Theorem mat2pmatlin 22853

Description: The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 21125. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 21112, see lmhmsca 21120. (Contributed by AV, 13-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.)

Hypotheses

Ref	Expression
mat2pmatbas.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
mat2pmatbas.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mat2pmatbas.b	⊢ 𝐵 = (Base‘𝐴)
mat2pmatbas.p	⊢ 𝑃 = (Poly1‘𝑅)
mat2pmatbas.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
mat2pmatbas0.h	⊢ 𝐻 = (Base‘𝐶)
mat2pmatlin.k	⊢ 𝐾 = (Base‘𝑅)
mat2pmatlin.s	⊢ 𝑆 = (algSc‘𝑃)
mat2pmatlin.m	⊢ · = ( ·𝑠 ‘𝐴)
mat2pmatlin.n	⊢ × = ( ·𝑠 ‘𝐶)

Assertion

Ref	Expression
mat2pmatlin	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑇‘(𝑋 · 𝑌)) = ((𝑆‘𝑋) × (𝑇‘𝑌)))

Proof of Theorem mat2pmatlin

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 489	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ CRing)
2		mat2pmatbas.p	. . . . . . . . . . 11 ⊢ 𝑃 = (Poly1‘𝑅)
3	2	ply1assa 22319	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
4		mat2pmatlin.s	. . . . . . . . . . 11 ⊢ 𝑆 = (algSc‘𝑃)
5		eqid 2765	. . . . . . . . . . 11 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
6	4, 5	asclrhm 22000	. . . . . . . . . 10 ⊢ (𝑃 ∈ AssAlg → 𝑆 ∈ ((Scalar‘𝑃) RingHom 𝑃))
7	1, 3, 6	3syl 19	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ ((Scalar‘𝑃) RingHom 𝑃))
8	2	ply1sca 22372	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
9	8	adantl 486	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
10	9	oveq1d 7415	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 RingHom 𝑃) = ((Scalar‘𝑃) RingHom 𝑃))
11	7, 10	eleqtrrd 2868	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ (𝑅 RingHom 𝑃))
12	11	adantr 485	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑆 ∈ (𝑅 RingHom 𝑃))
13	12	adantr 485	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑆 ∈ (𝑅 RingHom 𝑃))
14		mat2pmatlin.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑅)
15	14	eleq2i 2857	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐾 ↔ 𝑋 ∈ (Base‘𝑅))
16	15	birani 508	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑅))
17	16	ad2antlr 739	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑋 ∈ (Base‘𝑅))
18		mat2pmatbas.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
19		eqid 2765	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
20		mat2pmatbas.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
21		simprl 782	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
22		simpr 489	. . . . . . . 8 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
23	22	adantl 486	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
24		simplrr 789	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑌 ∈ 𝐵)
25	18, 19, 20, 21, 23, 24	matecld 22544	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑌𝑗) ∈ (Base‘𝑅))
26		eqid 2765	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
27		eqid 2765	. . . . . . 7 ⊢ (.r‘𝑃) = (.r‘𝑃)
28	19, 26, 27	rhmmul 20559	. . . . . 6 ⊢ ((𝑆 ∈ (𝑅 RingHom 𝑃) ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑖𝑌𝑗) ∈ (Base‘𝑅)) → (𝑆‘(𝑋(.r‘𝑅)(𝑖𝑌𝑗))) = ((𝑆‘𝑋)(.r‘𝑃)(𝑆‘(𝑖𝑌𝑗))))
29	13, 17, 25, 28	syl3anc 1394	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑆‘(𝑋(.r‘𝑅)(𝑖𝑌𝑗))) = ((𝑆‘𝑋)(.r‘𝑃)(𝑆‘(𝑖𝑌𝑗))))
30		crngring 20318	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
31	30	ad2antlr 739	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑅 ∈ Ring)
32	31	adantr 485	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
33		simpr 489	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵))
34	33	adantr 485	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵))
35		simpr 489	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
36		mat2pmatlin.m	. . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝐴)
37	18, 20, 14, 36, 26	matvscacell 22554	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑋 · 𝑌)𝑗) = (𝑋(.r‘𝑅)(𝑖𝑌𝑗)))
38	32, 34, 35, 37	syl3anc 1394	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑋 · 𝑌)𝑗) = (𝑋(.r‘𝑅)(𝑖𝑌𝑗)))
39	38	fveq2d 6875	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)) = (𝑆‘(𝑋(.r‘𝑅)(𝑖𝑌𝑗))))
40	30	anim2i 628	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
41		simpr 489	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
42	40, 41	anim12i 624	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐵))
43		df-3an 1103	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐵))
44	42, 43	sylibr 237	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵))
45		mat2pmatbas.t	. . . . . . . 8 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
46	45, 18, 20, 2, 4	mat2pmatvalel 22843	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘𝑌)𝑗) = (𝑆‘(𝑖𝑌𝑗)))
47	44, 46	sylan 591	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘𝑌)𝑗) = (𝑆‘(𝑖𝑌𝑗)))
48	47	oveq2d 7416	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑆‘𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑌)𝑗)) = ((𝑆‘𝑋)(.r‘𝑃)(𝑆‘(𝑖𝑌𝑗))))
49	29, 39, 48	3eqtr4d 2810	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)) = ((𝑆‘𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑌)𝑗)))
50		simpll 778	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑁 ∈ Fin)
51	50	adantr 485	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
52	14, 18, 20, 36	matvscl 22549	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑋 · 𝑌) ∈ 𝐵)
53	40, 52	sylan 591	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑋 · 𝑌) ∈ 𝐵)
54	53	adantr 485	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑋 · 𝑌) ∈ 𝐵)
55	45, 18, 20, 2, 4	mat2pmatvalel 22843	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)))
56	51, 32, 54, 35, 55	syl31anc 1396	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)))
57	2	ply1ring 22367	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
58	30, 57	syl 18	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
59	58	ad2antlr 739	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑃 ∈ Ring)
60	59	adantr 485	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑃 ∈ Ring)
61	30	adantl 486	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
62		simpl 487	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐾)
63		eqid 2765	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑃)
64	2, 4, 14, 63	ply1sclcl 22407	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝑆‘𝑋) ∈ (Base‘𝑃))
65	61, 62, 64	syl2an 607	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑆‘𝑋) ∈ (Base‘𝑃))
66		mat2pmatbas.c	. . . . . . . . 9 ⊢ 𝐶 = (𝑁 Mat 𝑃)
67		mat2pmatbas0.h	. . . . . . . . 9 ⊢ 𝐻 = (Base‘𝐶)
68	45, 18, 20, 2, 66, 67	mat2pmatbas0 22845	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) → (𝑇‘𝑌) ∈ 𝐻)
69	44, 68	syl 18	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑇‘𝑌) ∈ 𝐻)
70	65, 69	jca 520	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ((𝑆‘𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑌) ∈ 𝐻))
71	70	adantr 485	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑆‘𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑌) ∈ 𝐻))
72		mat2pmatlin.n	. . . . . 6 ⊢ × = ( ·𝑠 ‘𝐶)
73	66, 67, 63, 72, 27	matvscacell 22554	. . . . 5 ⊢ ((𝑃 ∈ Ring ∧ ((𝑆‘𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑌) ∈ 𝐻) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑆‘𝑋) × (𝑇‘𝑌))𝑗) = ((𝑆‘𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑌)𝑗)))
74	60, 71, 35, 73	syl3anc 1394	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑆‘𝑋) × (𝑇‘𝑌))𝑗) = ((𝑆‘𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑌)𝑗)))
75	49, 56, 74	3eqtr4d 2810	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆‘𝑋) × (𝑇‘𝑌))𝑗))
76	75	ralrimivva 3208	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆‘𝑋) × (𝑇‘𝑌))𝑗))
77	45, 18, 20, 2, 66, 67	mat2pmatbas0 22845	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝐵) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
78	50, 31, 53, 77	syl3anc 1394	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
79	63, 66, 67, 72	matvscl 22549	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝑆‘𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑌) ∈ 𝐻)) → ((𝑆‘𝑋) × (𝑇‘𝑌)) ∈ 𝐻)
80	50, 59, 70, 79	syl21anc 850	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ((𝑆‘𝑋) × (𝑇‘𝑌)) ∈ 𝐻)
81	66, 67	eqmat 22542	. . 3 ⊢ (((𝑇‘(𝑋 · 𝑌)) ∈ 𝐻 ∧ ((𝑆‘𝑋) × (𝑇‘𝑌)) ∈ 𝐻) → ((𝑇‘(𝑋 · 𝑌)) = ((𝑆‘𝑋) × (𝑇‘𝑌)) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆‘𝑋) × (𝑇‘𝑌))𝑗)))
82	78, 80, 81	syl2anc 595	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ((𝑇‘(𝑋 · 𝑌)) = ((𝑆‘𝑋) × (𝑇‘𝑌)) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆‘𝑋) × (𝑇‘𝑌))𝑗)))
83	76, 82	mpbird 260	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑇‘(𝑋 · 𝑌)) = ((𝑆‘𝑋) × (𝑇‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ‘cfv 6525  (class class class)co 7400  Fincfn 8931  Basecbs 17259  ._rcmulr 17301  Scalarcsca 17303   ·_𝑠 cvsca 17304  Ringcrg 20306  CRingccrg 20307   RingHom crh 20542  AssAlgcasa 21960  algSccascl 21962  Poly₁cpl1 22297   Mat cmat 22525   matToPolyMat cmat2pmat 22822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-ofr 7665  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-er 8682  df-map 8814  df-pm 8815  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-sup 9390  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-z 12583  df-dec 12703  df-uz 12854  df-fz 13527  df-fzo 13674  df-seq 14029  df-hash 14358  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-sca 17316  df-vsca 17317  df-ip 17318  df-tset 17319  df-ple 17320  df-ds 17322  df-hom 17324  df-cco 17325  df-0g 17484  df-gsum 17485  df-prds 17490  df-pws 17492  df-mre 17628  df-mrc 17629  df-acs 17631  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-mhm 18831  df-submnd 18832  df-grp 18993  df-minusg 18994  df-sbg 18995  df-mulg 19125  df-subg 19180  df-ghm 19275  df-cntz 19378  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-cring 20309  df-rhm 20545  df-subrng 20622  df-subrg 20646  df-lmod 20952  df-lss 21022  df-sra 21263  df-rgmod 21264  df-dsmm 21842  df-frlm 21857  df-assa 21963  df-ascl 21965  df-psr 22019  df-mpl 22021  df-opsr 22023  df-psr1 22300  df-ply1 22302  df-mat 22526  df-mat2pmat 22825

This theorem is referenced by:  cpmidgsumm2pm  22987