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Theorem pmatcollpw3fi1lem2 22611
Description: Lemma 2 for pmatcollpw3fi1 22612. (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)
Assertion
Ref Expression
pmatcollpw3fi1lem2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (∃𝑓 ∈ (𝐷m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))
Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑃,𝑛   𝑅,𝑛   𝑛,𝑋   ,𝑛   𝐵,𝑠,𝑛   𝐶,𝑛   𝑀,𝑠   𝑁,𝑠   𝑅,𝑠   𝐵,𝑓   𝐶,𝑓,𝑛   𝐷,𝑓   𝑓,𝑀   𝑓,𝑁   𝑅,𝑓   𝑇,𝑓   𝑓,𝑋   ,𝑓   ,𝑓,𝑠   𝐷,𝑛   𝐴,𝑓,𝑛,𝑠   𝐶,𝑠   𝐷,𝑠   𝑇,𝑠   𝑋,𝑠   ,𝑠   ,𝑠
Allowed substitution hints:   𝑃(𝑓,𝑠)   𝑇(𝑛)   (𝑛)

Proof of Theorem pmatcollpw3fi1lem2
Dummy variables 𝑙 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 6880 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝑛) = (𝑔𝑛))
21fveq2d 6885 . . . . . . 7 (𝑓 = 𝑔 → (𝑇‘(𝑓𝑛)) = (𝑇‘(𝑔𝑛)))
32oveq2d 7417 . . . . . 6 (𝑓 = 𝑔 → ((𝑛 𝑋) (𝑇‘(𝑓𝑛))) = ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))
43mpteq2dv 5240 . . . . 5 (𝑓 = 𝑔 → (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))
54oveq2d 7417 . . . 4 (𝑓 = 𝑔 → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))))
65eqeq2d 2735 . . 3 (𝑓 = 𝑔 → (𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))))
76cbvrexvw 3227 . 2 (∃𝑓 ∈ (𝐷m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ ∃𝑔 ∈ (𝐷m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))))
8 crngring 20140 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
98anim2i 616 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
1093adant3 1129 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
1110ad2antrr 723 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
12 simplr 766 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → 𝑔 ∈ (𝐷m {0}))
13 simpr 484 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))))
14 pmatcollpw.p . . . . . 6 𝑃 = (Poly1𝑅)
15 pmatcollpw.c . . . . . 6 𝐶 = (𝑁 Mat 𝑃)
16 pmatcollpw.b . . . . . 6 𝐵 = (Base‘𝐶)
17 pmatcollpw.m . . . . . 6 = ( ·𝑠𝐶)
18 pmatcollpw.e . . . . . 6 = (.g‘(mulGrp‘𝑃))
19 pmatcollpw.x . . . . . 6 𝑋 = (var1𝑅)
20 pmatcollpw.t . . . . . 6 𝑇 = (𝑁 matToPolyMat 𝑅)
21 pmatcollpw3.a . . . . . 6 𝐴 = (𝑁 Mat 𝑅)
22 pmatcollpw3.d . . . . . 6 𝐷 = (Base‘𝐴)
23 eqid 2724 . . . . . 6 (0g𝐴) = (0g𝐴)
24 eqid 2724 . . . . . 6 (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))
2514, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24pmatcollpw3fi1lem1 22610 . . . . 5 (((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‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛))))))
2611, 12, 13, 25syl3anc 1368 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛))))))
27 1nn 12220 . . . . . 6 1 ∈ ℕ
2827a1i 11 . . . . 5 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛)))))) → 1 ∈ ℕ)
29 oveq2 7409 . . . . . . . 8 (𝑠 = 1 → (0...𝑠) = (0...1))
3029oveq2d 7417 . . . . . . 7 (𝑠 = 1 → (𝐷m (0...𝑠)) = (𝐷m (0...1)))
3129mpteq1d 5233 . . . . . . . . 9 (𝑠 = 1 → (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))
3231oveq2d 7417 . . . . . . . 8 (𝑠 = 1 → (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))
3332eqeq2d 2735 . . . . . . 7 (𝑠 = 1 → (𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))
3430, 33rexeqbidv 3335 . . . . . 6 (𝑠 = 1 → (∃𝑓 ∈ (𝐷m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ ∃𝑓 ∈ (𝐷m (0...1))𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))
3534adantl 481 . . . . 5 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛)))))) ∧ 𝑠 = 1) → (∃𝑓 ∈ (𝐷m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ ∃𝑓 ∈ (𝐷m (0...1))𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))
36 elmapi 8839 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝐷m {0}) → 𝑔:{0}⟶𝐷)
37 c0ex 11205 . . . . . . . . . . . . . . . . . 18 0 ∈ V
3837snid 4656 . . . . . . . . . . . . . . . . 17 0 ∈ {0}
3938a1i 11 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...1) → 0 ∈ {0})
40 ffvelcdm 7073 . . . . . . . . . . . . . . . 16 ((𝑔:{0}⟶𝐷 ∧ 0 ∈ {0}) → (𝑔‘0) ∈ 𝐷)
4139, 40sylan2 592 . . . . . . . . . . . . . . 15 ((𝑔:{0}⟶𝐷𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
4241ex 412 . . . . . . . . . . . . . 14 (𝑔:{0}⟶𝐷 → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
4336, 42syl 17 . . . . . . . . . . . . 13 (𝑔 ∈ (𝐷m {0}) → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
4443adantl 481 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
4544imp 406 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
4621matring 22267 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
478, 46sylan2 592 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
48473adant3 1129 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
4922, 23ring0cl 20156 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → (0g𝐴) ∈ 𝐷)
5048, 49syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (0g𝐴) ∈ 𝐷)
5150ad2antrr 723 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → (0g𝐴) ∈ 𝐷)
5245, 51ifcld 4566 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝑔‘0), (0g𝐴)) ∈ 𝐷)
5352fmpttd 7106 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))):(0...1)⟶𝐷)
5422fvexi 6895 . . . . . . . . . . 11 𝐷 ∈ V
55 ovex 7434 . . . . . . . . . . 11 (0...1) ∈ V
5654, 55pm3.2i 470 . . . . . . . . . 10 (𝐷 ∈ V ∧ (0...1) ∈ V)
57 elmapg 8829 . . . . . . . . . 10 ((𝐷 ∈ V ∧ (0...1) ∈ V) → ((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) ∈ (𝐷m (0...1)) ↔ (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))):(0...1)⟶𝐷))
5856, 57mp1i 13 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) → ((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) ∈ (𝐷m (0...1)) ↔ (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))):(0...1)⟶𝐷))
5953, 58mpbird 257 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) ∈ (𝐷m (0...1)))
6059adantr 480 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) ∈ (𝐷m (0...1)))
61 fveq1 6880 . . . . . . . . . . . . 13 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → (𝑓𝑛) = ((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛))
6261fveq2d 6885 . . . . . . . . . . . 12 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → (𝑇‘(𝑓𝑛)) = (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛)))
6362oveq2d 7417 . . . . . . . . . . 11 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → ((𝑛 𝑋) (𝑇‘(𝑓𝑛))) = ((𝑛 𝑋) (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛))))
6463mpteq2dv 5240 . . . . . . . . . 10 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛)))))
6564oveq2d 7417 . . . . . . . . 9 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛))))))
6665eqeq2d 2735 . . . . . . . 8 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → (𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛)))))))
6766adantl 481 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) ∧ 𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))) → (𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛)))))))
6860, 67rspcedv 3597 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → (𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛))))) → ∃𝑓 ∈ (𝐷m (0...1))𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))
6968imp 406 . . . . 5 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛)))))) → ∃𝑓 ∈ (𝐷m (0...1))𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))
7028, 35, 69rspcedvd 3606 . . . 4 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛)))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))
7126, 70mpdan 684 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))
7271rexlimdva2 3149 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (∃𝑔 ∈ (𝐷m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))
737, 72biimtrid 241 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (∃𝑓 ∈ (𝐷m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wrex 3062  Vcvv 3466  ifcif 4520  {csn 4620  cmpt 5221  wf 6529  cfv 6533  (class class class)co 7401  m cmap 8816  Fincfn 8935  0cc0 11106  1c1 11107  cn 12209  ...cfz 13481  Basecbs 17143   ·𝑠 cvsca 17200  0gc0g 17384   Σg cgsu 17385  .gcmg 18985  mulGrpcmgp 20029  Ringcrg 20128  CRingccrg 20129  var1cv1 22018  Poly1cpl1 22019   Mat cmat 22229   matToPolyMat cmat2pmat 22528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-ot 4629  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-ofr 7664  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-fzo 13625  df-seq 13964  df-hash 14288  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-hom 17220  df-cco 17221  df-0g 17386  df-gsum 17387  df-prds 17392  df-pws 17394  df-mre 17529  df-mrc 17530  df-acs 17532  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18703  df-submnd 18704  df-grp 18856  df-minusg 18857  df-sbg 18858  df-mulg 18986  df-subg 19040  df-ghm 19129  df-cntz 19223  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-cring 20131  df-subrng 20436  df-subrg 20461  df-lmod 20698  df-lss 20769  df-sra 21011  df-rgmod 21012  df-dsmm 21595  df-frlm 21610  df-ascl 21718  df-psr 21771  df-mvr 21772  df-mpl 21773  df-opsr 21775  df-psr1 22022  df-vr1 22023  df-ply1 22024  df-mamu 22208  df-mat 22230  df-mat2pmat 22531
This theorem is referenced by:  pmatcollpw3fi1  22612
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