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Theorem cpmatacl 22722

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 2737	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2737	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2737	. . . . . 6 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
7	1, 2, 3, 4, 5, 6	cpmatelimp2 22720	. . . . 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 22720	. . . . . . 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 3138	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
11	5, 10	ringacl 20275	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+_g‘𝑅)𝑏) ∈ (Base‘𝑅))
12	11	3expb 1121	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+_g‘𝑅)𝑏) ∈ (Base‘𝑅))
13	2	ply1sca 22254	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
14	13	eqcomd 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
15	14	fveq2d 6910	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑅 ∈ Ring → (+_g‘(Scalar‘𝑃)) = (+_g‘𝑅))
16	15	oveqd 7448	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑅 ∈ Ring → (𝑎(+_g‘(Scalar‘𝑃))𝑏) = (𝑎(+_g‘𝑅)𝑏))
17	16	eleq1d 2826	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ Ring → ((𝑎(+_g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅) ↔ (𝑎(+_g‘𝑅)𝑏) ∈ (Base‘𝑅)))
18	17	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+_g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
20	19	ad5ant25 762	. . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑎(+_g‘(Scalar‘𝑃))𝑏) → ((algSc‘𝑃)‘𝑐) = ((algSc‘𝑃)‘(𝑎(+_g‘(Scalar‘𝑃))𝑏)))
23	22	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
26	25	ancomd 461	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)))
27	26	anim1i 615	. . . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (+_g‘𝐶) = (+_g‘𝐶)
30		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
31	3, 4, 29, 30	matplusgcell 22439	. . . . . . . . . . . . . . . . . . . . . . . 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 7440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ (𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ((𝑖𝑥𝑗)(+_g‘𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+_g‘𝑃)((algSc‘𝑃)‘𝑏)))
34	33	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ((𝑖𝑥𝑗)(+_g‘𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+_g‘𝑃)((algSc‘𝑃)‘𝑏)))
35		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
36	2	ply1ring 22249	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
37	36	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 22253	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
39	38	ad4antlr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 21903	. . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
42	41	fveq2d 6910	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
43	42	eleq2d 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑃))))
44	43	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
45	44	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 = (Scalar‘𝑃))
49	48	fveq2d 6910	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
50	49	eleq2d 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
55		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (+_g‘(Scalar‘𝑃)) = (+_g‘(Scalar‘𝑃))
56	54, 55, 30	ghmlin 19239	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . . . . . 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 3624	. . . . . . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . 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 3206	. . . . . . . . . . . . . 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 243	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 410	. . . . . . . . . 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 412	. . . . . . . . 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 410	. . . . . . 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 418	. . 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 482	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin)
81	80	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑁 ∈ Fin)
82		simpr 484	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
83	82	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑅 ∈ Ring)
84	2, 3	pmatring 22698	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
85	84	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐶 ∈ Ring)
86		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆)
87	86	anim2i 617	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆))
88		df-3an 1089	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑆) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝑆))
89	87, 88	sylibr 234	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑆))
90	1, 2, 3, 4	cpmatpmat 22716	. . . . . 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 484	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆)
93	92	anim2i 617	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝑆))
94		df-3an 1089	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑆) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝑆))
95	93, 94	sylibr 234	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑆))
96	1, 2, 3, 4	cpmatpmat 22716	. . . . . 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 20275	. . . . 5 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(+_g‘𝐶)𝑦) ∈ (Base‘𝐶))
99	85, 91, 97, 98	syl3anc 1373	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(+_g‘𝐶)𝑦) ∈ (Base‘𝐶))
100	1, 2, 3, 4, 5, 6	cpmatel2 22719	. . . 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 1373	. . 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 257	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆)
103	102	ralrimivva 3202	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 Basecbs 17247 +_gcplusg 17297 Scalarcsca 17300 GrpHom cghm 19230 Ringcrg 20230 LModclmod 20858 algSccascl 21872 Poly₁cpl1 22178 Mat cmat 22411 ConstPolyMat ccpmat 22709

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-coe1 22184 df-mamu 22395 df-mat 22412 df-cpmat 22712

This theorem is referenced by: cpmatsubgpmat 22726

W3C validator