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Theorem cpmatacl 21325
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 2801 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2801 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
6 eqid 2801 . . . . . 6 (algSc‘𝑃) = (algSc‘𝑃)
71, 2, 3, 4, 5, 6cpmatelimp2 21323 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))))
81, 2, 3, 4, 5, 6cpmatelimp2 21323 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏))))
9 r19.26-2 3141 . . . . . . . . . . . . . 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 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (+g𝑅) = (+g𝑅)
115, 10ringacl 19328 . . . . . . . . . . . . . . . . . . . . . . . . . 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 20886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
1413eqcomd 2807 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
1514fveq2d 6653 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → (+g‘(Scalar‘𝑃)) = (+g𝑅))
1615oveqd 7156 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑅 ∈ Ring → (𝑎(+g‘(Scalar‘𝑃))𝑏) = (𝑎(+g𝑅)𝑏))
1716eleq1d 2877 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑅 ∈ Ring → ((𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅) ↔ (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅)))
1817adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅) ↔ (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅)))
1912, 18mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
2019ad5ant25 761 . . . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . 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 6649 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (𝑎(+g‘(Scalar‘𝑃))𝑏) → ((algSc‘𝑃)‘𝑐) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏)))
2322eqeq2d 2812 . . . . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
2625ancomd 465 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)))
2726anim1i 617 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑖𝑁𝑗𝑁)))
2827ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . 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 2801 . . . . . . . . . . . . . . . . . . . . . . . . 25 (+g𝐶) = (+g𝐶)
30 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . 25 (+g𝑃) = (+g𝑃)
313, 4, 29, 30matplusgcell 21042 . . . . . . . . . . . . . . . . . . . . . . . 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 7148 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ (𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+g𝑃)((algSc‘𝑃)‘𝑏)))
3433ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+g𝑃)((algSc‘𝑃)‘𝑏)))
35 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Scalar‘𝑃) = (Scalar‘𝑃)
362ply1ring 20881 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
3736ad4antlr 732 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ Ring)
382ply1lmod 20885 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
3938ad4antlr 732 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ LMod)
406, 35, 37, 39asclghm 20573 . . . . . . . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
4241fveq2d 6653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
4342eleq2d 2878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4443biimpd 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4544ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4645adantrd 495 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4746imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑃)))
4813ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 = (Scalar‘𝑃))
4948fveq2d 6653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
5049eleq2d 2878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑏 ∈ (Base‘𝑅) ↔ 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5150biimpd 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑏 ∈ (Base‘𝑅) → 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5251adantld 494 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5352imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘(Scalar‘𝑃)))
54 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
55 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (+g‘(Scalar‘𝑃)) = (+g‘(Scalar‘𝑃))
5654, 55, 30ghmlin 18359 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2807 . . . . . . . . . . . . . . . . . . . . . . . 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 2858 . . . . . . . . . . . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . . . . . . . . . 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 3577 . . . . . . . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . . . . 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 3246 . . . . . . . . . . . . . . . . . 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 3246 . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . 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 3149 . . . . . . . . . . . . . 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, 68syl5bir 246 . . . . . . . . . . . . 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 419 . . . . . . . . . . . 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 460 . . . . . . . . . . 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 414 . . . . . . . . . 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 416 . . . . . . . . 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 414 . . . . . . 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 422 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))
80 simpl 486 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin)
8180adantr 484 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑁 ∈ Fin)
82 simpr 488 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
8382adantr 484 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑅 ∈ Ring)
842, 3pmatring 21301 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
8584adantr 484 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝐶 ∈ Ring)
86 simpl 486 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → 𝑥𝑆)
8786anim2i 619 . . . . . . 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 237 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝑆))
901, 2, 3, 4cpmatpmat 21319 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐶))
9189, 90syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑥 ∈ (Base‘𝐶))
92 simpr 488 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → 𝑦𝑆)
9392anim2i 619 . . . . . . 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 237 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝑆))
961, 2, 3, 4cpmatpmat 21319 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝐶))
9795, 96syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑦 ∈ (Base‘𝐶))
984, 29ringacl 19328 . . . . 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 21322 . . . 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 260 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(+g𝐶)𝑦) ∈ 𝑆)
103102ralrimivva 3159 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝐶)𝑦) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110  cfv 6328  (class class class)co 7139  Fincfn 8496  Basecbs 16479  +gcplusg 16561  Scalarcsca 16564   GrpHom cghm 18351  Ringcrg 19294  LModclmod 19631  algSccascl 20545  Poly1cpl1 20810   Mat cmat 21016   ConstPolyMat ccpmat 21312
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-ot 4537  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-ofr 7394  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-fzo 13033  df-seq 13369  df-hash 13691  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-mulr 16575  df-sca 16577  df-vsca 16578  df-ip 16579  df-tset 16580  df-ple 16581  df-ds 16583  df-hom 16585  df-cco 16586  df-0g 16711  df-gsum 16712  df-prds 16717  df-pws 16719  df-mre 16853  df-mrc 16854  df-acs 16856  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-mhm 17952  df-submnd 17953  df-grp 18102  df-minusg 18103  df-sbg 18104  df-mulg 18221  df-subg 18272  df-ghm 18352  df-cntz 18443  df-cmn 18904  df-abl 18905  df-mgp 19237  df-ur 19249  df-srg 19253  df-ring 19296  df-subrg 19530  df-lmod 19633  df-lss 19701  df-sra 19941  df-rgmod 19942  df-dsmm 20425  df-frlm 20440  df-ascl 20548  df-psr 20598  df-mvr 20599  df-mpl 20600  df-opsr 20602  df-psr1 20813  df-vr1 20814  df-ply1 20815  df-coe1 20816  df-mamu 20995  df-mat 21017  df-cpmat 21315
This theorem is referenced by:  cpmatsubgpmat  21329
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