Theorem cpmadugsumlemB 22852

Description: Lemma B for cpmadugsum 22856. (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 20220	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		cpmadugsum.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly1‘𝑅)
3	2	ply1ring 22224	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
5	4	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
6		eqid 2737	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
7	6	ringmgp 20214	. . . . . . . . . 10 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
8	5, 7	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑃) ∈ Mnd)
9	8	ad2antrr 727	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (mulGrp‘𝑃) ∈ Mnd)
10		elfznn0 13568	. . . . . . . . 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 12447	. . . . . . . . 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 1135	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
15		cpmadugsum.x	. . . . . . . . . . 11 ⊢ 𝑋 = (var1‘𝑅)
16		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
17	15, 2, 16	vr1cl 22194	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
18	14, 17	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑃))
19	18	ad2antrr 727	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑋 ∈ (Base‘𝑃))
20	6, 16	mgpbas 20120	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
21		cpmadugsum.e	. . . . . . . . 9 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
22		eqid 2737	. . . . . . . . . 10 ⊢ (.r‘𝑃) = (.r‘𝑃)
23	6, 22	mgpplusg 20119	. . . . . . . . 9 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘𝑃))
24	20, 21, 23	mulgnn0dir 19074	. . . . . . . 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 1375	. . . . . . 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 22175	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
27	26	anim2i 618	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
28	27	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
29		cpmadugsum.y	. . . . . . . . . . . 12 ⊢ 𝑌 = (𝑁 Mat 𝑃)
30	29	matsca2 22398	. . . . . . . . . . 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 727	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑃 = (Scalar‘𝑌))
33	32	fveq2d 6839	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (.r‘𝑃) = (.r‘(Scalar‘𝑌)))
34		eqidd 2738	. . . . . . . 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 19051	. . . . . . . . . 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 727	. . . . . . . 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 7382	. . . . . . 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 2772	. . . . . 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 618	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
41	40	3adant3 1133	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
42	29	matring 22421	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ Ring)
43	41, 42	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
44	43	ad2antrr 727	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ Ring)
45		simpll1 1214	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin)
46	14	ad2antrr 727	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑅 ∈ Ring)
47		simplrl 777	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑠 ∈ ℕ0)
48		simprr 773	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑m (0...𝑠)))
49	48	anim1i 616	. . . . . . . . 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 22725	. . . . . . . . 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 1376	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
55		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
56		cpmadugsum.r	. . . . . . . . 9 ⊢ × = (.r‘𝑌)
57		cpmadugsum.1	. . . . . . . . 9 ⊢ 1 = (1r‘𝑌)
58	55, 56, 57	ringlidm 20244	. . . . . . . 8 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) → ( 1 × (𝑇‘(𝑏‘𝑖))) = (𝑇‘(𝑏‘𝑖)))
59	44, 54, 58	syl2anc 585	. . . . . . 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 2743	. . . . . 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 7379	. . . . 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 22422	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑌 ∈ AssAlg)
63	27, 62	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ AssAlg)
64	63	3adant3 1133	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ AssAlg)
65	64	ad2antrr 727	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ AssAlg)
66	31	eqcomd 2743	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑌) = 𝑃)
67	66	fveq2d 6839	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
68	18, 67	eleqtrrd 2840	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
69	68	ad2antrr 727	. . . . . . 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 19065	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 ↑ 𝑋) ∈ (Base‘𝑃))
71	67	ad2antrr 727	. . . . . . . 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 2840	. . . . . . 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 1133	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
75	55, 57	ringidcl 20240	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 1 ∈ (Base‘𝑌))
76	74, 75	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ (Base‘𝑌))
77	76	ad2antrr 727	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 1 ∈ (Base‘𝑌))
78		eqid 2737	. . . . . . . 8 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
79		eqid 2737	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
80		eqid 2737	. . . . . . . 8 ⊢ (.r‘(Scalar‘𝑌)) = (.r‘(Scalar‘𝑌))
81		cpmadugsum.m	. . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑌)
82	55, 78, 79, 80, 81, 56	assa2ass 21856	. . . . . . 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 1382	. . . . . 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 2743	. . . . 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 2772	. . . 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 5179	. . 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 7377	. 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 2737	. . 3 ⊢ (0g‘𝑌) = (0g‘𝑌)
89	74	adantr 480	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑌 ∈ Ring)
90		ovexd 7396	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (0...𝑠) ∈ V)
91	29	matlmod 22407	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ LMod)
92	40, 91	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
93	92	3adant3 1133	. . . . 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 2743	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑌) = 𝑃)
98	97	fveq2d 6839	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
99	95, 98	eleqtrrd 2840	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
100	99	3adant3 1133	. . . . 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 20867	. . . . 5 ⊢ ((𝑌 ∈ LMod ∧ 𝑋 ∈ (Base‘(Scalar‘𝑌)) ∧ 1 ∈ (Base‘𝑌)) → (𝑋 · 1 ) ∈ (Base‘𝑌))
103	93, 100, 101, 102	syl3anc 1374	. . . 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 727	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ LMod)
106	30	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → (Scalar‘𝑌) = 𝑃)
107	106	fveq2d 6839	. . . . . . . 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 2823	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑖 ↑ 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 ↑ 𝑋) ∈ (Base‘𝑃)))
110	109	ad2antrr 727	. . . . 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 20867	. . . 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 1374	. . 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 1193	. . . 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 771	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑠 ∈ ℕ0)
117		eqid 2737	. . . . 5 ⊢ (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))
118		fzfid 13929	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0...𝑠) ∈ Fin)
119		ovexd 7396	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))) ∈ V)
120		fvexd 6850	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0g‘𝑌) ∈ V)
121	117, 118, 119, 120	fsuppmptdm 9283	. . . 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 1376	. . 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 20290	. 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 2773	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))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430   class class class wbr 5086   ↦ cmpt 5167  ‘cfv 6493  (class class class)co 7361   ↑_m cmap 8767  Fincfn 8887   finSupp cfsupp 9268  0cc0 11032  1c1 11033   + caddc 11035  ℕ₀cn0 12431  ...cfz 13455  Basecbs 17173  ._rcmulr 17215  Scalarcsca 17217   ·_𝑠 cvsca 17218  0_gc0g 17396   Σ_g cgsu 17397  Mndcmnd 18696  ._gcmg 19037  mulGrpcmgp 20115  1_rcur 20156  Ringcrg 20208  CRingccrg 20209  LModclmod 20849  AssAlgcasa 21843  var₁cv1 22152  Poly₁cpl1 22153   Mat cmat 22385   matToPolyMat cmat2pmat 22682

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7625  df-ofr 7626  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-sup 9349  df-oi 9419  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-fz 13456  df-fzo 13603  df-seq 13958  df-hash 14287  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-hom 17238  df-cco 17239  df-0g 17398  df-gsum 17399  df-prds 17404  df-pws 17406  df-mre 17542  df-mrc 17543  df-acs 17545  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mulg 19038  df-subg 19093  df-ghm 19182  df-cntz 19286  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-cring 20211  df-subrng 20517  df-subrg 20541  df-lmod 20851  df-lss 20921  df-sra 21163  df-rgmod 21164  df-dsmm 21725  df-frlm 21740  df-assa 21846  df-ascl 21848  df-psr 21902  df-mvr 21903  df-mpl 21904  df-opsr 21906  df-psr1 22156  df-vr1 22157  df-ply1 22158  df-mamu 22369  df-mat 22386  df-mat2pmat 22685

This theorem is referenced by:  cpmadugsumlemF  22854