Theorem mat2pmatlin 21346

Description: The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 19810. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 19797, see lmhmsca 19805. (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 487	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ CRing)
2		mat2pmatbas.p	. . . . . . . . . . 11 ⊢ 𝑃 = (Poly1‘𝑅)
3	2	ply1assa 20370	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
4		mat2pmatlin.s	. . . . . . . . . . 11 ⊢ 𝑆 = (algSc‘𝑃)
5		eqid 2824	. . . . . . . . . . 11 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
6	4, 5	asclrhm 20122	. . . . . . . . . 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 20424	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
9	8	adantl 484	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
10	9	oveq1d 7174	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 RingHom 𝑃) = ((Scalar‘𝑃) RingHom 𝑃))
11	7, 10	eleqtrrd 2919	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ (𝑅 RingHom 𝑃))
12	11	adantr 483	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑆 ∈ (𝑅 RingHom 𝑃))
13	12	adantr 483	. . . . . 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 2907	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐾 ↔ 𝑋 ∈ (Base‘𝑅))
16	15	biimpi 218	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐾 → 𝑋 ∈ (Base‘𝑅))
17	16	adantr 483	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑅))
18	17	ad2antlr 725	. . . . . 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 2824	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		mat2pmatbas.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
22		simprl 769	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
23		simpr 487	. . . . . . . 8 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
24	23	adantl 484	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
25		simplrr 776	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑌 ∈ 𝐵)
26	19, 20, 21, 22, 24, 25	matecld 21038	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑌𝑗) ∈ (Base‘𝑅))
27		eqid 2824	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
28		eqid 2824	. . . . . . 7 ⊢ (.r‘𝑃) = (.r‘𝑃)
29	20, 27, 28	rhmmul 19482	. . . . . 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 1367	. . . . 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 19311	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
32	31	ad2antlr 725	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑅 ∈ Ring)
33	32	adantr 483	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
34		simpr 487	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵))
35	34	adantr 483	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵))
36		simpr 487	. . . . . . 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 21048	. . . . . . 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 1367	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑋 · 𝑌)𝑗) = (𝑋(.r‘𝑅)(𝑖𝑌𝑗)))
40	39	fveq2d 6677	. . . . 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 618	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
42		simpr 487	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
43	41, 42	anim12i 614	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐵))
44		df-3an 1085	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌 ∈ 𝐵))
45	43, 44	sylibr 236	. . . . . . 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 21336	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘𝑌)𝑗) = (𝑆‘(𝑖𝑌𝑗)))
48	45, 47	sylan 582	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘𝑌)𝑗) = (𝑆‘(𝑖𝑌𝑗)))
49	48	oveq2d 7175	. . . . 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 2869	. . . 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 765	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑁 ∈ Fin)
52	51	adantr 483	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
53	14, 19, 21, 37	matvscl 21043	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑋 · 𝑌) ∈ 𝐵)
54	41, 53	sylan 582	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑋 · 𝑌) ∈ 𝐵)
55	54	adantr 483	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑋 · 𝑌) ∈ 𝐵)
56	46, 19, 21, 2, 4	mat2pmatvalel 21336	. . . . 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 1369	. . . 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 20419	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
59	31, 58	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
60	59	ad2antlr 725	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → 𝑃 ∈ Ring)
61	60	adantr 483	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑃 ∈ Ring)
62	31	adantl 484	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
63		simpl 485	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐾)
64		eqid 2824	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑃)
65	2, 4, 14, 64	ply1sclcl 20457	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝑆‘𝑋) ∈ (Base‘𝑃))
66	62, 63, 65	syl2an 597	. . . . . . 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 21338	. . . . . . . 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 514	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ((𝑆‘𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑌) ∈ 𝐻))
72	71	adantr 483	. . . . 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 21048	. . . . 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 1367	. . . 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 2869	. . 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 3194	. 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 21338	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝐵) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
79	51, 32, 54, 78	syl3anc 1367	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
80	64, 67, 68, 73	matvscl 21043	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝑆‘𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑌) ∈ 𝐻)) → ((𝑆‘𝑋) × (𝑇‘𝑌)) ∈ 𝐻)
81	51, 60, 71, 80	syl21anc 835	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → ((𝑆‘𝑋) × (𝑇‘𝑌)) ∈ 𝐻)
82	67, 68	eqmat 21036	. . 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 586	. 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 259	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑇‘(𝑋 · 𝑌)) = ((𝑆‘𝑋) × (𝑇‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ‘cfv 6358  (class class class)co 7159  Fincfn 8512  Basecbs 16486  ._rcmulr 16569  Scalarcsca 16571   ·_𝑠 cvsca 16572  Ringcrg 19300  CRingccrg 19301   RingHom crh 19467  AssAlgcasa 20085  algSccascl 20087  Poly₁cpl1 20348   Mat cmat 21019   matToPolyMat cmat2pmat 21315

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-ot 4579  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-ofr 7413  df-om 7584  df-1st 7692  df-2nd 7693  df-supp 7834  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-ixp 8465  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fsupp 8837  df-sup 8909  df-oi 8977  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-fzo 13037  df-seq 13373  df-hash 13694  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-mulr 16582  df-sca 16584  df-vsca 16585  df-ip 16586  df-tset 16587  df-ple 16588  df-ds 16590  df-hom 16592  df-cco 16593  df-0g 16718  df-gsum 16719  df-prds 16724  df-pws 16726  df-mre 16860  df-mrc 16861  df-acs 16863  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-mhm 17959  df-submnd 17960  df-grp 18109  df-minusg 18110  df-sbg 18111  df-mulg 18228  df-subg 18279  df-ghm 18359  df-cntz 18450  df-cmn 18911  df-abl 18912  df-mgp 19243  df-ur 19255  df-ring 19302  df-cring 19303  df-rnghom 19470  df-subrg 19536  df-lmod 19639  df-lss 19707  df-sra 19947  df-rgmod 19948  df-assa 20088  df-ascl 20090  df-psr 20139  df-mpl 20141  df-opsr 20143  df-psr1 20351  df-ply1 20353  df-dsmm 20879  df-frlm 20894  df-mat 21020  df-mat2pmat 21318

This theorem is referenced by:  cpmidgsumm2pm  21480