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Theorem pmatcollpw3fi1lem2 21330
Description: Lemma 2 for pmatcollpw3fi1 21331. (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 6668 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝑛) = (𝑔𝑛))
21fveq2d 6673 . . . . . . 7 (𝑓 = 𝑔 → (𝑇‘(𝑓𝑛)) = (𝑇‘(𝑔𝑛)))
32oveq2d 7166 . . . . . 6 (𝑓 = 𝑔 → ((𝑛 𝑋) (𝑇‘(𝑓𝑛))) = ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))
43mpteq2dv 5159 . . . . 5 (𝑓 = 𝑔 → (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))
54oveq2d 7166 . . . 4 (𝑓 = 𝑔 → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))))
65eqeq2d 2837 . . 3 (𝑓 = 𝑔 → (𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))))
76cbvrexvw 3456 . 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 19244 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
98anim2i 616 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
1093adant3 1126 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
1110ad2antrr 722 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
12 simplr 765 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → 𝑔 ∈ (𝐷m {0}))
13 simpr 485 . . . . 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 2826 . . . . . 6 (0g𝐴) = (0g𝐴)
24 eqid 2826 . . . . . 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 21329 . . . . 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 1365 . . . 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 11643 . . . . . 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 7158 . . . . . . . 8 (𝑠 = 1 → (0...𝑠) = (0...1))
3029oveq2d 7166 . . . . . . 7 (𝑠 = 1 → (𝐷m (0...𝑠)) = (𝐷m (0...1)))
3129mpteq1d 5152 . . . . . . . . 9 (𝑠 = 1 → (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))
3231oveq2d 7166 . . . . . . . 8 (𝑠 = 1 → (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))
3332eqeq2d 2837 . . . . . . 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 3408 . . . . . 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 482 . . . . 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 8423 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝐷m {0}) → 𝑔:{0}⟶𝐷)
37 c0ex 10629 . . . . . . . . . . . . . . . . . 18 0 ∈ V
3837snid 4598 . . . . . . . . . . . . . . . . 17 0 ∈ {0}
3938a1i 11 . . . . . . . . . . . . . . . 16 (𝑙 ∈ (0...1) → 0 ∈ {0})
40 ffvelrn 6847 . . . . . . . . . . . . . . . 16 ((𝑔:{0}⟶𝐷 ∧ 0 ∈ {0}) → (𝑔‘0) ∈ 𝐷)
4139, 40sylan2 592 . . . . . . . . . . . . . . 15 ((𝑔:{0}⟶𝐷𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
4241ex 413 . . . . . . . . . . . . . 14 (𝑔:{0}⟶𝐷 → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
4336, 42syl 17 . . . . . . . . . . . . 13 (𝑔 ∈ (𝐷m {0}) → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
4443adantl 482 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
4544imp 407 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
4621matring 20987 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
478, 46sylan2 592 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
48473adant3 1126 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
4922, 23ring0cl 19255 . . . . . . . . . . . . 13 (𝐴 ∈ Ring → (0g𝐴) ∈ 𝐷)
5048, 49syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (0g𝐴) ∈ 𝐷)
5150ad2antrr 722 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → (0g𝐴) ∈ 𝐷)
5245, 51ifcld 4515 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝑔‘0), (0g𝐴)) ∈ 𝐷)
5352fmpttd 6877 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))):(0...1)⟶𝐷)
5422fvexi 6683 . . . . . . . . . . 11 𝐷 ∈ V
55 ovex 7183 . . . . . . . . . . 11 (0...1) ∈ V
5654, 55pm3.2i 471 . . . . . . . . . 10 (𝐷 ∈ V ∧ (0...1) ∈ V)
57 elmapg 8414 . . . . . . . . . 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 258 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷m {0})) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) ∈ (𝐷m (0...1)))
6059adantr 481 . . . . . . 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 6668 . . . . . . . . . . . . 13 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → (𝑓𝑛) = ((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛))
6261fveq2d 6673 . . . . . . . . . . . 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 7166 . . . . . . . . . . 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 5159 . . . . . . . . . 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 7166 . . . . . . . . 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 2837 . . . . . . . 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 482 . . . . . . 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 3620 . . . . . 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 407 . . . . 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 3630 . . . 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 683 . . 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 3292 . 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, 72syl5bi 243 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wrex 3144  Vcvv 3500  ifcif 4470  {csn 4564  cmpt 5143  wf 6350  cfv 6354  (class class class)co 7150  m cmap 8401  Fincfn 8503  0cc0 10531  1c1 10532  cn 11632  ...cfz 12887  Basecbs 16478   ·𝑠 cvsca 16564  0gc0g 16708   Σg cgsu 16709  .gcmg 18169  mulGrpcmgp 19175  Ringcrg 19233  CRingccrg 19234  var1cv1 20279  Poly1cpl1 20280   Mat cmat 20951   matToPolyMat cmat2pmat 21247
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-ot 4573  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-ofr 7404  df-om 7574  df-1st 7685  df-2nd 7686  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8284  df-map 8403  df-pm 8404  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-sup 8900  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12888  df-fzo 13029  df-seq 13365  df-hash 13686  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-hom 16584  df-cco 16585  df-0g 16710  df-gsum 16711  df-prds 16716  df-pws 16718  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-submnd 17952  df-grp 18051  df-minusg 18052  df-sbg 18053  df-mulg 18170  df-subg 18221  df-ghm 18301  df-cntz 18392  df-cmn 18844  df-abl 18845  df-mgp 19176  df-ur 19188  df-ring 19235  df-cring 19236  df-subrg 19469  df-lmod 19572  df-lss 19640  df-sra 19880  df-rgmod 19881  df-ascl 20022  df-psr 20071  df-mvr 20072  df-mpl 20073  df-opsr 20075  df-psr1 20283  df-vr1 20284  df-ply1 20285  df-dsmm 20811  df-frlm 20826  df-mamu 20930  df-mat 20952  df-mat2pmat 21250
This theorem is referenced by:  pmatcollpw3fi1  21331
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