Theorem mat2pmatlin 22673

Description: The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 20993. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 20980, see lmhmsca 20988. (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 484	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ CRing)
2		mat2pmatbas.p	. . . . . . . . . . 11 ⊢ 𝑃 = (Poly1‘𝑅)
3	2	ply1assa 22135	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
4		mat2pmatlin.s	. . . . . . . . . . 11 ⊢ 𝑆 = (algSc‘𝑃)
5		eqid 2735	. . . . . . . . . . 11 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
6	4, 5	asclrhm 21850	. . . . . . . . . 10 ⊢ (𝑃 ∈ AssAlg → 𝑆 ∈ ((Scalar‘𝑃) RingHom 𝑃))
7	1, 3, 6	3syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ ((Scalar‘𝑃) RingHom 𝑃))
8	2	ply1sca 22188	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
9	8	adantl 481	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
10	9	oveq1d 7420	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 RingHom 𝑃) = ((Scalar‘𝑃) RingHom 𝑃))
11	7, 10	eleqtrrd 2837	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ (𝑅 RingHom 𝑃))
12	11	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑆 ∈ (𝑅 RingHom 𝑃))
13	12	adantr 480	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑆 ∈ (𝑅 RingHom 𝑃))
14		mat2pmatlin.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝑅)
15	14	eleq2i 2826	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐾 ↔ 𝑋 ∈ (Base‘𝑅))
16	15	biimpi 216	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐾 → 𝑋 ∈ (Base‘𝑅))
17	16	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑅))
18	17	ad2antlr 727	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑋 ∈ (Base‘𝑅))
19		mat2pmatbas.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
20		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		mat2pmatbas.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
22		simprl 770	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
23		simpr 484	. . . . . . . 8 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
24	23	adantl 481	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
25		simplrr 777	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑌 ∈ 𝐵)
26	19, 20, 21, 22, 24, 25	matecld 22364	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑌𝑗) ∈ (Base‘𝑅))
27		eqid 2735	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
28		eqid 2735	. . . . . . 7 ⊢ (.r‘𝑃) = (.r‘𝑃)
29	20, 27, 28	rhmmul 20446	. . . . . 6 ⊢ ((𝑆 ∈ (𝑅 RingHom 𝑃) ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑖𝑌𝑗) ∈ (Base‘𝑅)) → (𝑆‘(𝑋(.r‘𝑅)(𝑖𝑌𝑗))) = ((𝑆‘𝑋)(.r‘𝑃)(𝑆‘(𝑖𝑌𝑗))))
30	13, 18, 26, 29	syl3anc 1373	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑆‘(𝑋(.r‘𝑅)(𝑖𝑌𝑗))) = ((𝑆‘𝑋)(.r‘𝑃)(𝑆‘(𝑖𝑌𝑗))))
31		crngring 20205	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
32	31	ad2antlr 727	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑅 ∈ Ring)
33	32	adantr 480	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
34		simpr 484	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵))
35	34	adantr 480	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵))
36		simpr 484	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
37		mat2pmatlin.m	. . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝐴)
38	19, 21, 14, 37, 27	matvscacell 22374	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑋 · 𝑌)𝑗) = (𝑋(.r‘𝑅)(𝑖𝑌𝑗)))
39	33, 35, 36, 38	syl3anc 1373	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑋 · 𝑌)𝑗) = (𝑋(.r‘𝑅)(𝑖𝑌𝑗)))
40	39	fveq2d 6880	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)) = (𝑆‘(𝑋(.r‘𝑅)(𝑖𝑌𝑗))))
41	31	anim2i 617	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
42		simpr 484	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
43	41, 42	anim12i 613	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐵))
44		df-3an 1088	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐵))
45	43, 44	sylibr 234	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵))
46		mat2pmatbas.t	. . . . . . . 8 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
47	46, 19, 21, 2, 4	mat2pmatvalel 22663	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘𝑌)𝑗) = (𝑆‘(𝑖𝑌𝑗)))
48	45, 47	sylan 580	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘𝑌)𝑗) = (𝑆‘(𝑖𝑌𝑗)))
49	48	oveq2d 7421	. . . . 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))))
50	30, 40, 49	3eqtr4d 2780	. . . 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)))
51		simpll 766	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑁 ∈ Fin)
52	51	adantr 480	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
53	14, 19, 21, 37	matvscl 22369	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑋 · 𝑌) ∈ 𝐵)
54	41, 53	sylan 580	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑋 · 𝑌) ∈ 𝐵)
55	54	adantr 480	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑋 · 𝑌) ∈ 𝐵)
56	46, 19, 21, 2, 4	mat2pmatvalel 22663	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)))
57	52, 33, 55, 36, 56	syl31anc 1375	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)))
58	2	ply1ring 22183	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
59	31, 58	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
60	59	ad2antlr 727	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑃 ∈ Ring)
61	60	adantr 480	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑃 ∈ Ring)
62	31	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
63		simpl 482	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐾)
64		eqid 2735	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑃)
65	2, 4, 14, 64	ply1sclcl 22223	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝑆‘𝑋) ∈ (Base‘𝑃))
66	62, 63, 65	syl2an 596	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑆‘𝑋) ∈ (Base‘𝑃))
67		mat2pmatbas.c	. . . . . . . . 9 ⊢ 𝐶 = (𝑁 Mat 𝑃)
68		mat2pmatbas0.h	. . . . . . . . 9 ⊢ 𝐻 = (Base‘𝐶)
69	46, 19, 21, 2, 67, 68	mat2pmatbas0 22665	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) → (𝑇‘𝑌) ∈ 𝐻)
70	45, 69	syl 17	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑇‘𝑌) ∈ 𝐻)
71	66, 70	jca 511	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ((𝑆‘𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑌) ∈ 𝐻))
72	71	adantr 480	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑆‘𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑌) ∈ 𝐻))
73		mat2pmatlin.n	. . . . . 6 ⊢ × = ( ·𝑠 ‘𝐶)
74	67, 68, 64, 73, 28	matvscacell 22374	. . . . 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)))
75	61, 72, 36, 74	syl3anc 1373	. . . 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)))
76	50, 57, 75	3eqtr4d 2780	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆‘𝑋) × (𝑇‘𝑌))𝑗))
77	76	ralrimivva 3187	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆‘𝑋) × (𝑇‘𝑌))𝑗))
78	46, 19, 21, 2, 67, 68	mat2pmatbas0 22665	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝐵) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
79	51, 32, 54, 78	syl3anc 1373	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
80	64, 67, 68, 73	matvscl 22369	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝑆‘𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑌) ∈ 𝐻)) → ((𝑆‘𝑋) × (𝑇‘𝑌)) ∈ 𝐻)
81	51, 60, 71, 80	syl21anc 837	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ((𝑆‘𝑋) × (𝑇‘𝑌)) ∈ 𝐻)
82	67, 68	eqmat 22362	. . 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)))
83	79, 81, 82	syl2anc 584	. 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)))
84	77, 83	mpbird 257	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑇‘(𝑋 · 𝑌)) = ((𝑆‘𝑋) × (𝑇‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ‘cfv 6531  (class class class)co 7405  Fincfn 8959  Basecbs 17228  ._rcmulr 17272  Scalarcsca 17274   ·_𝑠 cvsca 17275  Ringcrg 20193  CRingccrg 20194   RingHom crh 20429  AssAlgcasa 21810  algSccascl 21812  Poly₁cpl1 22112   Mat cmat 22345   matToPolyMat cmat2pmat 22642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-ofr 7672  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-pm 8843  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-sup 9454  df-oi 9524  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-fzo 13672  df-seq 14020  df-hash 14349  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-hom 17295  df-cco 17296  df-0g 17455  df-gsum 17456  df-prds 17461  df-pws 17463  df-mre 17598  df-mrc 17599  df-acs 17601  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-grp 18919  df-minusg 18920  df-sbg 18921  df-mulg 19051  df-subg 19106  df-ghm 19196  df-cntz 19300  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-cring 20196  df-rhm 20432  df-subrng 20506  df-subrg 20530  df-lmod 20819  df-lss 20889  df-sra 21131  df-rgmod 21132  df-dsmm 21692  df-frlm 21707  df-assa 21813  df-ascl 21815  df-psr 21869  df-mpl 21871  df-opsr 21873  df-psr1 22115  df-ply1 22117  df-mat 22346  df-mat2pmat 22645

This theorem is referenced by:  cpmidgsumm2pm  22807