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Theorem mat2pmatghm 22554

Description: The transformation of matrices into polynomial matrices is an additive group homomorphism. (Contributed by AV, 28-Oct-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)

**Assertion**
Ref	Expression
mat2pmatghm	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶))

Proof of Theorem mat2pmatghm Dummy variables 𝑥 𝑦 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mat2pmatbas.b	. 2 ⊢ 𝐵 = (Base‘𝐴)
2		mat2pmatbas0.h	. 2 ⊢ 𝐻 = (Base‘𝐶)
3		eqid 2724	. 2 ⊢ (+_g‘𝐴) = (+_g‘𝐴)
4		eqid 2724	. 2 ⊢ (+_g‘𝐶) = (+_g‘𝐶)
5		mat2pmatbas.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	5	matgrp 22254	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Grp)
7		mat2pmatbas.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
8		mat2pmatbas.c	. . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃)
9	7, 8	pmatring 22516	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
10		ringgrp 20133	. . 3 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Grp)
11	9, 10	syl 17	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Grp)
12		mat2pmatbas.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
13	12, 5, 1, 7, 8, 2	mat2pmatf 22552	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝐻)
14		eqid 2724	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
15		simpl 482	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin)
16	15	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ Fin)
17	7	ply1ring 22089	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
18	17	ad2antlr 724	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ Ring)
19		simp1lr 1234	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
20		eqid 2724	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		simp2 1134	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
22		simp3 1135	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
23		simp1rl 1235	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑥 ∈ 𝐵)
24	5, 20, 1, 21, 22, 23	matecld 22250	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑥𝑗) ∈ (Base‘𝑅))
25		eqid 2724	. . . . . . . 8 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
26	7, 25, 20, 14	ply1sclcl 22127	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑥𝑗) ∈ (Base‘𝑅)) → ((algSc‘𝑃)‘(𝑖𝑥𝑗)) ∈ (Base‘𝑃))
27	19, 24, 26	syl2anc 583	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑖𝑥𝑗)) ∈ (Base‘𝑃))
28	8, 14, 2, 16, 18, 27	matbas2d 22247	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗))) ∈ 𝐻)
29		simp1rr 1236	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑦 ∈ 𝐵)
30	5, 20, 1, 21, 22, 29	matecld 22250	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑦𝑗) ∈ (Base‘𝑅))
31	7, 25, 20, 14	ply1sclcl 22127	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑦𝑗) ∈ (Base‘𝑅)) → ((algSc‘𝑃)‘(𝑖𝑦𝑗)) ∈ (Base‘𝑃))
32	19, 30, 31	syl2anc 583	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑖𝑦𝑗)) ∈ (Base‘𝑃))
33	8, 14, 2, 16, 18, 32	matbas2d 22247	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗))) ∈ 𝐻)
34		eqid 2724	. . . . . 6 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
35	8, 2, 4, 34	matplusg2 22251	. . . . 5 ⊢ (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗))) ∈ 𝐻 ∧ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗))) ∈ 𝐻) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗)))(+_g‘𝐶)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗)))) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗))) ∘_f (+_g‘𝑃)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗)))))
36	28, 33, 35	syl2anc 583	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗)))(+_g‘𝐶)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗)))) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗))) ∘_f (+_g‘𝑃)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗)))))
37		fvexd 6896	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑖𝑥𝑗)) ∈ V)
38		fvexd 6896	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑖𝑦𝑗)) ∈ V)
39		eqidd 2725	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗))))
40		eqidd 2725	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗))))
41	16, 16, 37, 38, 39, 40	offval22 8068	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗))) ∘_f (+_g‘𝑃)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗)))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((algSc‘𝑃)‘(𝑖𝑥𝑗))(+_g‘𝑃)((algSc‘𝑃)‘(𝑖𝑦𝑗)))))
42		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
43	42	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
44		3simpc 1147	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
45		eqid 2724	. . . . . . . . . . 11 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
46	5, 1, 3, 45	matplusgcell 22257	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑥(+_g‘𝐴)𝑦)𝑗) = ((𝑖𝑥𝑗)(+_g‘𝑅)(𝑖𝑦𝑗)))
47	43, 44, 46	syl2anc 583	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑥(+_g‘𝐴)𝑦)𝑗) = ((𝑖𝑥𝑗)(+_g‘𝑅)(𝑖𝑦𝑗)))
48	7	ply1sca 22094	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
49	48	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
50	49	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (+_g‘𝑅) = (+_g‘(Scalar‘𝑃)))
51	50	oveqd 7418	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑖𝑥𝑗)(+_g‘𝑅)(𝑖𝑦𝑗)) = ((𝑖𝑥𝑗)(+_g‘(Scalar‘𝑃))(𝑖𝑦𝑗)))
52	51	adantr 480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖𝑥𝑗)(+_g‘𝑅)(𝑖𝑦𝑗)) = ((𝑖𝑥𝑗)(+_g‘(Scalar‘𝑃))(𝑖𝑦𝑗)))
53	52	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑥𝑗)(+_g‘𝑅)(𝑖𝑦𝑗)) = ((𝑖𝑥𝑗)(+_g‘(Scalar‘𝑃))(𝑖𝑦𝑗)))
54	47, 53	eqtrd 2764	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑥(+_g‘𝐴)𝑦)𝑗) = ((𝑖𝑥𝑗)(+_g‘(Scalar‘𝑃))(𝑖𝑦𝑗)))
55	54	fveq2d 6885	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑖(𝑥(+_g‘𝐴)𝑦)𝑗)) = ((algSc‘𝑃)‘((𝑖𝑥𝑗)(+_g‘(Scalar‘𝑃))(𝑖𝑦𝑗))))
56		eqid 2724	. . . . . . . . 9 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
57	18	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ Ring)
58	7	ply1lmod 22093	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
59	58	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ LMod)
60	59	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ LMod)
61	25, 56, 57, 60	asclghm 21745	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (algSc‘𝑃) ∈ ((Scalar‘𝑃) GrpHom 𝑃))
62	49	eqcomd 2730	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝑃) = 𝑅)
63	62	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
64	63	eleq2d 2811	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑖𝑥𝑗) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑖𝑥𝑗) ∈ (Base‘𝑅)))
65	64	adantr 480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖𝑥𝑗) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑖𝑥𝑗) ∈ (Base‘𝑅)))
66	65	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑥𝑗) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑖𝑥𝑗) ∈ (Base‘𝑅)))
67	24, 66	mpbird 257	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑥𝑗) ∈ (Base‘(Scalar‘𝑃)))
68	63	eleq2d 2811	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑖𝑦𝑗) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑖𝑦𝑗) ∈ (Base‘𝑅)))
69	68	adantr 480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖𝑦𝑗) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑖𝑦𝑗) ∈ (Base‘𝑅)))
70	69	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑦𝑗) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑖𝑦𝑗) ∈ (Base‘𝑅)))
71	30, 70	mpbird 257	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑦𝑗) ∈ (Base‘(Scalar‘𝑃)))
72		eqid 2724	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73		eqid 2724	. . . . . . . . 9 ⊢ (+_g‘(Scalar‘𝑃)) = (+_g‘(Scalar‘𝑃))
74	72, 73, 34	ghmlin 19136	. . . . . . . 8 ⊢ (((algSc‘𝑃) ∈ ((Scalar‘𝑃) GrpHom 𝑃) ∧ (𝑖𝑥𝑗) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑖𝑦𝑗) ∈ (Base‘(Scalar‘𝑃))) → ((algSc‘𝑃)‘((𝑖𝑥𝑗)(+_g‘(Scalar‘𝑃))(𝑖𝑦𝑗))) = (((algSc‘𝑃)‘(𝑖𝑥𝑗))(+_g‘𝑃)((algSc‘𝑃)‘(𝑖𝑦𝑗))))
75	61, 67, 71, 74	syl3anc 1368	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘𝑃)‘((𝑖𝑥𝑗)(+_g‘(Scalar‘𝑃))(𝑖𝑦𝑗))) = (((algSc‘𝑃)‘(𝑖𝑥𝑗))(+_g‘𝑃)((algSc‘𝑃)‘(𝑖𝑦𝑗))))
76	55, 75	eqtr2d 2765	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (((algSc‘𝑃)‘(𝑖𝑥𝑗))(+_g‘𝑃)((algSc‘𝑃)‘(𝑖𝑦𝑗))) = ((algSc‘𝑃)‘(𝑖(𝑥(+_g‘𝐴)𝑦)𝑗)))
77	76	mpoeq3dva 7478	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (((algSc‘𝑃)‘(𝑖𝑥𝑗))(+_g‘𝑃)((algSc‘𝑃)‘(𝑖𝑦𝑗)))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑥(+_g‘𝐴)𝑦)𝑗))))
78	41, 77	eqtrd 2764	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗))) ∘_f (+_g‘𝑃)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗)))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑥(+_g‘𝐴)𝑦)𝑗))))
79	36, 78	eqtr2d 2765	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑥(+_g‘𝐴)𝑦)𝑗))) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗)))(+_g‘𝐶)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗)))))
80		simpl 482	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
81	5	matring 22267	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
82		ringmnd 20138	. . . . . . . . 9 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ Mnd)
83	81, 82	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Mnd)
84	83	anim1i 614	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐴 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
85		3anass 1092	. . . . . . 7 ⊢ ((𝐴 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝐴 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
86	84, 85	sylibr 233	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐴 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
87	1, 3	mndcl 18665	. . . . . 6 ⊢ ((𝐴 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+_g‘𝐴)𝑦) ∈ 𝐵)
88	86, 87	syl 17	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐴)𝑦) ∈ 𝐵)
89		df-3an 1086	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥(+_g‘𝐴)𝑦) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥(+_g‘𝐴)𝑦) ∈ 𝐵))
90	80, 88, 89	sylanbrc 582	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥(+_g‘𝐴)𝑦) ∈ 𝐵))
91	12, 5, 1, 7, 25	mat2pmatval 22548	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥(+_g‘𝐴)𝑦) ∈ 𝐵) → (𝑇‘(𝑥(+_g‘𝐴)𝑦)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑥(+_g‘𝐴)𝑦)𝑗))))
92	90, 91	syl 17	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘(𝑥(+_g‘𝐴)𝑦)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑥(+_g‘𝐴)𝑦)𝑗))))
93		simpl 482	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
94	93	anim2i 616	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝐵))
95		df-3an 1086	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝐵))
96	94, 95	sylibr 233	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
97	12, 5, 1, 7, 25	mat2pmatval 22548	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗))))
98	96, 97	syl 17	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘𝑥) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗))))
99		simpr 484	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
100	99	anim2i 616	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐵))
101		df-3an 1086	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐵))
102	100, 101	sylibr 233	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
103	12, 5, 1, 7, 25	mat2pmatval 22548	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑇‘𝑦) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗))))
104	102, 103	syl 17	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘𝑦) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗))))
105	98, 104	oveq12d 7419	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑇‘𝑥)(+_g‘𝐶)(𝑇‘𝑦)) = ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑥𝑗)))(+_g‘𝐶)(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖𝑦𝑗)))))
106	79, 92, 105	3eqtr4d 2774	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘(𝑥(+_g‘𝐴)𝑦)) = ((𝑇‘𝑥)(+_g‘𝐶)(𝑇‘𝑦)))
107	1, 2, 3, 4, 6, 11, 13, 106	isghmd 19140	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ‘cfv 6533 (class class class)co 7401 ∈ cmpo 7403 ∘_f cof 7661 Fincfn 8935 Basecbs 17143 +_gcplusg 17196 Scalarcsca 17199 Mndcmnd 18657 Grpcgrp 18853 GrpHom cghm 19128 Ringcrg 20128 LModclmod 20696 algSccascl 21715 Poly₁cpl1 22019 Mat cmat 22229 matToPolyMat cmat2pmat 22528

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-ot 4629 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-subrng 20436 df-subrg 20461 df-lmod 20698 df-lss 20769 df-sra 21011 df-rgmod 21012 df-dsmm 21595 df-frlm 21610 df-ascl 21718 df-psr 21771 df-mpl 21773 df-opsr 21775 df-psr1 22022 df-ply1 22024 df-mamu 22208 df-mat 22230 df-mat2pmat 22531

This theorem is referenced by: mat2pmatrhm 22558 0mat2pmat 22560 m2cpmghm 22568 pm2mp 22649 cayhamlem4 22712

W3C validator