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Theorem cpmatacl 22065
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 = (Poly1𝑅)
cpmatsrngpmat.c 𝐶 = (𝑁 Mat 𝑃)
Assertion
Ref Expression
cpmatacl ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝐶)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.
StepHypRef Expression
1 cpmatsrngpmat.s . . . . . 6 𝑆 = (𝑁 ConstPolyMat 𝑅)
2 cpmatsrngpmat.p . . . . . 6 𝑃 = (Poly1𝑅)
3 cpmatsrngpmat.c . . . . . 6 𝐶 = (𝑁 Mat 𝑃)
4 eqid 2736 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2736 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
6 eqid 2736 . . . . . 6 (algSc‘𝑃) = (algSc‘𝑃)
71, 2, 3, 4, 5, 6cpmatelimp2 22063 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))))
81, 2, 3, 4, 5, 6cpmatelimp2 22063 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏))))
9 r19.26-2 3135 . . . . . . . . . . . . . 14 (∀𝑖𝑁𝑗𝑁 (∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) ↔ (∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ ∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)))
10 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (+g𝑅) = (+g𝑅)
115, 10ringacl 19999 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅))
12113expb 1120 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅))
132ply1sca 21624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
1413eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
1514fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → (+g‘(Scalar‘𝑃)) = (+g𝑅))
1615oveqd 7374 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑅 ∈ Ring → (𝑎(+g‘(Scalar‘𝑃))𝑏) = (𝑎(+g𝑅)𝑏))
1716eleq1d 2822 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑅 ∈ Ring → ((𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅) ↔ (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅)))
1817adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅) ↔ (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅)))
1912, 18mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
2019ad5ant25 760 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
2120adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → (𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
22 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (𝑎(+g‘(Scalar‘𝑃))𝑏) → ((algSc‘𝑃)‘𝑐) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏)))
2322eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = (𝑎(+g‘(Scalar‘𝑃))𝑏) → ((𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐) ↔ (𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏))))
2423adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) ∧ 𝑐 = (𝑎(+g‘(Scalar‘𝑃))𝑏)) → ((𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐) ↔ (𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏))))
25 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
2625ancomd 462 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)))
2726anim1i 615 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑖𝑁𝑗𝑁)))
2827ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑖𝑁𝑗𝑁)))
29 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (+g𝐶) = (+g𝐶)
30 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (+g𝑃) = (+g𝑃)
313, 4, 29, 30matplusgcell 21782 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)))
3228, 31syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → (𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)))
33 oveq12 7366 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ (𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+g𝑃)((algSc‘𝑃)‘𝑏)))
3433ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+g𝑃)((algSc‘𝑃)‘𝑏)))
35 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Scalar‘𝑃) = (Scalar‘𝑃)
362ply1ring 21619 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
3736ad4antlr 731 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ Ring)
382ply1lmod 21623 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
3938ad4antlr 731 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ LMod)
406, 35, 37, 39asclghm 21286 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (algSc‘𝑃) ∈ ((Scalar‘𝑃) GrpHom 𝑃))
4113adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
4241fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
4342eleq2d 2823 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4443biimpd 228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4544ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4645adantrd 492 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4746imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑃)))
4813ad3antlr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 = (Scalar‘𝑃))
4948fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
5049eleq2d 2823 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑏 ∈ (Base‘𝑅) ↔ 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5150biimpd 228 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑏 ∈ (Base‘𝑅) → 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5251adantld 491 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5352imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘(Scalar‘𝑃)))
54 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
55 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (+g‘(Scalar‘𝑃)) = (+g‘(Scalar‘𝑃))
5654, 55, 30ghmlin 19013 . . . . . . . . . . . . . . . . . . . . . . . . . 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)(+g𝑃)((algSc‘𝑃)‘𝑏)))
5740, 47, 53, 56syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏)) = (((algSc‘𝑃)‘𝑎)(+g𝑃)((algSc‘𝑃)‘𝑏)))
5857eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (((algSc‘𝑃)‘𝑎)(+g𝑃)((algSc‘𝑃)‘𝑏)) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏)))
5934, 58sylan9eqr 2798 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏)))
6032, 59eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → (𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏)))
6121, 24, 60rspcedvd 3583 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) ∧ ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))
6261exp32 421 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
6362anassrs 468 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ (Base‘𝑅)) → ((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
6463rexlimdva 3152 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) → (∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
6564com23 86 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑎 ∈ (Base‘𝑅)) → ((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → (∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
6665rexlimdva 3152 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → (∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
6766impd 411 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ∃𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
6867ralimdvva 3201 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (∀𝑖𝑁𝑗𝑁 (∃𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ ∃𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
699, 68biimtrrid 242 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → ((∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ ∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
7069expd 416 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → (∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
7170expr 457 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥 ∈ (Base‘𝐶) → (∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) → (∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))))
7271impd 411 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → (∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
7372ex 413 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ (Base‘𝐶) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → (∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))))
7473com34 91 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦 ∈ (Base‘𝐶) → (∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))))
7574impd 411 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑦 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
768, 75syld 47 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝑆 → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
7776com23 86 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → (𝑦𝑆 → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
787, 77syld 47 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑦𝑆 → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
7978imp32 419 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))
80 simpl 483 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin)
8180adantr 481 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑁 ∈ Fin)
82 simpr 485 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
8382adantr 481 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑅 ∈ Ring)
842, 3pmatring 22041 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
8584adantr 481 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝐶 ∈ Ring)
86 simpl 483 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → 𝑥𝑆)
8786anim2i 617 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝑆))
88 df-3an 1089 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝑆) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝑆))
8987, 88sylibr 233 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝑆))
901, 2, 3, 4cpmatpmat 22059 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐶))
9189, 90syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑥 ∈ (Base‘𝐶))
92 simpr 485 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → 𝑦𝑆)
9392anim2i 617 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝑆))
94 df-3an 1089 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝑆) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝑆))
9593, 94sylibr 233 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝑆))
961, 2, 3, 4cpmatpmat 22059 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝐶))
9795, 96syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑦 ∈ (Base‘𝐶))
984, 29ringacl 19999 . . . . 5 ((𝐶 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(+g𝐶)𝑦) ∈ (Base‘𝐶))
9985, 91, 97, 98syl3anc 1371 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(+g𝐶)𝑦) ∈ (Base‘𝐶))
1001, 2, 3, 4, 5, 6cpmatel2 22062 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥(+g𝐶)𝑦) ∈ (Base‘𝐶)) → ((𝑥(+g𝐶)𝑦) ∈ 𝑆 ↔ ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
10181, 83, 99, 100syl3anc 1371 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ((𝑥(+g𝐶)𝑦) ∈ 𝑆 ↔ ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
10279, 101mpbird 256 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(+g𝐶)𝑦) ∈ 𝑆)
103102ralrimivva 3197 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝐶)𝑦) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  cfv 6496  (class class class)co 7357  Fincfn 8883  Basecbs 17083  +gcplusg 17133  Scalarcsca 17136   GrpHom cghm 19005  Ringcrg 19964  LModclmod 20322  algSccascl 21258  Poly1cpl1 21548   Mat cmat 21754   ConstPolyMat ccpmat 22052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-srg 19918  df-ring 19966  df-subrg 20220  df-lmod 20324  df-lss 20393  df-sra 20633  df-rgmod 20634  df-dsmm 21138  df-frlm 21153  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-vr1 21552  df-ply1 21553  df-coe1 21554  df-mamu 21733  df-mat 21755  df-cpmat 22055
This theorem is referenced by:  cpmatsubgpmat  22069
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