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Theorem pmatcollpw3fi1lem1 21324
Description: Lemma 1 for pmatcollpw3fi1 21326. (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 21231 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
5 ringmnd 19237 . . . . . . . . . 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 19262 . . . . . . . . . . 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 8583 . . . . . . . . . 10 {0} ∈ Fin
1312a1i 11 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → {0} ∈ Fin)
14 simplll 771 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑁 ∈ Fin)
15 simpllr 772 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → 𝑅 ∈ Ring)
16 elmapi 8418 . . . . . . . . . . . . 13 (𝐺 ∈ (𝐷m {0}) → 𝐺:{0}⟶𝐷)
1716adantl 482 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐺:{0}⟶𝐷)
1817ffvelrnda 6844 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
19 elsni 4576 . . . . . . . . . . . . 13 (𝑛 ∈ {0} → 𝑛 = 0)
20 0nn0 11901 . . . . . . . . . . . . 13 0 ∈ ℕ0
2119, 20syl6eqel 2921 . . . . . . . . . . . 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 21278 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐺𝑛) ∈ 𝐷𝑛 ∈ ℕ0)) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
3014, 15, 18, 22, 29syl22anc 834 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
3130ralrimiva 3182 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → ∀𝑛 ∈ {0} ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
328, 11, 13, 31gsummptcl 19018 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))) ∈ 𝐵)
33 eqid 2821 . . . . . . . . 9 (+g𝐶) = (+g𝐶)
34 eqid 2821 . . . . . . . . 9 (0g𝐶) = (0g𝐶)
358, 33, 34mndrid 17922 . . . . . . . 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 12997 . . . . . . . . . . . 12 (0...0) = {0}
3837eqcomi 2830 . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑛 = 0)
4341, 42eqtrd 2856 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 0)
4443iftrued 4473 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘0))
45 fveq2 6664 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → (𝐺𝑛) = (𝐺‘0))
4645eqcomd 2827 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → (𝐺‘0) = (𝐺𝑛))
4719, 46syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {0} → (𝐺‘0) = (𝐺𝑛))
4847ad2antlr 723 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → (𝐺‘0) = (𝐺𝑛))
4944, 48eqtrd 2856 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺𝑛))
50 1nn0 11902 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℕ0
5150a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 0 → 1 ∈ ℕ0)
52 nn0uz 12269 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
5351, 52eleqtrdi 2923 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → 1 ∈ (ℤ‘0))
54 eluzfz1 12904 . . . . . . . . . . . . . . . . . 18 (1 ∈ (ℤ‘0) → 0 ∈ (0...1))
5553, 54syl 17 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → 0 ∈ (0...1))
56 eleq1 2900 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → (𝑛 ∈ (0...1) ↔ 0 ∈ (0...1)))
5755, 56mpbird 258 . . . . . . . . . . . . . . . 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 ffvelrn 6842 . . . . . . . . . . . . . . . . . 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 6769 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐻𝑛) = (𝐺𝑛))
6665eqcomd 2827 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) = (𝐻𝑛))
6766fveq2d 6668 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → (𝑇‘(𝐺𝑛)) = (𝑇‘(𝐻𝑛)))
6867oveq2d 7161 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) = ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))
6939, 68mpteq12dva 5142 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))) = (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))
7069oveq2d 7161 . . . . . . . 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 7180 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (0 + 1) ∈ V)
728, 34mndidcl 17916 . . . . . . . . . . . 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 11748 . . . . . . . . . . . . . . . . . . . . 21 (0 + 1) = 1
7675eqeq2i 2834 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = (0 + 1) ↔ 𝑛 = 1)
77 ax-1ne0 10595 . . . . . . . . . . . . . . . . . . . . . 22 1 ≠ 0
7877neii 3018 . . . . . . . . . . . . . . . . . . . . 21 ¬ 1 = 0
79 eqeq1 2825 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → (𝑛 = 0 ↔ 1 = 0))
8078, 79mtbiri 328 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → ¬ 𝑛 = 0)
8176, 80sylbi 218 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (0 + 1) → ¬ 𝑛 = 0)
8281ad2antlr 723 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑛 = 0)
83 eqeq1 2825 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
8483notbid 319 . . . . . . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑙 = 0)
8786iffalsed 4476 . . . . . . . . . . . . . . . 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, 88syl6eq 2872 . . . . . . . . . . . . . . 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 2923 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → 1 ∈ (ℤ‘0))
92 eluzfz2 12905 . . . . . . . . . . . . . . . . . . 19 (1 ∈ (ℤ‘0) → 1 ∈ (0...1))
9391, 92syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → 1 ∈ (0...1))
94 eleq1 2900 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (𝑛 ∈ (0...1) ↔ 1 ∈ (0...1)))
9593, 94mpbird 258 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → 𝑛 ∈ (0...1))
9676, 95sylbi 218 . . . . . . . . . . . . . . . 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 6679 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (0g𝐴) ∈ V)
9940, 89, 97, 98fvmptd2 6769 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝐻𝑛) = (0g𝐴))
10099fveq2d 6668 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻𝑛)) = (𝑇‘(0g𝐴)))
10123fveq2i 6667 . . . . . . . . . . . . . . . 16 (0g𝐴) = (0g‘(𝑁 Mat 𝑅))
1023fveq2i 6667 . . . . . . . . . . . . . . . 16 (0g𝐶) = (0g‘(𝑁 Mat 𝑃))
10325, 2, 101, 1020mat2pmat 21274 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘(0g𝐴)) = (0g𝐶))
104103ancoms 459 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(0g𝐴)) = (0g𝐶))
105104ad2antrr 722 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(0g𝐴)) = (0g𝐶))
106100, 105eqtrd 2856 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻𝑛)) = (0g𝐶))
107106oveq2d 7161 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) = ((𝑛 𝑋) (0g𝐶)))
1082, 3pmatlmod 21232 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
109108ad2antrr 722 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝐶 ∈ LMod)
110 simpllr 772 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝑅 ∈ Ring)
111 eleq1 2900 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (𝑛 ∈ ℕ0 ↔ 1 ∈ ℕ0))
11290, 111mpbird 258 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → 𝑛 ∈ ℕ0)
11376, 112sylbi 218 . . . . . . . . . . . . . . 15 (𝑛 = (0 + 1) → 𝑛 ∈ ℕ0)
114113adantl 482 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ ℕ0)
115 eqid 2821 . . . . . . . . . . . . . . 15 (mulGrp‘𝑃) = (mulGrp‘𝑃)
116 eqid 2821 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
1172, 28, 115, 27, 116ply1moncl 20369 . . . . . . . . . . . . . 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 20346 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
1203matsca2 20959 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
121119, 120sylan2 592 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
122121eqcomd 2827 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐶) = 𝑃)
123122fveq2d 6668 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
124123eleq2d 2898 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 𝑋) ∈ (Base‘𝑃)))
125124ad2antrr 722 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 𝑋) ∈ (Base‘𝑃)))
126118, 125mpbird 258 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)))
127 eqid 2821 . . . . . . . . . . . . 13 (Scalar‘𝐶) = (Scalar‘𝐶)
128 eqid 2821 . . . . . . . . . . . . 13 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
129127, 26, 128, 34lmodvs0 19599 . . . . . . . . . . . 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 2856 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) = (0g𝐶))
1328, 7, 71, 74, 131gsumsnd 19003 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = (0g𝐶))
133132eqcomd 2827 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → (0g𝐶) = (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))))
13470, 133oveq12d 7163 . . . . . . 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 2858 . . . . . 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 2856 . . . 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 1102 . . 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 771 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑁 ∈ Fin)
141 simpllr 772 . . . . . 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 10624 . . . . . . . . . . . . . . 15 0 ∈ V
144143snid 4593 . . . . . . . . . . . . . 14 0 ∈ {0}
145144a1i 11 . . . . . . . . . . . . 13 (𝐺:{0}⟶𝐷 → 0 ∈ {0})
146142, 145ffvelrnd 6845 . . . . . . . . . . . 12 (𝐺:{0}⟶𝐷 → (𝐺‘0) ∈ 𝐷)
14716, 146syl 17 . . . . . . . . . . 11 (𝐺 ∈ (𝐷m {0}) → (𝐺‘0) ∈ 𝐷)
148147ad2antlr 723 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → (𝐺‘0) ∈ 𝐷)
14923matring 20982 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
15024, 88ring0cl 19250 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 0𝐷)
151149, 150syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0𝐷)
152151ad2antrr 722 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → 0𝐷)
153148, 152ifcld 4510 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝐺‘0), 0 ) ∈ 𝐷)
154153, 40fmptd 6871 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐻:(0...1)⟶𝐷)
15575oveq2i 7156 . . . . . . . . 9 (0...(0 + 1)) = (0...1)
156155feq2i 6500 . . . . . . . 8 (𝐻:(0...(0 + 1))⟶𝐷𝐻:(0...1)⟶𝐷)
157154, 156sylibr 235 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) → 𝐻:(0...(0 + 1))⟶𝐷)
158157ffvelrnda 6844 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → (𝐻𝑛) ∈ 𝐷)
159 elfznn0 12990 . . . . . . 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 21278 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐻𝑛) ∈ 𝐷𝑛 ∈ ℕ0)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) ∈ 𝐵)
162140, 141, 158, 160, 161syl22anc 834 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) ∈ 𝐵)
1638, 33, 11, 139, 162gsummptfzsplit 18983 . . . 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 1124 . . 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 2859 . 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 5148 . . 3 (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))
167166oveq2i 7156 . 2 (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))
168165, 167syl6eq 2872 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  Vcvv 3495  ifcif 4465  {csn 4559  cmpt 5138  wf 6345  cfv 6349  (class class class)co 7145  m cmap 8396  Fincfn 8498  0cc0 10526  1c1 10527   + caddc 10529  0cn0 11886  cuz 12232  ...cfz 12882  Basecbs 16473  +gcplusg 16555  Scalarcsca 16558   ·𝑠 cvsca 16559  0gc0g 16703   Σg cgsu 16704  Mndcmnd 17901  .gcmg 18164  CMndccmn 18837  mulGrpcmgp 19170  Ringcrg 19228  LModclmod 19565  var1cv1 20274  Poly1cpl1 20275   Mat cmat 20946   matToPolyMat cmat2pmat 21242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-ot 4568  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-ofr 7399  df-om 7569  df-1st 7680  df-2nd 7681  df-supp 7822  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-2o 8094  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-ixp 8451  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fsupp 8823  df-sup 8895  df-oi 8963  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-fzo 13024  df-seq 13360  df-hash 13681  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-mulr 16569  df-sca 16571  df-vsca 16572  df-ip 16573  df-tset 16574  df-ple 16575  df-ds 16577  df-hom 16579  df-cco 16580  df-0g 16705  df-gsum 16706  df-prds 16711  df-pws 16713  df-mre 16847  df-mrc 16848  df-acs 16850  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-mhm 17946  df-submnd 17947  df-grp 18046  df-minusg 18047  df-sbg 18048  df-mulg 18165  df-subg 18216  df-ghm 18296  df-cntz 18387  df-cmn 18839  df-abl 18840  df-mgp 19171  df-ur 19183  df-ring 19230  df-subrg 19464  df-lmod 19567  df-lss 19635  df-sra 19875  df-rgmod 19876  df-ascl 20017  df-psr 20066  df-mvr 20067  df-mpl 20068  df-opsr 20070  df-psr1 20278  df-vr1 20279  df-ply1 20280  df-dsmm 20806  df-frlm 20821  df-mamu 20925  df-mat 20947  df-mat2pmat 21245
This theorem is referenced by:  pmatcollpw3fi1lem2  21325
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