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Theorem pmatcollpw3fi1lem1 22135
Description: Lemma 1 for pmatcollpw3fi1 22137. (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 485 . . . . 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 22041 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
5 ringmnd 19974 . . . . . . . . . 10 (𝐶 ∈ Ring → 𝐶 ∈ Mnd)
64, 5syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Mnd)
76adantr 481 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐶 ∈ Mnd)
8 pmatcollpw.b . . . . . . . . 9 𝐵 = (Base‘𝐶)
9 ringcmn 20003 . . . . . . . . . . 11 (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
104, 9syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ CMnd)
1110adantr 481 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐶 ∈ CMnd)
12 snfi 8988 . . . . . . . . . 10 {0} ∈ Fin
1312a1i 11 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → {0} ∈ Fin)
14 simplll 773 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑁 ∈ Fin)
15 simpllr 774 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑅 ∈ Ring)
16 elmapi 8787 . . . . . . . . . . . . 13 (𝐺 ∈ (𝐷m {0}) → 𝐺:{0}⟶𝐷)
1716adantl 482 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐺:{0}⟶𝐷)
1817ffvelcdmda 7035 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
19 elsni 4603 . . . . . . . . . . . . 13 (𝑛 ∈ {0} → 𝑛 = 0)
20 0nn0 12428 . . . . . . . . . . . . 13 0 ∈ ℕ0
2119, 20eqeltrdi 2846 . . . . . . . . . . . 12 (𝑛 ∈ {0} → 𝑛 ∈ ℕ0)
2221adantl 482 . . . . . . . . . . 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 22089 . . . . . . . . . . 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 3143 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → ∀𝑛 ∈ {0} ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
328, 11, 13, 31gsummptcl 19744 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))) ∈ 𝐵)
33 eqid 2736 . . . . . . . . 9 (+g𝐶) = (+g𝐶)
34 eqid 2736 . . . . . . . . 9 (0g𝐶) = (0g𝐶)
358, 33, 34mndrid 18577 . . . . . . . 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 13541 . . . . . . . . . . . 12 (0...0) = {0}
3837eqcomi 2745 . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 𝑛)
4219ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑛 = 0)
4341, 42eqtrd 2776 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 0)
4443iftrued 4494 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘0))
45 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → (𝐺𝑛) = (𝐺‘0))
4645eqcomd 2742 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → (𝐺‘0) = (𝐺𝑛))
4719, 46syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {0} → (𝐺‘0) = (𝐺𝑛))
4847ad2antlr 725 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → (𝐺‘0) = (𝐺𝑛))
4944, 48eqtrd 2776 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺𝑛))
50 1nn0 12429 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℕ0
5150a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 0 → 1 ∈ ℕ0)
52 nn0uz 12805 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
5351, 52eleqtrdi 2848 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → 1 ∈ (ℤ‘0))
54 eluzfz1 13448 . . . . . . . . . . . . . . . . . 18 (1 ∈ (ℤ‘0) → 0 ∈ (0...1))
5553, 54syl 17 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → 0 ∈ (0...1))
56 eleq1 2825 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → (𝑛 ∈ (0...1) ↔ 0 ∈ (0...1)))
5755, 56mpbird 256 . . . . . . . . . . . . . . . 16 (𝑛 = 0 → 𝑛 ∈ (0...1))
5819, 57syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ {0} → 𝑛 ∈ (0...1))
5958adantl 482 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑛 ∈ (0...1))
60 ffvelcdm 7032 . . . . . . . . . . . . . . . . . 18 ((𝐺:{0}⟶𝐷𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
6160ex 413 . . . . . . . . . . . . . . . . 17 (𝐺:{0}⟶𝐷 → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6216, 61syl 17 . . . . . . . . . . . . . . . 16 (𝐺 ∈ (𝐷m {0}) → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6362adantl 482 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6463imp 407 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
6540, 49, 59, 64fvmptd2 6956 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐻𝑛) = (𝐺𝑛))
6665eqcomd 2742 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) = (𝐻𝑛))
6766fveq2d 6846 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝑇‘(𝐺𝑛)) = (𝑇‘(𝐻𝑛)))
6867oveq2d 7373 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) = ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))
6939, 68mpteq12dva 5194 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))) = (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))
7069oveq2d 7373 . . . . . . . 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 7392 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (0 + 1) ∈ V)
728, 34mndidcl 18571 . . . . . . . . . . . 12 (𝐶 ∈ Mnd → (0g𝐶) ∈ 𝐵)
736, 72syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐶) ∈ 𝐵)
7473adantr 481 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (0g𝐶) ∈ 𝐵)
75 0p1e1 12275 . . . . . . . . . . . . . . . . . . . . 21 (0 + 1) = 1
7675eqeq2i 2749 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = (0 + 1) ↔ 𝑛 = 1)
77 ax-1ne0 11120 . . . . . . . . . . . . . . . . . . . . . 22 1 ≠ 0
7877neii 2945 . . . . . . . . . . . . . . . . . . . . 21 ¬ 1 = 0
79 eqeq1 2740 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → (𝑛 = 0 ↔ 1 = 0))
8078, 79mtbiri 326 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → ¬ 𝑛 = 0)
8176, 80sylbi 216 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (0 + 1) → ¬ 𝑛 = 0)
8281ad2antlr 725 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑛 = 0)
83 eqeq1 2740 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
8483notbid 317 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑛 → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
8584adantl 482 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
8682, 85mpbird 256 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑙 = 0)
8786iffalsed 4497 . . . . . . . . . . . . . . . 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 2792 . . . . . . . . . . . . . . 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 2848 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → 1 ∈ (ℤ‘0))
92 eluzfz2 13449 . . . . . . . . . . . . . . . . . . 19 (1 ∈ (ℤ‘0) → 1 ∈ (0...1))
9391, 92syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → 1 ∈ (0...1))
94 eleq1 2825 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (𝑛 ∈ (0...1) ↔ 1 ∈ (0...1)))
9593, 94mpbird 256 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → 𝑛 ∈ (0...1))
9676, 95sylbi 216 . . . . . . . . . . . . . . . 16 (𝑛 = (0 + 1) → 𝑛 ∈ (0...1))
9796adantl 482 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ (0...1))
98 fvexd 6857 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (0g𝐴) ∈ V)
9940, 89, 97, 98fvmptd2 6956 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝐻𝑛) = (0g𝐴))
10099fveq2d 6846 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻𝑛)) = (𝑇‘(0g𝐴)))
10123fveq2i 6845 . . . . . . . . . . . . . . . 16 (0g𝐴) = (0g‘(𝑁 Mat 𝑅))
1023fveq2i 6845 . . . . . . . . . . . . . . . 16 (0g𝐶) = (0g‘(𝑁 Mat 𝑃))
10325, 2, 101, 1020mat2pmat 22085 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘(0g𝐴)) = (0g𝐶))
104103ancoms 459 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(0g𝐴)) = (0g𝐶))
105104ad2antrr 724 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(0g𝐴)) = (0g𝐶))
106100, 105eqtrd 2776 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻𝑛)) = (0g𝐶))
107106oveq2d 7373 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) = ((𝑛 𝑋) (0g𝐶)))
1082, 3pmatlmod 22042 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
109108ad2antrr 724 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝐶 ∈ LMod)
110 simpllr 774 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝑅 ∈ Ring)
111 eleq1 2825 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (𝑛 ∈ ℕ0 ↔ 1 ∈ ℕ0))
11290, 111mpbird 256 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → 𝑛 ∈ ℕ0)
11376, 112sylbi 216 . . . . . . . . . . . . . . 15 (𝑛 = (0 + 1) → 𝑛 ∈ ℕ0)
114113adantl 482 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ ℕ0)
115 eqid 2736 . . . . . . . . . . . . . . 15 (mulGrp‘𝑃) = (mulGrp‘𝑃)
116 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
1172, 28, 115, 27, 116ply1moncl 21642 . . . . . . . . . . . . . 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 21619 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
1203matsca2 21769 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
121119, 120sylan2 593 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
122121eqcomd 2742 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐶) = 𝑃)
123122fveq2d 6846 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
124123eleq2d 2823 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 𝑋) ∈ (Base‘𝑃)))
125124ad2antrr 724 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 𝑋) ∈ (Base‘𝑃)))
126118, 125mpbird 256 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)))
127 eqid 2736 . . . . . . . . . . . . 13 (Scalar‘𝐶) = (Scalar‘𝐶)
128 eqid 2736 . . . . . . . . . . . . 13 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
129127, 26, 128, 34lmodvs0 20356 . . . . . . . . . . . 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 2776 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) = (0g𝐶))
1328, 7, 71, 74, 131gsumsnd 19729 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = (0g𝐶))
133132eqcomd 2742 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (0g𝐶) = (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))))
13470, 133oveq12d 7375 . . . . . . 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 2778 . . . . . 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 481 . . . . 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 2776 . . . 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 773 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑁 ∈ Fin)
141 simpllr 774 . . . . . 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 11149 . . . . . . . . . . . . . . 15 0 ∈ V
144143snid 4622 . . . . . . . . . . . . . 14 0 ∈ {0}
145144a1i 11 . . . . . . . . . . . . 13 (𝐺:{0}⟶𝐷 → 0 ∈ {0})
146142, 145ffvelcdmd 7036 . . . . . . . . . . . 12 (𝐺:{0}⟶𝐷 → (𝐺‘0) ∈ 𝐷)
14716, 146syl 17 . . . . . . . . . . 11 (𝐺 ∈ (𝐷m {0}) → (𝐺‘0) ∈ 𝐷)
148147ad2antlr 725 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → (𝐺‘0) ∈ 𝐷)
14923matring 21792 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
15024, 88ring0cl 19990 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 0𝐷)
151149, 150syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0𝐷)
152151ad2antrr 724 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → 0𝐷)
153148, 152ifcld 4532 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝐺‘0), 0 ) ∈ 𝐷)
154153, 40fmptd 7062 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐻:(0...1)⟶𝐷)
15575oveq2i 7368 . . . . . . . . 9 (0...(0 + 1)) = (0...1)
156155feq2i 6660 . . . . . . . 8 (𝐻:(0...(0 + 1))⟶𝐷𝐻:(0...1)⟶𝐷)
157154, 156sylibr 233 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐻:(0...(0 + 1))⟶𝐷)
158157ffvelcdmda 7035 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → (𝐻𝑛) ∈ 𝐷)
159 elfznn0 13534 . . . . . . 7 (𝑛 ∈ (0...(0 + 1)) → 𝑛 ∈ ℕ0)
160159adantl 482 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑛 ∈ ℕ0)
16123, 24, 25, 2, 3, 8, 26, 27, 28mat2pmatscmxcl 22089 . . . . . 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 19709 . . . 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 1132 . . 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 2779 . 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 5201 . . 3 (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))
167166oveq2i 7368 . 2 (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))
168165, 167eqtrdi 2792 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  ifcif 4486  {csn 4586  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  Fincfn 8883  0cc0 11051  1c1 11052   + caddc 11054  0cn0 12413  cuz 12763  ...cfz 13424  Basecbs 17083  +gcplusg 17133  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321   Σg cgsu 17322  Mndcmnd 18556  .gcmg 18872  CMndccmn 19562  mulGrpcmgp 19896  Ringcrg 19964  LModclmod 20322  var1cv1 21547  Poly1cpl1 21548   Mat cmat 21754   matToPolyMat cmat2pmat 22053
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-subrg 20220  df-lmod 20324  df-lss 20393  df-sra 20633  df-rgmod 20634  df-dsmm 21138  df-frlm 21153  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-vr1 21552  df-ply1 21553  df-mamu 21733  df-mat 21755  df-mat2pmat 22056
This theorem is referenced by:  pmatcollpw3fi1lem2  22136
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