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Theorem pmatcollpw3fi1lem1 20639
 Description: Lemma 1 for pmatcollpw3fi1 20641. (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 476 . . . . 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 20546 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
5 ringmnd 18602 . . . . . . . . . 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 18627 . . . . . . . . . . 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 8079 . . . . . . . . . 10 {0} ∈ Fin
1312a1i 11 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → {0} ∈ Fin)
14 simplll 813 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) → 𝑁 ∈ Fin)
15 simpllr 815 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) → 𝑅 ∈ Ring)
16 elmapi 7921 . . . . . . . . . . . . 13 (𝐺 ∈ (𝐷𝑚 {0}) → 𝐺:{0}⟶𝐷)
1716adantl 481 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → 𝐺:{0}⟶𝐷)
1817ffvelrnda 6399 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
19 elsni 4227 . . . . . . . . . . . . 13 (𝑛 ∈ {0} → 𝑛 = 0)
20 0nn0 11345 . . . . . . . . . . . . 13 0 ∈ ℕ0
2119, 20syl6eqel 2738 . . . . . . . . . . . 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 20593 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐺𝑛) ∈ 𝐷𝑛 ∈ ℕ0)) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
3014, 15, 18, 22, 29syl22anc 1367 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
3130ralrimiva 2995 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → ∀𝑛 ∈ {0} ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) ∈ 𝐵)
328, 11, 13, 31gsummptcl 18412 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛))))) ∈ 𝐵)
33 eqid 2651 . . . . . . . . 9 (+g𝐶) = (+g𝐶)
34 eqid 2651 . . . . . . . . 9 (0g𝐶) = (0g𝐶)
358, 33, 34mndrid 17359 . . . . . . . 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 694 . . . . . . 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 0z 11426 . . . . . . . . . . . . 13 0 ∈ ℤ
38 fzsn 12421 . . . . . . . . . . . . 13 (0 ∈ ℤ → (0...0) = {0})
3937, 38ax-mp 5 . . . . . . . . . . . 12 (0...0) = {0}
4039eqcomi 2660 . . . . . . . . . . 11 {0} = (0...0)
4140a1i 11 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → {0} = (0...0))
42 pmatcollpw3fi1lem1.h . . . . . . . . . . . . . . 15 𝐻 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝐺‘0), 0 ))
4342a1i 11 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) → 𝐻 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝐺‘0), 0 )))
44 simpr 476 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 𝑛)
4519ad2antlr 763 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑛 = 0)
4644, 45eqtrd 2685 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 0)
4746iftrued 4127 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘0))
48 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → (𝐺𝑛) = (𝐺‘0))
4948eqcomd 2657 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → (𝐺‘0) = (𝐺𝑛))
5019, 49syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {0} → (𝐺‘0) = (𝐺𝑛))
5150ad2antlr 763 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → (𝐺‘0) = (𝐺𝑛))
5247, 51eqtrd 2685 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺𝑛))
53 1nn0 11346 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℕ0
5453a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 0 → 1 ∈ ℕ0)
55 nn0uz 11760 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
5654, 55syl6eleq 2740 . . . . . . . . . . . . . . . . . 18 (𝑛 = 0 → 1 ∈ (ℤ‘0))
57 eluzfz1 12386 . . . . . . . . . . . . . . . . . 18 (1 ∈ (ℤ‘0) → 0 ∈ (0...1))
5856, 57syl 17 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → 0 ∈ (0...1))
59 eleq1 2718 . . . . . . . . . . . . . . . . 17 (𝑛 = 0 → (𝑛 ∈ (0...1) ↔ 0 ∈ (0...1)))
6058, 59mpbird 247 . . . . . . . . . . . . . . . 16 (𝑛 = 0 → 𝑛 ∈ (0...1))
6119, 60syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ {0} → 𝑛 ∈ (0...1))
6261adantl 481 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) → 𝑛 ∈ (0...1))
63 ffvelrn 6397 . . . . . . . . . . . . . . . . . 18 ((𝐺:{0}⟶𝐷𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
6463ex 449 . . . . . . . . . . . . . . . . 17 (𝐺:{0}⟶𝐷 → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6516, 64syl 17 . . . . . . . . . . . . . . . 16 (𝐺 ∈ (𝐷𝑚 {0}) → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6665adantl 481 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → (𝑛 ∈ {0} → (𝐺𝑛) ∈ 𝐷))
6766imp 444 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) ∈ 𝐷)
6843, 52, 62, 67fvmptd 6327 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) → (𝐻𝑛) = (𝐺𝑛))
6968eqcomd 2657 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) → (𝐺𝑛) = (𝐻𝑛))
7069fveq2d 6233 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) → (𝑇‘(𝐺𝑛)) = (𝑇‘(𝐻𝑛)))
7170oveq2d 6706 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 𝑋) (𝑇‘(𝐺𝑛))) = ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))
7241, 71mpteq12dva 4765 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))) = (𝑛 ∈ (0...0) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))
7372oveq2d 6706 . . . . . . . 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))))))
74 ovexd 6720 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → (0 + 1) ∈ V)
758, 34mndidcl 17355 . . . . . . . . . . . 12 (𝐶 ∈ Mnd → (0g𝐶) ∈ 𝐵)
766, 75syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐶) ∈ 𝐵)
7776adantr 480 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → (0g𝐶) ∈ 𝐵)
7842a1i 11 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → 𝐻 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝐺‘0), 0 )))
79 0p1e1 11170 . . . . . . . . . . . . . . . . . . . . 21 (0 + 1) = 1
8079eqeq2i 2663 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = (0 + 1) ↔ 𝑛 = 1)
81 ax-1ne0 10043 . . . . . . . . . . . . . . . . . . . . . 22 1 ≠ 0
8281neii 2825 . . . . . . . . . . . . . . . . . . . . 21 ¬ 1 = 0
83 eqeq1 2655 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → (𝑛 = 0 ↔ 1 = 0))
8482, 83mtbiri 316 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → ¬ 𝑛 = 0)
8580, 84sylbi 207 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (0 + 1) → ¬ 𝑛 = 0)
8685ad2antlr 763 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑛 = 0)
87 eqeq1 2655 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
8887notbid 307 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑛 → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
8988adantl 481 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
9086, 89mpbird 247 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑙 = 0)
9190iffalsed 4130 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = 0 )
92 pmatcollpw3fi1lem1.0 . . . . . . . . . . . . . . . 16 0 = (0g𝐴)
9391, 92syl6eq 2701 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (0g𝐴))
9453a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → 1 ∈ ℕ0)
9594, 55syl6eleq 2740 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → 1 ∈ (ℤ‘0))
96 eluzfz2 12387 . . . . . . . . . . . . . . . . . . 19 (1 ∈ (ℤ‘0) → 1 ∈ (0...1))
9795, 96syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → 1 ∈ (0...1))
98 eleq1 2718 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (𝑛 ∈ (0...1) ↔ 1 ∈ (0...1)))
9997, 98mpbird 247 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → 𝑛 ∈ (0...1))
10080, 99sylbi 207 . . . . . . . . . . . . . . . 16 (𝑛 = (0 + 1) → 𝑛 ∈ (0...1))
101100adantl 481 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ (0...1))
102 fvexd 6241 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → (0g𝐴) ∈ V)
10378, 93, 101, 102fvmptd 6327 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → (𝐻𝑛) = (0g𝐴))
104103fveq2d 6233 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻𝑛)) = (𝑇‘(0g𝐴)))
10523fveq2i 6232 . . . . . . . . . . . . . . . 16 (0g𝐴) = (0g‘(𝑁 Mat 𝑅))
1063fveq2i 6232 . . . . . . . . . . . . . . . 16 (0g𝐶) = (0g‘(𝑁 Mat 𝑃))
10725, 2, 105, 1060mat2pmat 20589 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘(0g𝐴)) = (0g𝐶))
108107ancoms 468 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(0g𝐴)) = (0g𝐶))
109108ad2antrr 762 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(0g𝐴)) = (0g𝐶))
110104, 109eqtrd 2685 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻𝑛)) = (0g𝐶))
111110oveq2d 6706 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) = ((𝑛 𝑋) (0g𝐶)))
1122, 3pmatlmod 20547 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
113112ad2antrr 762 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → 𝐶 ∈ LMod)
114 simpllr 815 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → 𝑅 ∈ Ring)
115 eleq1 2718 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (𝑛 ∈ ℕ0 ↔ 1 ∈ ℕ0))
11694, 115mpbird 247 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → 𝑛 ∈ ℕ0)
11780, 116sylbi 207 . . . . . . . . . . . . . . 15 (𝑛 = (0 + 1) → 𝑛 ∈ ℕ0)
118117adantl 481 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ ℕ0)
119 eqid 2651 . . . . . . . . . . . . . . 15 (mulGrp‘𝑃) = (mulGrp‘𝑃)
120 eqid 2651 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
1212, 28, 119, 27, 120ply1moncl 19689 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ (Base‘𝑃))
122114, 118, 121syl2anc 694 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 𝑋) ∈ (Base‘𝑃))
1232ply1ring 19666 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
1243matsca2 20274 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
125123, 124sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
126125eqcomd 2657 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐶) = 𝑃)
127126fveq2d 6233 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
128127eleq2d 2716 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 𝑋) ∈ (Base‘𝑃)))
129128ad2antrr 762 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 𝑋) ∈ (Base‘𝑃)))
130122, 129mpbird 247 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶)))
131 eqid 2651 . . . . . . . . . . . . 13 (Scalar‘𝐶) = (Scalar‘𝐶)
132 eqid 2651 . . . . . . . . . . . . 13 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
133131, 26, 132, 34lmodvs0 18945 . . . . . . . . . . . 12 ((𝐶 ∈ LMod ∧ (𝑛 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 𝑋) (0g𝐶)) = (0g𝐶))
134113, 130, 133syl2anc 694 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (0g𝐶)) = (0g𝐶))
135111, 134eqtrd 2685 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) = (0g𝐶))
1368, 7, 74, 77, 135gsumsnd 18398 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = (0g𝐶))
137136eqcomd 2657 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → (0g𝐶) = (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))))
13873, 137oveq12d 6708 . . . . . . 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)))))))
13936, 138eqtr3d 2687 . . . . . 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)))))))
140139adantr 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)))))))
1411, 140eqtrd 2685 . . . 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)))))))
1421413impa 1278 . . 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)))))))
14320a1i 11 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → 0 ∈ ℕ0)
144 simplll 813 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑁 ∈ Fin)
145 simpllr 815 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑅 ∈ Ring)
146 id 22 . . . . . . . . . . . . 13 (𝐺:{0}⟶𝐷𝐺:{0}⟶𝐷)
147 c0ex 10072 . . . . . . . . . . . . . . 15 0 ∈ V
148147snid 4241 . . . . . . . . . . . . . 14 0 ∈ {0}
149148a1i 11 . . . . . . . . . . . . 13 (𝐺:{0}⟶𝐷 → 0 ∈ {0})
150146, 149ffvelrnd 6400 . . . . . . . . . . . 12 (𝐺:{0}⟶𝐷 → (𝐺‘0) ∈ 𝐷)
15116, 150syl 17 . . . . . . . . . . 11 (𝐺 ∈ (𝐷𝑚 {0}) → (𝐺‘0) ∈ 𝐷)
152151ad2antlr 763 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑙 ∈ (0...1)) → (𝐺‘0) ∈ 𝐷)
15323matring 20297 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
15424, 92ring0cl 18615 . . . . . . . . . . . 12 (𝐴 ∈ Ring → 0𝐷)
155153, 154syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0𝐷)
156155ad2antrr 762 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑙 ∈ (0...1)) → 0𝐷)
157152, 156ifcld 4164 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝐺‘0), 0 ) ∈ 𝐷)
158157, 42fmptd 6425 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → 𝐻:(0...1)⟶𝐷)
15979oveq2i 6701 . . . . . . . . 9 (0...(0 + 1)) = (0...1)
160159feq2i 6075 . . . . . . . 8 (𝐻:(0...(0 + 1))⟶𝐷𝐻:(0...1)⟶𝐷)
161158, 160sylibr 224 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) → 𝐻:(0...(0 + 1))⟶𝐷)
162161ffvelrnda 6399 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → (𝐻𝑛) ∈ 𝐷)
163 elfznn0 12471 . . . . . . 7 (𝑛 ∈ (0...(0 + 1)) → 𝑛 ∈ ℕ0)
164163adantl 481 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑛 ∈ ℕ0)
16523, 24, 25, 2, 3, 8, 26, 27, 28mat2pmatscmxcl 20593 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐻𝑛) ∈ 𝐷𝑛 ∈ ℕ0)) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) ∈ 𝐵)
166144, 145, 162, 164, 165syl22anc 1367 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → ((𝑛 𝑋) (𝑇‘(𝐻𝑛))) ∈ 𝐵)
1678, 33, 11, 143, 166gsummptfzsplit 18378 . . . 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)))))))
1681673adant3 1101 . . 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)))))))
169142, 168eqtr4d 2688 . 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))))))
170 mpteq1 4770 . . . 4 ((0...(0 + 1)) = (0...1) → (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))
171159, 170ax-mp 5 . . 3 (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))
172171oveq2i 6701 . 2 (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛)))))
173169, 172syl6eq 2701 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 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ifcif 4119  {csn 4210   ↦ cmpt 4762  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899  Fincfn 7997  0cc0 9974  1c1 9975   + caddc 9977  ℕ0cn0 11330  ℤcz 11415  ℤ≥cuz 11725  ...cfz 12364  Basecbs 15904  +gcplusg 15988  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147   Σg cgsu 16148  Mndcmnd 17341  .gcmg 17587  CMndccmn 18239  mulGrpcmgp 18535  Ringcrg 18593  LModclmod 18911  var1cv1 19594  Poly1cpl1 19595   Mat cmat 20261   matToPolyMat cmat2pmat 20557 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-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-dsmm 20124  df-frlm 20139  df-mamu 20238  df-mat 20262  df-mat2pmat 20560 This theorem is referenced by:  pmatcollpw3fi1lem2  20640
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