i 616	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
28	27	3adant3 1130	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
29		cpmadugsum.y	. . . . . . . . . . . 12 ⊢ 𝑌 = (𝑁 Mat 𝑃)
30	29	matsca2 22309	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
31	28, 30	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝑌))
32	31	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑃 = (Scalar‘𝑌))
33	32	fveq2d 6895	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (._r‘𝑃) = (._r‘(Scalar‘𝑌)))
34		eqidd 2728	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) = (𝑖 ↑ 𝑋))
35	20, 21	mulg1 19027	. . . . . . . . . 10 ⊢ (𝑋 ∈ (Base‘𝑃) → (1 ↑ 𝑋) = 𝑋)
36	18, 35	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (1 ↑ 𝑋) = 𝑋)
37	36	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (1 ↑ 𝑋) = 𝑋)
38	33, 34, 37	oveq123d 7435	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋)(._r‘𝑃)(1 ↑ 𝑋)) = ((𝑖 ↑ 𝑋)(._r‘(Scalar‘𝑌))𝑋))
39	25, 38	eqtrd 2767	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 + 1) ↑ 𝑋) = ((𝑖 ↑ 𝑋)(._r‘(Scalar‘𝑌))𝑋))
40	4	anim2i 616	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
41	40	3adant3 1130	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
42	29	matring 22332	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ Ring)
43	41, 42	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
44	43	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ Ring)
45		simpll1 1210	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin)
46	14	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑅 ∈ Ring)
47		simplrl 776	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑠 ∈ ℕ₀)
48		simprr 772	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))
49	48	anim1i 614	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠)))
50		cpmadugsum.a	. . . . . . . . . 10 ⊢ 𝐴 = (𝑁 Mat 𝑅)
51		cpmadugsum.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐴)
52		cpmadugsum.t	. . . . . . . . . 10 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
53	50, 51, 2, 29, 52	m2pmfzmap 22636	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ₀) ∧ (𝑏 ∈ (𝐵 ↑_m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
54	45, 46, 47, 49, 53	syl31anc 1371	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
55		eqid 2727	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
56		cpmadugsum.r	. . . . . . . . 9 ⊢ × = (._r‘𝑌)
57		cpmadugsum.1	. . . . . . . . 9 ⊢ 1 = (1_r‘𝑌)
58	55, 56, 57	ringlidm 20194	. . . . . . . 8 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) → ( 1 × (𝑇‘(𝑏‘𝑖))) = (𝑇‘(𝑏‘𝑖)))
59	44, 54, 58	syl2anc 583	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ( 1 × (𝑇‘(𝑏‘𝑖))) = (𝑇‘(𝑏‘𝑖)))
60	59	eqcomd 2733	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏‘𝑖)) = ( 1 × (𝑇‘(𝑏‘𝑖))))
61	39, 60	oveq12d 7432	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) = (((𝑖 ↑ 𝑋)(._r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏‘𝑖)))))
62	29	matassa 22333	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑌 ∈ AssAlg)
63	27, 62	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ AssAlg)
64	63	3adant3 1130	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ AssAlg)
65	64	ad2antrr 725	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ AssAlg)
66	31	eqcomd 2733	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑌) = 𝑃)
67	66	fveq2d 6895	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
68	18, 67	eleqtrrd 2831	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
69	68	ad2antrr 725	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
70	20, 21, 9, 11, 19	mulgnn0cld 19041	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
71	67	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
72	70, 71	eleqtrrd 2831	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)))
73	40, 42	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
74	73	3adant3 1130	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
75	55, 57	ringidcl 20191	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 1 ∈ (Base‘𝑌))
76	74, 75	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ (Base‘𝑌))
77	76	ad2antrr 725	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 1 ∈ (Base‘𝑌))
78		eqid 2727	. . . . . . . 8 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
79		eqid 2727	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
80		eqid 2727	. . . . . . . 8 ⊢ (._r‘(Scalar‘𝑌)) = (._r‘(Scalar‘𝑌))
81		cpmadugsum.m	. . . . . . . 8 ⊢ · = ( ·_𝑠 ‘𝑌)
82	55, 78, 79, 80, 81, 56	assa2ass 21784	. . . . . . 7 ⊢ ((𝑌 ∈ AssAlg ∧ (𝑋 ∈ (Base‘(Scalar‘𝑌)) ∧ (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌))) ∧ ( 1 ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))) → ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (((𝑖 ↑ 𝑋)(._r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏‘𝑖)))))
83	65, 69, 72, 77, 54, 82	syl122anc 1377	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (((𝑖 ↑ 𝑋)(._r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏‘𝑖)))))
84	83	eqcomd 2733	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (((𝑖 ↑ 𝑋)(._r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏‘𝑖)))) = ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))
85	61, 84	eqtrd 2767	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) = ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))
86	85	mpteq2dva 5242	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))
87	86	oveq2d 7430	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) = (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))))
88		eqid 2727	. . 3 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
89	74	adantr 480	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑌 ∈ Ring)
90		ovexd 7449	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (0...𝑠) ∈ V)
91	29	matlmod 22318	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ LMod)
92	40, 91	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
93	92	3adant3 1130	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod)
94	1	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
95	94, 17	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘𝑃))
96	27, 30	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
97	96	eqcomd 2733	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑌) = 𝑃)
98	97	fveq2d 6895	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
99	95, 98	eleqtrrd 2831	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
100	99	3adant3 1130	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
101	43, 75	syl 17	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ (Base‘𝑌))
102	55, 78, 81, 79	lmodvscl 20750	. . . . 5 ⊢ ((𝑌 ∈ LMod ∧ 𝑋 ∈ (Base‘(Scalar‘𝑌)) ∧ 1 ∈ (Base‘𝑌)) → (𝑋 · 1 ) ∈ (Base‘𝑌))
103	93, 100, 101, 102	syl3anc 1369	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑋 · 1 ) ∈ (Base‘𝑌))
104	103	adantr 480	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑋 · 1 ) ∈ (Base‘𝑌))
105	93	ad2antrr 725	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ LMod)
106	30	eqcomd 2733	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → (Scalar‘𝑌) = 𝑃)
107	106	fveq2d 6895	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
108	28, 107	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
109	108	eleq2d 2814	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)))
110	109	ad2antrr 725	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)))
111	70, 110	mpbird 257	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)))
112	55, 78, 81, 79	lmodvscl 20750	. . . 4 ⊢ ((𝑌 ∈ LMod ∧ (𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
113	105, 111, 54, 112	syl3anc 1369	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
114		simpl1 1189	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑁 ∈ Fin)
115	14	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑅 ∈ Ring)
116		simprl 770	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑠 ∈ ℕ₀)
117		eqid 2727	. . . . 5 ⊢ (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))
118		fzfid 13962	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ₀) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (0...𝑠) ∈ Fin)
119		ovexd 7449	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ₀) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ V)
120		fvexd 6906	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ₀) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (0_g‘𝑌) ∈ V)
121	117, 118, 119, 120	fsuppmptdm 9391	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ₀) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) finSupp (0_g‘𝑌))
122	114, 115, 116, 48, 121	syl31anc 1371	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) finSupp (0_g‘𝑌))
123	55, 88, 56, 89, 90, 104, 113, 122	gsummulc2 20242	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = ((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))))
124	87, 123	eqtr2d 2768	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3469 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 ↑_m cmap 8836 Fincfn 8955 finSupp cfsupp 9377 0cc0 11130 1c1 11131 + caddc 11133 ℕ₀cn0 12494 ...cfz 13508 Basecbs 17171 ._rcmulr 17225 Scalarcsca 17227 ·_𝑠 cvsca 17228 0_gc0g 17412 Σ_g cgsu 17413 Mndcmnd 18685 ._gcmg 19014 mulGrpcmgp 20065 1_rcur 20112 Ringcrg 20164 CRingccrg 20165 LModclmod 20732 AssAlgcasa 21771 var₁cv1 22082 Poly₁cpl1 22083 Mat cmat 22294 matToPolyMat cmat2pmat 22593

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-fzo 13652 df-seq 13991 df-hash 14314 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-subrng 20472 df-subrg 20497 df-lmod 20734 df-lss 20805 df-sra 21047 df-rgmod 21048 df-dsmm 21653 df-frlm 21668 df-assa 21774 df-ascl 21776 df-psr 21829 df-mvr 21830 df-mpl 21831 df-opsr 21833 df-psr1 22086 df-vr1 22087 df-ply1 22088 df-mamu 22273 df-mat 22295 df-mat2pmat 22596

This theorem is referenced by: cpmadugsumlemF 22765

W3C validator