Theorem cpmadugsumlemB 23192

Description: Lemma B for cpmadugsum 23196. (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 = (1r‘𝑌)

Assertion

Ref	Expression
cpmadugsumlemB	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σg (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 20472	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		cpmadugsum.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly1‘𝑅)
3	2	ply1ring 22565	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 18	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
5	4	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
6		eqid 2761	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
7	6	ringmgp 20465	. . . . . . . . . 10 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
8	5, 7	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd)
9	8	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (mulGrp‘𝑃) ∈ Mnd)
10		elfznn0 13754	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ0)
11	10	adantl 487	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑖 ∈ ℕ0)
12		1nn0 12622	. . . . . . . . 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 1152	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
15		cpmadugsum.x	. . . . . . . . . . 11 ⊢ 𝑋 = (var1‘𝑅)
16		eqid 2761	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
17	15, 2, 16	vr1cl 22535	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
18	14, 17	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃))
19	18	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑋 ∈ (Base‘𝑃))
20	6, 16	mgpbas 20365	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
21		cpmadugsum.e	. . . . . . . . 9 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
22		eqid 2761	. . . . . . . . . 10 ⊢ (.r‘𝑃) = (.r‘𝑃)
23	6, 22	mgpplusg 20364	. . . . . . . . 9 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘𝑃))
24	20, 21, 23	mulgnn0dir 19314	. . . . . . . 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 1399	. . . . . . 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 22516	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
27	26	anim2i 629	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
28	27	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
29		cpmadugsum.y	. . . . . . . . . . . 12 ⊢ 𝑌 = (𝑁 Mat 𝑃)
30	29	matsca2 22735	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
31	28, 30	syl 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 = (Scalar‘𝑌))
32	31	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑃 = (Scalar‘𝑌))
33	32	fveq2d 6889	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (.r‘𝑃) = (.r‘(Scalar‘𝑌)))
34		eqidd 2762	. . . . . . . 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 19291	. . . . . . . . . 10 ⊢ (𝑋 ∈ (Base‘𝑃) → (1 ↑ 𝑋) = 𝑋)
36	18, 35	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (1 ↑ 𝑋) = 𝑋)
37	36	ad2antrr 739	. . . . . . . 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 7441	. . . . . . 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 2796	. . . . . 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 629	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
41	40	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
42	29	matring 22758	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ Ring)
43	41, 42	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
44	43	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ Ring)
45		simpll1 1231	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin)
46	14	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑅 ∈ Ring)
47		simplrl 789	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑠 ∈ ℕ0)
48		simprr 785	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑m (0...𝑠)))
49	48	anim1i 627	. . . . . . . . 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 23065	. . . . . . . . 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 1400	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
55		eqid 2761	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
56		cpmadugsum.r	. . . . . . . . 9 ⊢ × = (.r‘𝑌)
57		cpmadugsum.1	. . . . . . . . 9 ⊢ 1 = (1r‘𝑌)
58	55, 56, 57	ringlidm 20498	. . . . . . . 8 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) → ( 1 × (𝑇‘(𝑏‘𝑖))) = (𝑇‘(𝑏‘𝑖)))
59	44, 54, 58	syl2anc 596	. . . . . . 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 2767	. . . . . 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 7438	. . . . 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 22759	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑌 ∈ AssAlg)
63	27, 62	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ AssAlg)
64	63	3adant3 1150	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ AssAlg)
65	64	ad2antrr 739	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ AssAlg)
66	31	eqcomd 2767	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑌) = 𝑃)
67	66	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
68	18, 67	eleqtrrd 2864	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
69	68	ad2antrr 739	. . . . . . 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 19305	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
71	67	ad2antrr 739	. . . . . . . 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 2864	. . . . . . 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 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
74	73	3adant3 1150	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
75	55, 57	ringidcl 20494	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 1 ∈ (Base‘𝑌))
76	74, 75	syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ (Base‘𝑌))
77	76	ad2antrr 739	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 1 ∈ (Base‘𝑌))
78		eqid 2761	. . . . . . . 8 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
79		eqid 2761	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
80		eqid 2761	. . . . . . . 8 ⊢ (.r‘(Scalar‘𝑌)) = (.r‘(Scalar‘𝑌))
81		cpmadugsum.m	. . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑌)
82	55, 78, 79, 80, 81, 56	assa2ass 22171	. . . . . . 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 1406	. . . . . 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 2767	. . . . 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 2796	. . . 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 5198	. . 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 7436	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))))
88		eqid 2761	. . 3 ⊢ (0g‘𝑌) = (0g‘𝑌)
89	74	adantr 486	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Ring)
90		ovexd 7455	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (0...𝑠) ∈ V)
91	29	matlmod 22744	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ LMod)
92	40, 91	syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
93	92	3adant3 1150	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ LMod)
94	1	adantl 487	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
95	94, 17	syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘𝑃))
96	27, 30	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
97	96	eqcomd 2767	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑌) = 𝑃)
98	97	fveq2d 6889	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
99	95, 98	eleqtrrd 2864	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
100	99	3adant3 1150	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
101	43, 75	syl 18	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ (Base‘𝑌))
102	55, 78, 81, 79	lmodvscl 21153	. . . . 5 ⊢ ((𝑌 ∈ LMod ∧ 𝑋 ∈ (Base‘(Scalar‘𝑌)) ∧ 1 ∈ (Base‘𝑌)) → (𝑋 · 1 ) ∈ (Base‘𝑌))
103	93, 100, 101, 102	syl3anc 1398	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑋 · 1 ) ∈ (Base‘𝑌))
104	103	adantr 486	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑋 · 1 ) ∈ (Base‘𝑌))
105	93	ad2antrr 739	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ LMod)
106	30	eqcomd 2767	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → (Scalar‘𝑌) = 𝑃)
107	106	fveq2d 6889	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
108	28, 107	syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
109	108	eleq2d 2847	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)))
110	109	ad2antrr 739	. . . . 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 260	. . . 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 21153	. . . 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 1398	. . 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 1210	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑁 ∈ Fin)
115	14	adantr 486	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑅 ∈ Ring)
116		simprl 783	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑠 ∈ ℕ0)
117		eqid 2761	. . . . 5 ⊢ (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))
118		fzfid 14116	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0...𝑠) ∈ Fin)
119		ovexd 7455	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ V)
120		fvexd 6900	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0g‘𝑌) ∈ V)
121	117, 118, 119, 120	fsuppmptdm 9368	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) finSupp (0g‘𝑌))
122	114, 115, 116, 48, 121	syl31anc 1400	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) finSupp (0g‘𝑌))
123	55, 88, 56, 89, 90, 104, 113, 122	gsummulc2 20546	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑋 · 1 ) × ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = ((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))))
124	87, 123	eqtr2d 2797	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  Fincfn 8973   finSupp cfsupp 9353  0cc0 11200  1c1 11201   + caddc 11203  ℕ₀cn0 12606  ...cfz 13639  Basecbs 17387  ._rcmulr 17429  Scalarcsca 17431   ·_𝑠 cvsca 17432  0_gc0g 17610   Σ_g cgsu 17611  Mndcmnd 18923  ._gcmg 19277  mulGrpcmgp 20360  1_rcur 20407  Ringcrg 20459  CRingccrg 20460  LModclmod 21135  AssAlgcasa 22158  var₁cv1 22494  Poly₁cpl1 22495   Mat cmat 22722   matToPolyMat cmat2pmat 23022

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-ofr 7694  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-sup 9434  df-oi 9504  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-fzo 13789  df-seq 14145  df-hash 14475  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-hom 17452  df-cco 17453  df-0g 17612  df-gsum 17613  df-prds 17618  df-pws 17620  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-mulg 19278  df-subg 19333  df-ghm 19428  df-cntz 19531  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-cring 20462  df-subrng 20798  df-subrg 20822  df-lmod 21137  df-lss 21207  df-sra 21448  df-rgmod 21449  df-dsmm 22038  df-frlm 22053  df-assa 22161  df-ascl 22163  df-psr 22217  df-mvr 22218  df-mpl 22219  df-opsr 22221  df-psr1 22498  df-vr1 22499  df-ply1 22500  df-mamu 22706  df-mat 22723  df-mat2pmat 23025

This theorem is used by:  cpmadugsumlemF  23194