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Theorem cpmatacl 22573
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 2726 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2726 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
6 eqid 2726 . . . . . 6 (algSc‘𝑃) = (algSc‘𝑃)
71, 2, 3, 4, 5, 6cpmatelimp2 22571 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))))
81, 2, 3, 4, 5, 6cpmatelimp2 22571 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏))))
9 r19.26-2 3132 . . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (+g𝑅) = (+g𝑅)
115, 10ringacl 20177 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅))
12113expb 1117 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅))
132ply1sca 22126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
1413eqcomd 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
1514fveq2d 6889 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → (+g‘(Scalar‘𝑃)) = (+g𝑅))
1615oveqd 7422 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑅 ∈ Ring → (𝑎(+g‘(Scalar‘𝑃))𝑏) = (𝑎(+g𝑅)𝑏))
1716eleq1d 2812 . . . . . . . . . . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . . . . . . . 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 6885 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (𝑎(+g‘(Scalar‘𝑃))𝑏) → ((algSc‘𝑃)‘𝑐) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏)))
2322eqeq2d 2737 . . . . . . . . . . . . . . . . . . . . . . 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 614 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑖𝑁𝑗𝑁)))
2827ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (+g𝐶) = (+g𝐶)
30 eqid 2726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (+g𝑃) = (+g𝑃)
313, 4, 29, 30matplusgcell 22290 . . . . . . . . . . . . . . . . . . . . . . . 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 7414 . . . . . . . . . . . . . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Scalar‘𝑃) = (Scalar‘𝑃)
362ply1ring 22121 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
3736ad4antlr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ Ring)
382ply1lmod 22125 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
3938ad4antlr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ LMod)
406, 35, 37, 39asclghm 21777 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6889 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
4342eleq2d 2813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 = (Scalar‘𝑃))
4948fveq2d 6889 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
5049eleq2d 2813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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 2726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
55 eqid 2726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (+g‘(Scalar‘𝑃)) = (+g‘(Scalar‘𝑃))
5654, 55, 30ghmlin 19146 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . . . . . . 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 2788 . . . . . . . . . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . . . . . . . . . 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 3608 . . . . . . . . . . . . . . . . . . . . 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 3149 . . . . . . . . . . . . . . . . . 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 3149 . . . . . . . . . . . . . . . 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 3198 . . . . . . . . . . . . . 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 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 22549 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
8584adantr 480 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝐶 ∈ Ring)
86 simpl 482 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → 𝑥𝑆)
8786anim2i 616 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝑆))
88 df-3an 1086 . . . . . . 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 22567 . . . . . 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 616 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝑆))
94 df-3an 1086 . . . . . . 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 22567 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝐶))
9795, 96syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑦 ∈ (Base‘𝐶))
984, 29ringacl 20177 . . . . 5 ((𝐶 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(+g𝐶)𝑦) ∈ (Base‘𝐶))
9985, 91, 97, 98syl3anc 1368 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(+g𝐶)𝑦) ∈ (Base‘𝐶))
1001, 2, 3, 4, 5, 6cpmatel2 22570 . . . 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 1368 . . 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 3194 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝐶)𝑦) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  wrex 3064  cfv 6537  (class class class)co 7405  Fincfn 8941  Basecbs 17153  +gcplusg 17206  Scalarcsca 17209   GrpHom cghm 19138  Ringcrg 20138  LModclmod 20706  algSccascl 21747  Poly1cpl1 22051   Mat cmat 22262   ConstPolyMat ccpmat 22560
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-ofr 7668  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-sup 9439  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-fzo 13634  df-seq 13973  df-hash 14296  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-sca 17222  df-vsca 17223  df-ip 17224  df-tset 17225  df-ple 17226  df-ds 17228  df-hom 17230  df-cco 17231  df-0g 17396  df-gsum 17397  df-prds 17402  df-pws 17404  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18713  df-submnd 18714  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18996  df-subg 19050  df-ghm 19139  df-cntz 19233  df-cmn 19702  df-abl 19703  df-mgp 20040  df-rng 20058  df-ur 20087  df-srg 20092  df-ring 20140  df-subrng 20446  df-subrg 20471  df-lmod 20708  df-lss 20779  df-sra 21021  df-rgmod 21022  df-dsmm 21627  df-frlm 21642  df-ascl 21750  df-psr 21803  df-mvr 21804  df-mpl 21805  df-opsr 21807  df-psr1 22054  df-vr1 22055  df-ply1 22056  df-coe1 22057  df-mamu 22241  df-mat 22263  df-cpmat 22563
This theorem is referenced by:  cpmatsubgpmat  22577
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