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Theorem pmatcollpw3fi1lem1 22792
Description: Lemma 1 for pmatcollpw3fi1 22794. (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 484 . . . . 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 22698 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
5 ringmnd 20240 . . . . . . . . . 10 (𝐶 ∈ Ring → 𝐶 ∈ Mnd)
64, 5syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Mnd)
76adantr 480 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐶 ∈ Mnd)
8 pmatcollpw.b . . . . . . . . 9 𝐵 = (Base‘𝐶)
9 ringcmn 20279 . . . . . . . . . . 11 (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
104, 9syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ CMnd)
1110adantr 480 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐶 ∈ CMnd)
12 snfi 9083 . . . . . . . . . 10 {0} ∈ Fin
1312a1i 11 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → {0} ∈ Fin)
14 simplll 775 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑁 ∈ Fin)
15 simpllr 776 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑅 ∈ Ring)
16 elmapi 8889 . . . . . . . . . . . . 13 (𝐺 ∈ (𝐷m {0}) → 𝐺:{0}⟶𝐷)
1716adantl 481 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐺:{0}⟶𝐷)
1817ffvelcdmda 7104 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
19 elsni 4643 . . . . . . . . . . . . 13 (𝑛 ∈ {0} → 𝑛 = 0)
20 0nn0 12541 . . . . . . . . . . . . 13 0 ∈ ℕ0
2119, 20eqeltrdi 2849 . . . . . . . . . . . 12 (𝑛 ∈ {0} → 𝑛 ∈ ℕ0)
2221adantl 481 . . . . . . . . . . 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 22746 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐺𝑛) ∈ 𝐷𝑛 ∈ ℕ0)) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
3014, 15, 18, 22, 29syl22anc 839 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
3130ralrimiva 3146 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → ∀𝑛 ∈ {0} ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
328, 11, 13, 31gsummptcl 19985 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))) ∈ 𝐵)
33 eqid 2737 . . . . . . . . 9 (+g𝐶) = (+g𝐶)
34 eqid 2737 . . . . . . . . 9 (0g𝐶) = (0g𝐶)
358, 33, 34mndrid 18768 . . . . . . . 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 584 . . . . . . 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 13667 . . . . . . . . . . . 12 (0...0) = {0}
3837eqcomi 2746 . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 𝑛)
4219ad2antlr 727 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑛 = 0)
4341, 42eqtrd 2777 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 0)
4443iftrued 4533 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘0))
45 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → (𝐺𝑛) = (𝐺‘0))
4645eqcomd 2743 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → (𝐺‘0) = (𝐺𝑛))
4719, 46syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {0} → (𝐺‘0) = (𝐺𝑛))
4847ad2antlr 727 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → (𝐺‘0) = (𝐺𝑛))
4944, 48eqtrd 2777 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺𝑛))
50 1nn0 12542 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℕ0
5150a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 0 → 1 ∈ ℕ0)
52 nn0uz 12920 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
5351, 52eleqtrdi 2851 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → 1 ∈ (ℤ‘0))
54 eluzfz1 13571 . . . . . . . . . . . . . . . . . 18 (1 ∈ (ℤ‘0) → 0 ∈ (0...1))
5553, 54syl 17 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → 0 ∈ (0...1))
56 eleq1 2829 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → (𝑛 ∈ (0...1) ↔ 0 ∈ (0...1)))
5755, 56mpbird 257 . . . . . . . . . . . . . . . 16 (𝑛 = 0 → 𝑛 ∈ (0...1))
5819, 57syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ {0} → 𝑛 ∈ (0...1))
5958adantl 481 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑛 ∈ (0...1))
60 ffvelcdm 7101 . . . . . . . . . . . . . . . . . 18 ((𝐺:{0}⟶𝐷𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
6160ex 412 . . . . . . . . . . . . . . . . 17 (𝐺:{0}⟶𝐷 → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6216, 61syl 17 . . . . . . . . . . . . . . . 16 (𝐺 ∈ (𝐷m {0}) → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6362adantl 481 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6463imp 406 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
6540, 49, 59, 64fvmptd2 7024 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐻𝑛) = (𝐺𝑛))
6665eqcomd 2743 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) = (𝐻𝑛))
6766fveq2d 6910 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝑇‘(𝐺𝑛)) = (𝑇‘(𝐻𝑛)))
6867oveq2d 7447 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) = ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))
6939, 68mpteq12dva 5231 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))) = (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))
7069oveq2d 7447 . . . . . . . 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 7466 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (0 + 1) ∈ V)
728, 34mndidcl 18762 . . . . . . . . . . . 12 (𝐶 ∈ Mnd → (0g𝐶) ∈ 𝐵)
736, 72syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐶) ∈ 𝐵)
7473adantr 480 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (0g𝐶) ∈ 𝐵)
75 0p1e1 12388 . . . . . . . . . . . . . . . . . . . . 21 (0 + 1) = 1
7675eqeq2i 2750 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = (0 + 1) ↔ 𝑛 = 1)
77 ax-1ne0 11224 . . . . . . . . . . . . . . . . . . . . . 22 1 ≠ 0
7877neii 2942 . . . . . . . . . . . . . . . . . . . . 21 ¬ 1 = 0
79 eqeq1 2741 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → (𝑛 = 0 ↔ 1 = 0))
8078, 79mtbiri 327 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → ¬ 𝑛 = 0)
8176, 80sylbi 217 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (0 + 1) → ¬ 𝑛 = 0)
8281ad2antlr 727 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑛 = 0)
83 eqeq1 2741 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
8483notbid 318 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑛 → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
8584adantl 481 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
8682, 85mpbird 257 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑙 = 0)
8786iffalsed 4536 . . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . . 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 2851 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → 1 ∈ (ℤ‘0))
92 eluzfz2 13572 . . . . . . . . . . . . . . . . . . 19 (1 ∈ (ℤ‘0) → 1 ∈ (0...1))
9391, 92syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → 1 ∈ (0...1))
94 eleq1 2829 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (𝑛 ∈ (0...1) ↔ 1 ∈ (0...1)))
9593, 94mpbird 257 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → 𝑛 ∈ (0...1))
9676, 95sylbi 217 . . . . . . . . . . . . . . . 16 (𝑛 = (0 + 1) → 𝑛 ∈ (0...1))
9796adantl 481 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ (0...1))
98 fvexd 6921 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (0g𝐴) ∈ V)
9940, 89, 97, 98fvmptd2 7024 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝐻𝑛) = (0g𝐴))
10099fveq2d 6910 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻𝑛)) = (𝑇‘(0g𝐴)))
10123fveq2i 6909 . . . . . . . . . . . . . . . 16 (0g𝐴) = (0g‘(𝑁 Mat 𝑅))
1023fveq2i 6909 . . . . . . . . . . . . . . . 16 (0g𝐶) = (0g‘(𝑁 Mat 𝑃))
10325, 2, 101, 1020mat2pmat 22742 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘(0g𝐴)) = (0g𝐶))
104103ancoms 458 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(0g𝐴)) = (0g𝐶))
105104ad2antrr 726 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(0g𝐴)) = (0g𝐶))
106100, 105eqtrd 2777 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻𝑛)) = (0g𝐶))
107106oveq2d 7447 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) = ((𝑛 𝑋) (0g𝐶)))
1082, 3pmatlmod 22699 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
109108ad2antrr 726 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝐶 ∈ LMod)
110 simpllr 776 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝑅 ∈ Ring)
111 eleq1 2829 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (𝑛 ∈ ℕ0 ↔ 1 ∈ ℕ0))
11290, 111mpbird 257 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → 𝑛 ∈ ℕ0)
11376, 112sylbi 217 . . . . . . . . . . . . . . 15 (𝑛 = (0 + 1) → 𝑛 ∈ ℕ0)
114113adantl 481 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ ℕ0)
115 eqid 2737 . . . . . . . . . . . . . . 15 (mulGrp‘𝑃) = (mulGrp‘𝑃)
116 eqid 2737 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
1172, 28, 115, 27, 116ply1moncl 22274 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ (Base‘𝑃))
118110, 114, 117syl2anc 584 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 𝑋) ∈ (Base‘𝑃))
1192ply1ring 22249 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
1203matsca2 22426 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
121119, 120sylan2 593 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
122121eqcomd 2743 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐶) = 𝑃)
123122fveq2d 6910 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
124123eleq2d 2827 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 𝑋) ∈ (Base‘𝑃)))
125124ad2antrr 726 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 𝑋) ∈ (Base‘𝑃)))
126118, 125mpbird 257 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)))
127 eqid 2737 . . . . . . . . . . . . 13 (Scalar‘𝐶) = (Scalar‘𝐶)
128 eqid 2737 . . . . . . . . . . . . 13 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
129127, 26, 128, 34lmodvs0 20894 . . . . . . . . . . . 12 ((𝐶 ∈ LMod ∧ (𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 𝑋) (0g𝐶)) = (0g𝐶))
130109, 126, 129syl2anc 584 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (0g𝐶)) = (0g𝐶))
131107, 130eqtrd 2777 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) = (0g𝐶))
1328, 7, 71, 74, 131gsumsnd 19970 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = (0g𝐶))
133132eqcomd 2743 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (0g𝐶) = (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))))
13470, 133oveq12d 7449 . . . . . . 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 2779 . . . . . 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 480 . . . . 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 2777 . . . 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 1110 . . 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 775 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑁 ∈ Fin)
141 simpllr 776 . . . . . 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 11255 . . . . . . . . . . . . . . 15 0 ∈ V
144143snid 4662 . . . . . . . . . . . . . 14 0 ∈ {0}
145144a1i 11 . . . . . . . . . . . . 13 (𝐺:{0}⟶𝐷 → 0 ∈ {0})
146142, 145ffvelcdmd 7105 . . . . . . . . . . . 12 (𝐺:{0}⟶𝐷 → (𝐺‘0) ∈ 𝐷)
14716, 146syl 17 . . . . . . . . . . 11 (𝐺 ∈ (𝐷m {0}) → (𝐺‘0) ∈ 𝐷)
148147ad2antlr 727 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → (𝐺‘0) ∈ 𝐷)
14923matring 22449 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
15024, 88ring0cl 20264 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 0𝐷)
151149, 150syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0𝐷)
152151ad2antrr 726 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → 0𝐷)
153148, 152ifcld 4572 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝐺‘0), 0 ) ∈ 𝐷)
154153, 40fmptd 7134 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐻:(0...1)⟶𝐷)
15575oveq2i 7442 . . . . . . . . 9 (0...(0 + 1)) = (0...1)
156155feq2i 6728 . . . . . . . 8 (𝐻:(0...(0 + 1))⟶𝐷𝐻:(0...1)⟶𝐷)
157154, 156sylibr 234 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐻:(0...(0 + 1))⟶𝐷)
158157ffvelcdmda 7104 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → (𝐻𝑛) ∈ 𝐷)
159 elfznn0 13660 . . . . . . 7 (𝑛 ∈ (0...(0 + 1)) → 𝑛 ∈ ℕ0)
160159adantl 481 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑛 ∈ ℕ0)
16123, 24, 25, 2, 3, 8, 26, 27, 28mat2pmatscmxcl 22746 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐻𝑛) ∈ 𝐷𝑛 ∈ ℕ0)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) ∈ 𝐵)
162140, 141, 158, 160, 161syl22anc 839 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) ∈ 𝐵)
1638, 33, 11, 139, 162gsummptfzsplit 19950 . . . 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 1133 . . 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 2780 . 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 5238 . . 3 (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))
167166oveq2i 7442 . 2 (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))
168165, 167eqtrdi 2793 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  ifcif 4525  {csn 4626  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  Fincfn 8985  0cc0 11155  1c1 11156   + caddc 11158  0cn0 12526  cuz 12878  ...cfz 13547  Basecbs 17247  +gcplusg 17297  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484   Σg cgsu 17485  Mndcmnd 18747  .gcmg 19085  CMndccmn 19798  mulGrpcmgp 20137  Ringcrg 20230  LModclmod 20858  var1cv1 22177  Poly1cpl1 22178   Mat cmat 22411   matToPolyMat cmat2pmat 22710
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-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-mamu 22395  df-mat 22412  df-mat2pmat 22713
This theorem is referenced by:  pmatcollpw3fi1lem2  22793
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