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Theorem pmatcollpw3fi1lem1 21395
Description: Lemma 1 for pmatcollpw3fi1 21397. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.)
Hypotheses
Ref Expression
pmatcollpw.p 𝑃 = (Poly1𝑅)
pmatcollpw.c 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw.b 𝐵 = (Base‘𝐶)
pmatcollpw.m = ( ·𝑠𝐶)
pmatcollpw.e = (.g‘(mulGrp‘𝑃))
pmatcollpw.x 𝑋 = (var1𝑅)
pmatcollpw.t 𝑇 = (𝑁 matToPolyMat 𝑅)
pmatcollpw3.a 𝐴 = (𝑁 Mat 𝑅)
pmatcollpw3.d 𝐷 = (Base‘𝐴)
pmatcollpw3fi1lem1.0 0 = (0g𝐴)
pmatcollpw3fi1lem1.h 𝐻 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝐺‘0), 0 ))
Assertion
Ref Expression
pmatcollpw3fi1lem1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))))
Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑃,𝑛   𝑅,𝑛   𝑛,𝑋   ,𝑛   𝐶,𝑛   𝐵,𝑙   𝑀,𝑙   𝑁,𝑙   𝑅,𝑙   𝐷,𝑙,𝑛   𝐴,𝑙   𝐺,𝑙,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝐶(𝑙)   𝑃(𝑙)   𝑇(𝑛,𝑙)   (𝑙)   𝐻(𝑛,𝑙)   (𝑛,𝑙)   𝑋(𝑙)   0 (𝑛,𝑙)

Proof of Theorem pmatcollpw3fi1lem1
StepHypRef Expression
1 simpr 488 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))))
2 pmatcollpw.p . . . . . . . . . . 11 𝑃 = (Poly1𝑅)
3 pmatcollpw.c . . . . . . . . . . 11 𝐶 = (𝑁 Mat 𝑃)
42, 3pmatring 21301 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
5 ringmnd 19304 . . . . . . . . . 10 (𝐶 ∈ Ring → 𝐶 ∈ Mnd)
64, 5syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Mnd)
76adantr 484 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐶 ∈ Mnd)
8 pmatcollpw.b . . . . . . . . 9 𝐵 = (Base‘𝐶)
9 ringcmn 19331 . . . . . . . . . . 11 (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
104, 9syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ CMnd)
1110adantr 484 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐶 ∈ CMnd)
12 snfi 8581 . . . . . . . . . 10 {0} ∈ Fin
1312a1i 11 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → {0} ∈ Fin)
14 simplll 774 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑁 ∈ Fin)
15 simpllr 775 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑅 ∈ Ring)
16 elmapi 8415 . . . . . . . . . . . . 13 (𝐺 ∈ (𝐷m {0}) → 𝐺:{0}⟶𝐷)
1716adantl 485 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐺:{0}⟶𝐷)
1817ffvelrnda 6832 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
19 elsni 4545 . . . . . . . . . . . . 13 (𝑛 ∈ {0} → 𝑛 = 0)
20 0nn0 11904 . . . . . . . . . . . . 13 0 ∈ ℕ0
2119, 20eqeltrdi 2901 . . . . . . . . . . . 12 (𝑛 ∈ {0} → 𝑛 ∈ ℕ0)
2221adantl 485 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑛 ∈ ℕ0)
23 pmatcollpw3.a . . . . . . . . . . . 12 𝐴 = (𝑁 Mat 𝑅)
24 pmatcollpw3.d . . . . . . . . . . . 12 𝐷 = (Base‘𝐴)
25 pmatcollpw.t . . . . . . . . . . . 12 𝑇 = (𝑁 matToPolyMat 𝑅)
26 pmatcollpw.m . . . . . . . . . . . 12 = ( ·𝑠𝐶)
27 pmatcollpw.e . . . . . . . . . . . 12 = (.g‘(mulGrp‘𝑃))
28 pmatcollpw.x . . . . . . . . . . . 12 𝑋 = (var1𝑅)
2923, 24, 25, 2, 3, 8, 26, 27, 28mat2pmatscmxcl 21349 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐺𝑛) ∈ 𝐷𝑛 ∈ ℕ0)) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
3014, 15, 18, 22, 29syl22anc 837 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
3130ralrimiva 3152 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → ∀𝑛 ∈ {0} ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
328, 11, 13, 31gsummptcl 19084 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))) ∈ 𝐵)
33 eqid 2801 . . . . . . . . 9 (+g𝐶) = (+g𝐶)
34 eqid 2801 . . . . . . . . 9 (0g𝐶) = (0g𝐶)
358, 33, 34mndrid 17928 . . . . . . . 8 ((𝐶 ∈ Mnd ∧ (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))) ∈ 𝐵) → ((𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))(+g𝐶)(0g𝐶)) = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))))
367, 32, 35syl2anc 587 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → ((𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))(+g𝐶)(0g𝐶)) = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))))
37 fz0sn 13006 . . . . . . . . . . . 12 (0...0) = {0}
3837eqcomi 2810 . . . . . . . . . . 11 {0} = (0...0)
3938a1i 11 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → {0} = (0...0))
40 pmatcollpw3fi1lem1.h . . . . . . . . . . . . . 14 𝐻 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝐺‘0), 0 ))
41 simpr 488 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 𝑛)
4219ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑛 = 0)
4341, 42eqtrd 2836 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 0)
4443iftrued 4436 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘0))
45 fveq2 6649 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → (𝐺𝑛) = (𝐺‘0))
4645eqcomd 2807 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → (𝐺‘0) = (𝐺𝑛))
4719, 46syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {0} → (𝐺‘0) = (𝐺𝑛))
4847ad2antlr 726 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → (𝐺‘0) = (𝐺𝑛))
4944, 48eqtrd 2836 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺𝑛))
50 1nn0 11905 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℕ0
5150a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 0 → 1 ∈ ℕ0)
52 nn0uz 12272 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
5351, 52eleqtrdi 2903 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → 1 ∈ (ℤ‘0))
54 eluzfz1 12913 . . . . . . . . . . . . . . . . . 18 (1 ∈ (ℤ‘0) → 0 ∈ (0...1))
5553, 54syl 17 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → 0 ∈ (0...1))
56 eleq1 2880 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → (𝑛 ∈ (0...1) ↔ 0 ∈ (0...1)))
5755, 56mpbird 260 . . . . . . . . . . . . . . . 16 (𝑛 = 0 → 𝑛 ∈ (0...1))
5819, 57syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ {0} → 𝑛 ∈ (0...1))
5958adantl 485 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑛 ∈ (0...1))
60 ffvelrn 6830 . . . . . . . . . . . . . . . . . 18 ((𝐺:{0}⟶𝐷𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
6160ex 416 . . . . . . . . . . . . . . . . 17 (𝐺:{0}⟶𝐷 → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6216, 61syl 17 . . . . . . . . . . . . . . . 16 (𝐺 ∈ (𝐷m {0}) → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6362adantl 485 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6463imp 410 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
6540, 49, 59, 64fvmptd2 6757 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐻𝑛) = (𝐺𝑛))
6665eqcomd 2807 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) = (𝐻𝑛))
6766fveq2d 6653 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝑇‘(𝐺𝑛)) = (𝑇‘(𝐻𝑛)))
6867oveq2d 7155 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) = ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))
6939, 68mpteq12dva 5117 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))) = (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))
7069oveq2d 7155 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))))
71 ovexd 7174 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (0 + 1) ∈ V)
728, 34mndidcl 17922 . . . . . . . . . . . 12 (𝐶 ∈ Mnd → (0g𝐶) ∈ 𝐵)
736, 72syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐶) ∈ 𝐵)
7473adantr 484 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (0g𝐶) ∈ 𝐵)
75 0p1e1 11751 . . . . . . . . . . . . . . . . . . . . 21 (0 + 1) = 1
7675eqeq2i 2814 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = (0 + 1) ↔ 𝑛 = 1)
77 ax-1ne0 10599 . . . . . . . . . . . . . . . . . . . . . 22 1 ≠ 0
7877neii 2992 . . . . . . . . . . . . . . . . . . . . 21 ¬ 1 = 0
79 eqeq1 2805 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → (𝑛 = 0 ↔ 1 = 0))
8078, 79mtbiri 330 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → ¬ 𝑛 = 0)
8176, 80sylbi 220 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (0 + 1) → ¬ 𝑛 = 0)
8281ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑛 = 0)
83 eqeq1 2805 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
8483notbid 321 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑛 → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
8584adantl 485 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
8682, 85mpbird 260 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑙 = 0)
8786iffalsed 4439 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = 0 )
88 pmatcollpw3fi1lem1.0 . . . . . . . . . . . . . . . 16 0 = (0g𝐴)
8987, 88eqtrdi 2852 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (0g𝐴))
9050a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → 1 ∈ ℕ0)
9190, 52eleqtrdi 2903 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → 1 ∈ (ℤ‘0))
92 eluzfz2 12914 . . . . . . . . . . . . . . . . . . 19 (1 ∈ (ℤ‘0) → 1 ∈ (0...1))
9391, 92syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → 1 ∈ (0...1))
94 eleq1 2880 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (𝑛 ∈ (0...1) ↔ 1 ∈ (0...1)))
9593, 94mpbird 260 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → 𝑛 ∈ (0...1))
9676, 95sylbi 220 . . . . . . . . . . . . . . . 16 (𝑛 = (0 + 1) → 𝑛 ∈ (0...1))
9796adantl 485 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ (0...1))
98 fvexd 6664 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (0g𝐴) ∈ V)
9940, 89, 97, 98fvmptd2 6757 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝐻𝑛) = (0g𝐴))
10099fveq2d 6653 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻𝑛)) = (𝑇‘(0g𝐴)))
10123fveq2i 6652 . . . . . . . . . . . . . . . 16 (0g𝐴) = (0g‘(𝑁 Mat 𝑅))
1023fveq2i 6652 . . . . . . . . . . . . . . . 16 (0g𝐶) = (0g‘(𝑁 Mat 𝑃))
10325, 2, 101, 1020mat2pmat 21345 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘(0g𝐴)) = (0g𝐶))
104103ancoms 462 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(0g𝐴)) = (0g𝐶))
105104ad2antrr 725 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(0g𝐴)) = (0g𝐶))
106100, 105eqtrd 2836 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻𝑛)) = (0g𝐶))
107106oveq2d 7155 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) = ((𝑛 𝑋) (0g𝐶)))
1082, 3pmatlmod 21302 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
109108ad2antrr 725 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝐶 ∈ LMod)
110 simpllr 775 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝑅 ∈ Ring)
111 eleq1 2880 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (𝑛 ∈ ℕ0 ↔ 1 ∈ ℕ0))
11290, 111mpbird 260 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → 𝑛 ∈ ℕ0)
11376, 112sylbi 220 . . . . . . . . . . . . . . 15 (𝑛 = (0 + 1) → 𝑛 ∈ ℕ0)
114113adantl 485 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ ℕ0)
115 eqid 2801 . . . . . . . . . . . . . . 15 (mulGrp‘𝑃) = (mulGrp‘𝑃)
116 eqid 2801 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
1172, 28, 115, 27, 116ply1moncl 20904 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ (Base‘𝑃))
118110, 114, 117syl2anc 587 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 𝑋) ∈ (Base‘𝑃))
1192ply1ring 20881 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
1203matsca2 21029 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
121119, 120sylan2 595 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
122121eqcomd 2807 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐶) = 𝑃)
123122fveq2d 6653 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
124123eleq2d 2878 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 𝑋) ∈ (Base‘𝑃)))
125124ad2antrr 725 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 𝑋) ∈ (Base‘𝑃)))
126118, 125mpbird 260 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)))
127 eqid 2801 . . . . . . . . . . . . 13 (Scalar‘𝐶) = (Scalar‘𝐶)
128 eqid 2801 . . . . . . . . . . . . 13 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
129127, 26, 128, 34lmodvs0 19665 . . . . . . . . . . . 12 ((𝐶 ∈ LMod ∧ (𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 𝑋) (0g𝐶)) = (0g𝐶))
130109, 126, 129syl2anc 587 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (0g𝐶)) = (0g𝐶))
131107, 130eqtrd 2836 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) = (0g𝐶))
1328, 7, 71, 74, 131gsumsnd 19069 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = (0g𝐶))
133132eqcomd 2807 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (0g𝐶) = (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))))
13470, 133oveq12d 7157 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → ((𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))(+g𝐶)(0g𝐶)) = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))(+g𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))))
13536, 134eqtr3d 2838 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))) = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))(+g𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))))
136135adantr 484 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))) = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))(+g𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))))
1371, 136eqtrd 2836 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))) → 𝑀 = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))(+g𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))))
1381373impa 1107 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))) → 𝑀 = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))(+g𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))))
13920a1i 11 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 0 ∈ ℕ0)
140 simplll 774 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑁 ∈ Fin)
141 simpllr 775 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑅 ∈ Ring)
142 id 22 . . . . . . . . . . . . 13 (𝐺:{0}⟶𝐷𝐺:{0}⟶𝐷)
143 c0ex 10628 . . . . . . . . . . . . . . 15 0 ∈ V
144143snid 4564 . . . . . . . . . . . . . 14 0 ∈ {0}
145144a1i 11 . . . . . . . . . . . . 13 (𝐺:{0}⟶𝐷 → 0 ∈ {0})
146142, 145ffvelrnd 6833 . . . . . . . . . . . 12 (𝐺:{0}⟶𝐷 → (𝐺‘0) ∈ 𝐷)
14716, 146syl 17 . . . . . . . . . . 11 (𝐺 ∈ (𝐷m {0}) → (𝐺‘0) ∈ 𝐷)
148147ad2antlr 726 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → (𝐺‘0) ∈ 𝐷)
14923matring 21052 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
15024, 88ring0cl 19319 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 0𝐷)
151149, 150syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0𝐷)
152151ad2antrr 725 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → 0𝐷)
153148, 152ifcld 4473 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝐺‘0), 0 ) ∈ 𝐷)
154153, 40fmptd 6859 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐻:(0...1)⟶𝐷)
15575oveq2i 7150 . . . . . . . . 9 (0...(0 + 1)) = (0...1)
156155feq2i 6483 . . . . . . . 8 (𝐻:(0...(0 + 1))⟶𝐷𝐻:(0...1)⟶𝐷)
157154, 156sylibr 237 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐻:(0...(0 + 1))⟶𝐷)
158157ffvelrnda 6832 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → (𝐻𝑛) ∈ 𝐷)
159 elfznn0 12999 . . . . . . 7 (𝑛 ∈ (0...(0 + 1)) → 𝑛 ∈ ℕ0)
160159adantl 485 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑛 ∈ ℕ0)
16123, 24, 25, 2, 3, 8, 26, 27, 28mat2pmatscmxcl 21349 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐻𝑛) ∈ 𝐷𝑛 ∈ ℕ0)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) ∈ 𝐵)
162140, 141, 158, 160, 161syl22anc 837 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) ∈ 𝐵)
1638, 33, 11, 139, 162gsummptfzsplit 19049 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))(+g𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))))
1641633adant3 1129 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))) → (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))(+g𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))))
165138, 164eqtr4d 2839 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))))
166155mpteq1i 5123 . . 3 (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))
167166oveq2i 7150 . 2 (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))
168165, 167eqtrdi 2852 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  Vcvv 3444  ifcif 4428  {csn 4528  cmpt 5113  wf 6324  cfv 6328  (class class class)co 7139  m cmap 8393  Fincfn 8496  0cc0 10530  1c1 10531   + caddc 10533  0cn0 11889  cuz 12235  ...cfz 12889  Basecbs 16479  +gcplusg 16561  Scalarcsca 16564   ·𝑠 cvsca 16565  0gc0g 16709   Σg cgsu 16710  Mndcmnd 17907  .gcmg 18220  CMndccmn 18902  mulGrpcmgp 19236  Ringcrg 19294  LModclmod 19631  var1cv1 20809  Poly1cpl1 20810   Mat cmat 21016   matToPolyMat cmat2pmat 21313
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-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-mamu 20995  df-mat 21017  df-mat2pmat 21316
This theorem is referenced by:  pmatcollpw3fi1lem2  21396
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