Theorem cpmadugsumlemB 23247

Description: Lemma B for cpmadugsum 23251. (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 20523	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		cpmadugsum.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly1‘𝑅)
3	2	ply1ring 22620	. . . . . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
7	6	ringmgp 20516	. . . . . . . . . 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 13794	. . . . . . . . 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 12662	. . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
17	15, 2, 16	vr1cl 22590	. . . . . . . . . 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 20416	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
21		cpmadugsum.e	. . . . . . . . 9 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
22		eqid 2763	. . . . . . . . . 10 ⊢ (.r‘𝑃) = (.r‘𝑃)
23	6, 22	mgpplusg 20415	. . . . . . . . 9 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘𝑃))
24	20, 21, 23	mulgnn0dir 19364	. . . . . . . 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 1400	. . . . . . 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 22571	. . . . . . . . . . . . 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 22790	. . . . . . . . . . 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 6920	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (.r‘𝑃) = (.r‘(Scalar‘𝑌)))
34		eqidd 2764	. . . . . . . 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 19341	. . . . . . . . . 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 7472	. . . . . . 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 2798	. . . . . 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 22813	. . . . . . . . . 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 23120	. . . . . . . . 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 1401	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
55		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
56		cpmadugsum.r	. . . . . . . . 9 ⊢ × = (.r‘𝑌)
57		cpmadugsum.1	. . . . . . . . 9 ⊢ 1 = (1r‘𝑌)
58	55, 56, 57	ringlidm 20549	. . . . . . . 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 2769	. . . . . 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 7469	. . . . 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 22814	. . . . . . . . . 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 2769	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑌) = 𝑃)
67	66	fveq2d 6920	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
68	18, 67	eleqtrrd 2867	. . . . . . . 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 19355	. . . . . . . 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 2867	. . . . . . 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 20545	. . . . . . . . 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 2763	. . . . . . . 8 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
79		eqid 2763	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
80		eqid 2763	. . . . . . . 8 ⊢ (.r‘(Scalar‘𝑌)) = (.r‘(Scalar‘𝑌))
81		cpmadugsum.m	. . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑌)
82	55, 78, 79, 80, 81, 56	assa2ass 22226	. . . . . . 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 1407	. . . . . 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 2769	. . . . 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 2798	. . . 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 5204	. . 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 7467	. 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 2763	. . 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 7486	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (0...𝑠) ∈ V)
91	29	matlmod 22799	. . . . . . 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 2769	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑌) = 𝑃)
98	97	fveq2d 6920	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
99	95, 98	eleqtrrd 2867	. . . . . 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 21204	. . . . 5 ⊢ ((𝑌 ∈ LMod ∧ 𝑋 ∈ (Base‘(Scalar‘𝑌)) ∧ 1 ∈ (Base‘𝑌)) → (𝑋 · 1 ) ∈ (Base‘𝑌))
103	93, 100, 101, 102	syl3anc 1399	. . . 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 2769	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → (Scalar‘𝑌) = 𝑃)
107	106	fveq2d 6920	. . . . . . . 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 2850	. . . . . 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 21204	. . . 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 1399	. . 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 2763	. . . . 5 ⊢ (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))
118		fzfid 14156	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0...𝑠) ∈ Fin)
119		ovexd 7486	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ V)
120		fvexd 6931	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0g‘𝑌) ∈ V)
121	117, 118, 119, 120	fsuppmptdm 9405	. . . 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 1401	. . 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 20597	. 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 2799	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 2147  Vcvv 3454   class class class wbr 5108   ↦ cmpt 5192  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885  Fincfn 9011   finSupp cfsupp 9391  0cc0 11239  1c1 11240   + caddc 11242  ℕ₀cn0 12646  ...cfz 13679  Basecbs 17431  ._rcmulr 17473  Scalarcsca 17475   ·_𝑠 cvsca 17476  0_gc0g 17660   Σ_g cgsu 17661  Mndcmnd 18973  ._gcmg 19327  mulGrpcmgp 20411  1_rcur 20458  Ringcrg 20510  CRingccrg 20511  LModclmod 21186  AssAlgcasa 22213  var₁cv1 22549  Poly₁cpl1 22550   Mat cmat 22777   matToPolyMat cmat2pmat 23077

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-ofr 7726  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-pm 8888  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-sup 9471  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-fzo 13829  df-seq 14185  df-hash 14516  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-mulr 17487  df-sca 17489  df-vsca 17490  df-ip 17491  df-tset 17492  df-ple 17493  df-ds 17495  df-hom 17497  df-cco 17498  df-0g 17662  df-gsum 17663  df-prds 17668  df-pws 17670  df-mre 17806  df-mrc 17807  df-acs 17809  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-mhm 19028  df-submnd 19029  df-grp 19197  df-minusg 19198  df-sbg 19199  df-mulg 19328  df-subg 19383  df-ghm 19478  df-cntz 19582  df-cmn 20047  df-abl 20048  df-mgp 20412  df-rng 20426  df-ur 20459  df-ring 20512  df-cring 20513  df-subrng 20849  df-subrg 20873  df-lmod 21188  df-lss 21258  df-sra 21499  df-rgmod 21500  df-dsmm 22093  df-frlm 22108  df-assa 22216  df-ascl 22218  df-psr 22272  df-mvr 22273  df-mpl 22274  df-opsr 22276  df-psr1 22553  df-vr1 22554  df-ply1 22555  df-mamu 22761  df-mat 22778  df-mat2pmat 23080

This theorem is used by:  cpmadugsumlemF  23249