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Theorem cpmatacl 20569
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 2651 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2651 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
6 eqid 2651 . . . . . 6 (algSc‘𝑃) = (algSc‘𝑃)
71, 2, 3, 4, 5, 6cpmatelimp2 20567 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑎 ∈ (Base‘𝑅)(𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎))))
81, 2, 3, 4, 5, 6cpmatelimp2 20567 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑗𝑁𝑏 ∈ (Base‘𝑅)(𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏))))
9 r19.26-2 3094 . . . . . . . . . . . . . 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 2651 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (+g𝑅) = (+g𝑅)
115, 10ringacl 18624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅))
12113expb 1285 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅))
132ply1sca 19671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
1413eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
1514fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑅 ∈ Ring → (+g‘(Scalar‘𝑃)) = (+g𝑅))
1615oveqd 6707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑅 ∈ Ring → (𝑎(+g‘(Scalar‘𝑃))𝑏) = (𝑎(+g𝑅)𝑏))
1716eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑅 ∈ Ring → ((𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅) ↔ (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅)))
1817adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅) ↔ (𝑎(+g𝑅)𝑏) ∈ (Base‘𝑅)))
1912, 18mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
2019ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑅 ∈ Ring → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅)))
2120ad3antlr 767 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅)))
2221imp 444 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
2322adantr 480 . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑎(+g‘(Scalar‘𝑃))𝑏) ∈ (Base‘𝑅))
24 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = (𝑎(+g‘(Scalar‘𝑃))𝑏) → ((algSc‘𝑃)‘𝑐) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏)))
2524eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . . . . 24 (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))))
2625adantl 481 . . . . . . . . . . . . . . . . . . . . . . 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))) ∧ 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))))
27 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶)))
2827ancomd 466 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)))
2928anim1i 591 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑖𝑁𝑗𝑁)))
3029ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((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)))
31 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (+g𝐶) = (+g𝐶)
32 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (+g𝑃) = (+g𝑃)
333, 4, 31, 32matplusgcell 20287 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)))
3430, 33syl 17 . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)))
35 oveq12 6699 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎) ∧ (𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏)) → ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+g𝑃)((algSc‘𝑃)‘𝑏)))
3635ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑖𝑦𝑗) = ((algSc‘𝑃)‘𝑏) ∧ (𝑖𝑥𝑗) = ((algSc‘𝑃)‘𝑎)) → ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)) = (((algSc‘𝑃)‘𝑎)(+g𝑃)((algSc‘𝑃)‘𝑏)))
37 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Scalar‘𝑃) = (Scalar‘𝑃)
382ply1ring 19666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
3938ad4antlr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ Ring)
402ply1lmod 19670 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
4140ad4antlr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑃 ∈ LMod)
426, 37, 39, 41asclghm 19386 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (algSc‘𝑃) ∈ ((Scalar‘𝑃) GrpHom 𝑃))
4313adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
4443fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
4544eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4645biimpd 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4746ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑎 ∈ (Base‘𝑅) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4847adantrd 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑎 ∈ (Base‘(Scalar‘𝑃))))
4948imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑃)))
5013ad3antlr 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 = (Scalar‘𝑃))
5150fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
5251eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑏 ∈ (Base‘𝑅) ↔ 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5352biimpd 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑏 ∈ (Base‘𝑅) → 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5453adantld 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ (Base‘(Scalar‘𝑃))))
5554imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑥 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘(Scalar‘𝑃)))
56 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
57 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (+g‘(Scalar‘𝑃)) = (+g‘(Scalar‘𝑃))
5856, 57, 32ghmlin 17712 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((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)))
5942, 49, 55, 58syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((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)))
6059eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((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)))
6136, 60sylan9eqr 2707 . . . . . . . . . . . . . . . . . . . . . . . 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))) → ((𝑖𝑥𝑗)(+g𝑃)(𝑖𝑦𝑗)) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏)))
6234, 61eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . . 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) = ((algSc‘𝑃)‘(𝑎(+g‘(Scalar‘𝑃))𝑏)))
6323, 26, 62rspcedvd 3348 . . . . . . . . . . . . . . . . . . . . . 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 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))
6463ex 449 . . . . . . . . . . . . . . . . . . . . 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)))
6564expd 451 . . . . . . . . . . . . . . . . . . . 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))))
6665anassrs 681 . . . . . . . . . . . . . . . . . . 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))))
6766rexlimdva 3060 . . . . . . . . . . . . . . . . . 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))))
6867com23 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))))
6968rexlimdva 3060 . . . . . . . . . . . . . . . 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))))
7069impd 446 . . . . . . . . . . . . . . 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)))
7170ralimdvva 2993 . . . . . . . . . . . . . 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)))
729, 71syl5bir 233 . . . . . . . . . . . . 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)))
7372expd 451 . . . . . . . . . . . 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))))
7473expr 642 . . . . . . . . . . 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)))))
7574impd 446 . . . . . . . . . 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))))
7675ex 449 . . . . . . . . 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)))))
7776com34 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)))))
7877impd 446 . . . . . . 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))))
798, 78syld 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))))
8079com23 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))))
817, 80syld 47 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑦𝑆 → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))))
8281imp32 448 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐))
83 simpl 472 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin)
8483adantr 480 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑁 ∈ Fin)
85 simpr 476 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
8685adantr 480 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑅 ∈ Ring)
872, 3pmatring 20546 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
8887adantr 480 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝐶 ∈ Ring)
89 simpl 472 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → 𝑥𝑆)
9089anim2i 592 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝑆))
91 df-3an 1056 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝑆) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥𝑆))
9290, 91sylibr 224 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝑆))
931, 2, 3, 4cpmatpmat 20563 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝐶))
9492, 93syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑥 ∈ (Base‘𝐶))
95 simpr 476 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → 𝑦𝑆)
9695anim2i 592 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝑆))
97 df-3an 1056 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝑆) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝑆))
9896, 97sylibr 224 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝑆))
991, 2, 3, 4cpmatpmat 20563 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝐶))
10098, 99syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → 𝑦 ∈ (Base‘𝐶))
1014, 31ringacl 18624 . . . . 5 ((𝐶 ∈ Ring ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(+g𝐶)𝑦) ∈ (Base‘𝐶))
10288, 94, 100, 101syl3anc 1366 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(+g𝐶)𝑦) ∈ (Base‘𝐶))
1031, 2, 3, 4, 5, 6cpmatel2 20566 . . . 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)))
10484, 86, 102, 103syl3anc 1366 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ((𝑥(+g𝐶)𝑦) ∈ 𝑆 ↔ ∀𝑖𝑁𝑗𝑁𝑐 ∈ (Base‘𝑅)(𝑖(𝑥(+g𝐶)𝑦)𝑗) = ((algSc‘𝑃)‘𝑐)))
10582, 104mpbird 247 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(+g𝐶)𝑦) ∈ 𝑆)
106105ralrimivva 3000 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝐶)𝑦) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cfv 5926  (class class class)co 6690  Fincfn 7997  Basecbs 15904  +gcplusg 15988  Scalarcsca 15991   GrpHom cghm 17704  Ringcrg 18593  LModclmod 18911  algSccascl 19359  Poly1cpl1 19595   Mat cmat 20261   ConstPolyMat ccpmat 20556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-0g 16149  df-gsum 16150  df-prds 16155  df-pws 16157  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-srg 18552  df-ring 18595  df-subrg 18826  df-lmod 18913  df-lss 18981  df-sra 19220  df-rgmod 19221  df-ascl 19362  df-psr 19404  df-mvr 19405  df-mpl 19406  df-opsr 19408  df-psr1 19598  df-vr1 19599  df-ply1 19600  df-coe1 19601  df-dsmm 20124  df-frlm 20139  df-mamu 20238  df-mat 20262  df-cpmat 20559
This theorem is referenced by:  cpmatsubgpmat  20573
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