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Theorem pmatcollpw3fi1lem2 20683
 Description: Lemma 2 for pmatcollpw3fi1 20684. (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 6271 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝑛) = (𝑔𝑛))
21fveq2d 6276 . . . . . . 7 (𝑓 = 𝑔 → (𝑇‘(𝑓𝑛)) = (𝑇‘(𝑔𝑛)))
32oveq2d 6749 . . . . . 6 (𝑓 = 𝑔 → ((𝑛 𝑋) (𝑇‘(𝑓𝑛))) = ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))
43mpteq2dv 4821 . . . . 5 (𝑓 = 𝑔 → (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))
54oveq2d 6749 . . . 4 (𝑓 = 𝑔 → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))))
65eqeq2d 2702 . . 3 (𝑓 = 𝑔 → (𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))))
76cbvrexv 3243 . 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 18647 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
98anim2i 594 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
1093adant3 1124 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
1110ad2antrr 764 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
12 simplr 809 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → 𝑔 ∈ (𝐷𝑚 {0}))
13 simpr 479 . . . . . 6 ((((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 . . . . . . 7 𝑃 = (Poly1𝑅)
15 pmatcollpw.c . . . . . . 7 𝐶 = (𝑁 Mat 𝑃)
16 pmatcollpw.b . . . . . . 7 𝐵 = (Base‘𝐶)
17 pmatcollpw.m . . . . . . 7 = ( ·𝑠𝐶)
18 pmatcollpw.e . . . . . . 7 = (.g‘(mulGrp‘𝑃))
19 pmatcollpw.x . . . . . . 7 𝑋 = (var1𝑅)
20 pmatcollpw.t . . . . . . 7 𝑇 = (𝑁 matToPolyMat 𝑅)
21 pmatcollpw3.a . . . . . . 7 𝐴 = (𝑁 Mat 𝑅)
22 pmatcollpw3.d . . . . . . 7 𝐷 = (Base‘𝐴)
23 eqid 2692 . . . . . . 7 (0g𝐴) = (0g𝐴)
24 eqid 2692 . . . . . . 7 (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 20682 . . . . . 6 (((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 1407 . . . . 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))))))
27 1nn 11112 . . . . . . 7 1 ∈ ℕ
2827a1i 11 . . . . . 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)))))) → 1 ∈ ℕ)
29 oveq2 6741 . . . . . . . . 9 (𝑠 = 1 → (0...𝑠) = (0...1))
3029oveq2d 6749 . . . . . . . 8 (𝑠 = 1 → (𝐷𝑚 (0...𝑠)) = (𝐷𝑚 (0...1)))
3129mpteq1d 4814 . . . . . . . . . 10 (𝑠 = 1 → (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))
3231oveq2d 6749 . . . . . . . . 9 (𝑠 = 1 → (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))
3332eqeq2d 2702 . . . . . . . 8 (𝑠 = 1 → (𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))
3430, 33rexeqbidv 3224 . . . . . . 7 (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 473 . . . . . 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)))))) ∧ 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 7964 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝐷𝑚 {0}) → 𝑔:{0}⟶𝐷)
37 c0ex 10115 . . . . . . . . . . . . . . . . . . 19 0 ∈ V
3837snid 4284 . . . . . . . . . . . . . . . . . 18 0 ∈ {0}
3938a1i 11 . . . . . . . . . . . . . . . . 17 (𝑙 ∈ (0...1) → 0 ∈ {0})
40 ffvelrn 6440 . . . . . . . . . . . . . . . . 17 ((𝑔:{0}⟶𝐷 ∧ 0 ∈ {0}) → (𝑔‘0) ∈ 𝐷)
4139, 40sylan2 492 . . . . . . . . . . . . . . . 16 ((𝑔:{0}⟶𝐷𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
4241ex 449 . . . . . . . . . . . . . . 15 (𝑔:{0}⟶𝐷 → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
4336, 42syl 17 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝐷𝑚 {0}) → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
4443adantl 473 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
4544imp 444 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
4621matring 20340 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
478, 46sylan2 492 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
48473adant3 1124 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
4922, 23ring0cl 18658 . . . . . . . . . . . . . 14 (𝐴 ∈ Ring → (0g𝐴) ∈ 𝐷)
5048, 49syl 17 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (0g𝐴) ∈ 𝐷)
5150ad2antrr 764 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑙 ∈ (0...1)) → (0g𝐴) ∈ 𝐷)
5245, 51ifcld 4207 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝑔‘0), (0g𝐴)) ∈ 𝐷)
5352, 24fmptd 6468 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))):(0...1)⟶𝐷)
54 fvex 6282 . . . . . . . . . . . . 13 (Base‘𝐴) ∈ V
5522, 54eqeltri 2767 . . . . . . . . . . . 12 𝐷 ∈ V
56 ovex 6761 . . . . . . . . . . . 12 (0...1) ∈ V
5755, 56pm3.2i 470 . . . . . . . . . . 11 (𝐷 ∈ V ∧ (0...1) ∈ V)
58 elmapg 7955 . . . . . . . . . . 11 ((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))
5957, 58mp1i 13 . . . . . . . . . 10 (((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))
6053, 59mpbird 247 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) ∈ (𝐷𝑚 (0...1)))
6160adantr 472 . . . . . . . 8 ((((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)))
62 fveq1 6271 . . . . . . . . . . . . . 14 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → (𝑓𝑛) = ((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛))
6362fveq2d 6276 . . . . . . . . . . . . 13 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → (𝑇‘(𝑓𝑛)) = (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛)))
6463oveq2d 6749 . . . . . . . . . . . 12 (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))))
6564mpteq2dv 4821 . . . . . . . . . . 11 (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)))))
6665oveq2d 6749 . . . . . . . . . 10 (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))))))
6766eqeq2d 2702 . . . . . . . . 9 (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)))))))
6867adantl 473 . . . . . . . 8 (((((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)))))))
6961, 68rspcedv 3385 . . . . . . 7 ((((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)))))))
7069imp 444 . . . . . 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))))))
7128, 35, 70rspcedvd 3389 . . . . 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 ∈ ℕ ∃𝑓 ∈ (𝐷𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))
7226, 71mpdan 705 . . . 4 ((((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))))))
7372ex 449 . . 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)))))))
7473rexlimdva 3101 . 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)))))))
757, 74syl5bi 232 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 196   ∧ wa 383   ∧ w3a 1072   = wceq 1564   ∈ wcel 2071  ∃wrex 2983  Vcvv 3272  ifcif 4162  {csn 4253   ↦ cmpt 4805  ⟶wf 5965  ‘cfv 5969  (class class class)co 6733   ↑𝑚 cmap 7942  Fincfn 8040  0cc0 10017  1c1 10018  ℕcn 11101  ...cfz 12408  Basecbs 15948   ·𝑠 cvsca 16036  0gc0g 16191   Σg cgsu 16192  .gcmg 17630  mulGrpcmgp 18578  Ringcrg 18636  CRingccrg 18637  var1cv1 19637  Poly1cpl1 19638   Mat cmat 20304   matToPolyMat cmat2pmat 20600 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-rep 4847  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034  ax-inf2 8619  ax-cnex 10073  ax-resscn 10074  ax-1cn 10075  ax-icn 10076  ax-addcl 10077  ax-addrcl 10078  ax-mulcl 10079  ax-mulrcl 10080  ax-mulcom 10081  ax-addass 10082  ax-mulass 10083  ax-distr 10084  ax-i2m1 10085  ax-1ne0 10086  ax-1rid 10087  ax-rnegex 10088  ax-rrecex 10089  ax-cnre 10090  ax-pre-lttri 10091  ax-pre-lttrn 10092  ax-pre-ltadd 10093  ax-pre-mulgt0 10094 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1567  df-fal 1570  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-nel 2968  df-ral 2987  df-rex 2988  df-reu 2989  df-rmo 2990  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-pss 3664  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-tp 4258  df-op 4260  df-ot 4262  df-uni 4513  df-int 4552  df-iun 4598  df-iin 4599  df-br 4729  df-opab 4789  df-mpt 4806  df-tr 4829  df-id 5096  df-eprel 5101  df-po 5107  df-so 5108  df-fr 5145  df-se 5146  df-we 5147  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-pred 5761  df-ord 5807  df-on 5808  df-lim 5809  df-suc 5810  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-f1 5974  df-fo 5975  df-f1o 5976  df-fv 5977  df-isom 5978  df-riota 6694  df-ov 6736  df-oprab 6737  df-mpt2 6738  df-of 6982  df-ofr 6983  df-om 7151  df-1st 7253  df-2nd 7254  df-supp 7384  df-wrecs 7495  df-recs 7556  df-rdg 7594  df-1o 7648  df-2o 7649  df-oadd 7652  df-er 7830  df-map 7944  df-pm 7945  df-ixp 7994  df-en 8041  df-dom 8042  df-sdom 8043  df-fin 8044  df-fsupp 8360  df-sup 8432  df-oi 8499  df-card 8846  df-pnf 10157  df-mnf 10158  df-xr 10159  df-ltxr 10160  df-le 10161  df-sub 10349  df-neg 10350  df-nn 11102  df-2 11160  df-3 11161  df-4 11162  df-5 11163  df-6 11164  df-7 11165  df-8 11166  df-9 11167  df-n0 11374  df-z 11459  df-dec 11575  df-uz 11769  df-fz 12409  df-fzo 12549  df-seq 12885  df-hash 13201  df-struct 15950  df-ndx 15951  df-slot 15952  df-base 15954  df-sets 15955  df-ress 15956  df-plusg 16045  df-mulr 16046  df-sca 16048  df-vsca 16049  df-ip 16050  df-tset 16051  df-ple 16052  df-ds 16055  df-hom 16057  df-cco 16058  df-0g 16193  df-gsum 16194  df-prds 16199  df-pws 16201  df-mre 16337  df-mrc 16338  df-acs 16340  df-mgm 17332  df-sgrp 17374  df-mnd 17385  df-mhm 17425  df-submnd 17426  df-grp 17515  df-minusg 17516  df-sbg 17517  df-mulg 17631  df-subg 17681  df-ghm 17748  df-cntz 17839  df-cmn 18284  df-abl 18285  df-mgp 18579  df-ur 18591  df-ring 18638  df-cring 18639  df-subrg 18869  df-lmod 18956  df-lss 19024  df-sra 19263  df-rgmod 19264  df-ascl 19405  df-psr 19447  df-mvr 19448  df-mpl 19449  df-opsr 19451  df-psr1 19641  df-vr1 19642  df-ply1 19643  df-dsmm 20167  df-frlm 20182  df-mamu 20281  df-mat 20305  df-mat2pmat 20603 This theorem is referenced by:  pmatcollpw3fi1  20684
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