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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 = (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 2737 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2737 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
6 eqid 2737 . . . . . 6 (algSc‘𝑃) = (algSc‘𝑃)
71, 2, 3, 4, 5, 6cpmatelimp2 22720 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))))
81, 2, 3, 4, 5, 6cpmatelimp2 22720 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏))))
9 r19.26-2 3138 . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (+g𝑅) = (+g𝑅)
115, 10ringacl 20275 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅))
12113expb 1121 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅))
132ply1sca 22254 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
1413eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
1514fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → (+g‘(Scalar‘𝑃)) = (+g𝑅))
1615oveqd 7448 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑅 ∈ Ring → (𝑎(+g‘(Scalar‘𝑃))𝑏) = (𝑎(+g𝑅)𝑏))
1716eleq1d 2826 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑅 ∈ Ring → ((𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅) ↔ (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅)))
1817adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅) ↔ (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅)))
1912, 18mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
2019ad5ant25 762 . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (𝑎(+g‘(Scalar‘𝑃))𝑏) → ((algSc‘𝑃)‘𝑐) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏)))
2322eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . 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 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))) ∧ 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 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
2625ancomd 461 . . . . . . . . . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (+g𝐶) = (+g𝐶)
30 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (+g𝑃) = (+g𝑃)
313, 4, 29, 30matplusgcell 22439 . . . . . . . . . . . . . . . . . . . . . . . 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 7440 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ (𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+g𝑃)((algSc‘𝑃)‘𝑏)))
3433ancoms 458 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+g𝑃)((algSc‘𝑃)‘𝑏)))
35 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Scalar‘𝑃) = (Scalar‘𝑃)
362ply1ring 22249 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
3736ad4antlr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ Ring)
382ply1lmod 22253 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
3938ad4antlr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ LMod)
406, 35, 37, 39asclghm 21903 . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
4241fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
4342eleq2d 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4443biimpd 229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4544ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4645adantrd 491 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4746imp 406 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑃)))
4813ad3antlr 731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 = (Scalar‘𝑃))
4948fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
5049eleq2d 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑏 ∈ (Base‘𝑅) ↔ 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5150biimpd 229 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑏 ∈ (Base‘𝑅) → 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5251adantld 490 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5352imp 406 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (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))
5654, 55, 30ghmlin 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)(+g𝑃)((algSc‘𝑃)‘𝑏)))
5740, 47, 53, 56syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . . . . . . . . . . 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 2799 . . . . . . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . . . . . . 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 3624 . . . . . . . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . . . 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 3155 . . . . . . . . . . . . . . . . . 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 3155 . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . 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 3206 . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . 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 415 . . . . . . . . . . . 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 456 . . . . . . . . . . 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 410 . . . . . . . . . 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 412 . . . . . . . . 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 410 . . . . . . 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 418 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))
80 simpl 482 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin)
8180adantr 480 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑁 ∈ Fin)
82 simpr 484 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
8382adantr 480 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑅 ∈ Ring)
842, 3pmatring 22698 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
8584adantr 480 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝐶 ∈ Ring)
86 simpl 482 . . . . . . . 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 234 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝑆))
901, 2, 3, 4cpmatpmat 22716 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐶))
9189, 90syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑥 ∈ (Base‘𝐶))
92 simpr 484 . . . . . . . 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 234 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝑆))
961, 2, 3, 4cpmatpmat 22716 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝐶))
9795, 96syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑦 ∈ (Base‘𝐶))
984, 29ringacl 20275 . . . . 5 ((𝐶 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(+g𝐶)𝑦) ∈ (Base‘𝐶))
9985, 91, 97, 98syl3anc 1373 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(+g𝐶)𝑦) ∈ (Base‘𝐶))
1001, 2, 3, 4, 5, 6cpmatel2 22719 . . . 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 1373 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ((𝑥(+g𝐶)𝑦) ∈ 𝑆 ↔ ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
10279, 101mpbird 257 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(+g𝐶)𝑦) ∈ 𝑆)
103102ralrimivva 3202 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝐶)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
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