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Theorem cpmatacl 22756

Description: The set of all constant polynomial matrices over a ring 𝑅 is closed under addition. (Contributed by AV, 17-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.)

**Hypotheses**
Ref	Expression
cpmatsrngpmat.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmatsrngpmat.p	⊢ 𝑃 = (Poly₁‘𝑅)
cpmatsrngpmat.c	⊢ 𝐶 = (𝑁 Mat 𝑃)

**Assertion**
Ref	Expression
cpmatacl	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆)

Distinct variable groups: 𝑥,𝑁,𝑦 𝑥,𝑅,𝑦 𝑦,𝑆 Allowed substitution hints: 𝐶(𝑥,𝑦) 𝑃(𝑥,𝑦) 𝑆(𝑥)

Proof of Theorem cpmatacl Dummy variables 𝑖 𝑗 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cpmatsrngpmat.s	. . . . . 6 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
2		cpmatsrngpmat.p	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
3		cpmatsrngpmat.c	. . . . . 6 ⊢ 𝐶 = (𝑁 Mat 𝑃)
4		eqid 2761	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2761	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2761	. . . . . 6 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
7	1, 2, 3, 4, 5, 6	cpmatelimp2 22754	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))))
8	1, 2, 3, 4, 5, 6	cpmatelimp2 22754	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ 𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏))))
9		r19.26-2 3146	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) ↔ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)))
10		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
11	5, 10	ringacl 20307	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+_g‘𝑅)𝑏) ∈ (Base‘𝑅))
12	11	3expb 1132	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+_g‘𝑅)𝑏) ∈ (Base‘𝑅))
13	2	ply1sca 22294	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
14	13	eqcomd 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
15	14	fveq2d 6867	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑅 ∈ Ring → (+_g‘(Scalar‘𝑃)) = (+_g‘𝑅))
16	15	oveqd 7409	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑅 ∈ Ring → (𝑎(+_g‘(Scalar‘𝑃))𝑏) = (𝑎(+_g‘𝑅)𝑏))
17	16	eleq1d 2846	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ Ring → ((𝑎(+_g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅) ↔ (𝑎(+_g‘𝑅)𝑏) ∈ (Base‘𝑅)))
18	17	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝑎(+_g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅) ↔ (𝑎(+_g‘𝑅)𝑏) ∈ (Base‘𝑅)))
19	12, 18	mpbird 259	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+_g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
20	19	ad5ant25 771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+_g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
21	20	adantr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → (𝑎(+_g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
22		fveq2 6863	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑎(+_g‘(Scalar‘𝑃))𝑏) → ((algSc‘𝑃)‘𝑐) = ((algSc‘𝑃)‘(𝑎(+_g‘(Scalar‘𝑃))𝑏)))
23	22	eqeq2d 2772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = (𝑎(+_g‘(Scalar‘𝑃))𝑏) → ((𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐) ↔ (𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘(𝑎(+_g‘(Scalar‘𝑃))𝑏))))
24	23	adantl 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) ∧ 𝑐 = (𝑎(+_g‘(Scalar‘𝑃))𝑏)) → ((𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐) ↔ (𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘(𝑎(+_g‘(Scalar‘𝑃))𝑏))))
25		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
26	25	ancomd 465	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)))
27	26	anim1i 624	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)))
28	27	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)))
29		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (+_g‘𝐶) = (+_g‘𝐶)
30		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
31	3, 4, 29, 30	matplusgcell 22473	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((𝑖𝑥𝑗)(+_g‘𝑃)(𝑖𝑦𝑗)))
32	28, 31	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → (𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((𝑖𝑥𝑗)(+_g‘𝑃)(𝑖𝑦𝑗)))
33		oveq12 7401	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ (𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ((𝑖𝑥𝑗)(+_g‘𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+_g‘𝑃)((algSc‘𝑃)‘𝑏)))
34	33	ancoms 462	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ((𝑖𝑥𝑗)(+_g‘𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+_g‘𝑃)((algSc‘𝑃)‘𝑏)))
35		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
36	2	ply1ring 22289	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
37	36	ad4antlr 743	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ Ring)
38	2	ply1lmod 22293	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
39	38	ad4antlr 743	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ LMod)
40	6, 35, 37, 39	asclghm 21914	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (algSc‘𝑃) ∈ ((Scalar‘𝑃) GrpHom 𝑃))
41	13	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
42	41	fveq2d 6867	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
43	42	eleq2d 2847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑃))))
44	43	biimpd 231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
45	44	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
46	45	adantrd 495	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
47	46	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑃)))
48	13	ad3antlr 741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 = (Scalar‘𝑃))
49	48	fveq2d 6867	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
50	49	eleq2d 2847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑏 ∈ (Base‘𝑅) ↔ 𝑏 ∈ (Base‘(Scalar‘𝑃))))
51	50	biimpd 231	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑏 ∈ (Base‘𝑅) → 𝑏 ∈ (Base‘(Scalar‘𝑃))))
52	51	adantld 494	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ (Base‘(Scalar‘𝑃))))
53	52	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘(Scalar‘𝑃)))
54		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
55		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (+_g‘(Scalar‘𝑃)) = (+_g‘(Scalar‘𝑃))
56	54, 55, 30	ghmlin 19244	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((algSc‘𝑃) ∈ ((Scalar‘𝑃) GrpHom 𝑃) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑏 ∈ (Base‘(Scalar‘𝑃))) → ((algSc‘𝑃)‘(𝑎(+_g‘(Scalar‘𝑃))𝑏)) = (((algSc‘𝑃)‘𝑎)(+_g‘𝑃)((algSc‘𝑃)‘𝑏)))
57	40, 47, 53, 56	syl3anc 1389	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((algSc‘𝑃)‘(𝑎(+_g‘(Scalar‘𝑃))𝑏)) = (((algSc‘𝑃)‘𝑎)(+_g‘𝑃)((algSc‘𝑃)‘𝑏)))
58	57	eqcomd 2767	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (((algSc‘𝑃)‘𝑎)(+_g‘𝑃)((algSc‘𝑃)‘𝑏)) = ((algSc‘𝑃)‘(𝑎(+_g‘(Scalar‘𝑃))𝑏)))
59	34, 58	sylan9eqr 2818	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → ((𝑖𝑥𝑗)(+_g‘𝑃)(𝑖𝑦𝑗)) = ((algSc‘𝑃)‘(𝑎(+_g‘(Scalar‘𝑃))𝑏)))
60	32, 59	eqtrd 2796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → (𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘(𝑎(+_g‘(Scalar‘𝑃))𝑏)))
61	21, 24, 60	rspcedvd 3583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))
62	61	exp32 424	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
63	62	anassrs 471	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) → ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
64	63	rexlimdva 3162	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) → (∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
65	64	com23 86	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) → ((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → (∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
66	65	rexlimdva 3162	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → (∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
67	66	impd 414	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
68	67	ralimdvva 3208	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
69	9, 68	biimtrrid 245	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → ((∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
70	69	expd 419	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
71	70	expr 460	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝐶) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))))
72	71	impd 414	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
73	72	ex 416	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ (Base‘𝐶) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))))
74	73	com34 91	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ (Base‘𝐶) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))))
75	74	impd 414	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑦 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
76	8, 75	syld 47	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ 𝑆 → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
77	76	com23 86	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → (𝑦 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
78	7, 77	syld 47	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ 𝑆 → (𝑦 ∈ 𝑆 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
79	78	imp32 422	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))
80		simpl 486	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin)
81	80	adantr 484	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑁 ∈ Fin)
82		simpr 488	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
83	82	adantr 484	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑅 ∈ Ring)
84	2, 3	pmatring 22732	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
85	84	adantr 484	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐶 ∈ Ring)
86		simpl 486	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆)
87	86	anim2i 626	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆))
88		df-3an 1099	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑆) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆))
89	87, 88	sylibr 236	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑆))
90	1, 2, 3, 4	cpmatpmat 22750	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐶))
91	89, 90	syl 17	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ (Base‘𝐶))
92		simpr 488	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆)
93	92	anim2i 626	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝑆))
94		df-3an 1099	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑆) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝑆))
95	93, 94	sylibr 236	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑆))
96	1, 2, 3, 4	cpmatpmat 22750	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝐶))
97	95, 96	syl 17	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ (Base‘𝐶))
98	4, 29	ringacl 20307	. . . . 5 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(+_g‘𝐶)𝑦) ∈ (Base‘𝐶))
99	85, 91, 97, 98	syl3anc 1389	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(+_g‘𝐶)𝑦) ∈ (Base‘𝐶))
100	1, 2, 3, 4, 5, 6	cpmatel2 22753	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥(+_g‘𝐶)𝑦) ∈ (Base‘𝐶)) → ((𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
101	81, 83, 99, 100	syl3anc 1389	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+_g‘𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
102	79, 101	mpbird 259	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆)
103	102	ralrimivva 3204	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 ‘cfv 6517 (class class class)co 7392 Fincfn 8923 Basecbs 17228 +_gcplusg 17269 Scalarcsca 17272 GrpHom cghm 19236 Ringcrg 20262 LModclmod 20907 algSccascl 21884 Poly₁cpl1 22219 Mat cmat 22447 ConstPolyMat ccpmat 22743

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-ofr 7657 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-sup 9385 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-fz 13510 df-fzo 13657 df-seq 14012 df-hash 14341 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-hom 17293 df-cco 17294 df-0g 17453 df-gsum 17454 df-prds 17459 df-pws 17461 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-mhm 18800 df-submnd 18801 df-grp 18961 df-minusg 18962 df-sbg 18963 df-mulg 19093 df-subg 19148 df-ghm 19237 df-cntz 19340 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-srg 20216 df-ring 20264 df-subrng 20575 df-subrg 20599 df-lmod 20909 df-lss 20979 df-sra 21220 df-rgmod 21221 df-dsmm 21764 df-frlm 21779 df-ascl 21887 df-psr 21941 df-mvr 21942 df-mpl 21943 df-opsr 21945 df-psr1 22222 df-vr1 22223 df-ply1 22224 df-coe1 22225 df-mamu 22431 df-mat 22448 df-cpmat 22746

This theorem is referenced by: cpmatsubgpmat 22760

W3C validator