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Theorem pmatcollpw3fi1lem2 20640
Description: Lemma 2 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)
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 6228 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝑛) = (𝑔𝑛))
21fveq2d 6233 . . . . . . 7 (𝑓 = 𝑔 → (𝑇‘(𝑓𝑛)) = (𝑇‘(𝑔𝑛)))
32oveq2d 6706 . . . . . 6 (𝑓 = 𝑔 → ((𝑛 𝑋) (𝑇‘(𝑓𝑛))) = ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))
43mpteq2dv 4778 . . . . 5 (𝑓 = 𝑔 → (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))
54oveq2d 6706 . . . 4 (𝑓 = 𝑔 → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))))
65eqeq2d 2661 . . 3 (𝑓 = 𝑔 → (𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))))
76cbvrexv 3202 . 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 18604 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
98anim2i 592 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
1093adant3 1101 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
1110ad2antrr 762 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
12 simplr 807 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → 𝑔 ∈ (𝐷𝑚 {0}))
13 simpr 476 . . . . . 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 2651 . . . . . . 7 (0g𝐴) = (0g𝐴)
24 eqid 2651 . . . . . . 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 20639 . . . . . 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 1366 . . . . 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 11069 . . . . . . 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 6698 . . . . . . . . 9 (𝑠 = 1 → (0...𝑠) = (0...1))
3029oveq2d 6706 . . . . . . . 8 (𝑠 = 1 → (𝐷𝑚 (0...𝑠)) = (𝐷𝑚 (0...1)))
3129mpteq1d 4771 . . . . . . . . . 10 (𝑠 = 1 → (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))
3231oveq2d 6706 . . . . . . . . 9 (𝑠 = 1 → (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))
3332eqeq2d 2661 . . . . . . . 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 3183 . . . . . . 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 481 . . . . . 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 7921 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝐷𝑚 {0}) → 𝑔:{0}⟶𝐷)
37 c0ex 10072 . . . . . . . . . . . . . . . . . . 19 0 ∈ V
3837snid 4241 . . . . . . . . . . . . . . . . . 18 0 ∈ {0}
3938a1i 11 . . . . . . . . . . . . . . . . 17 (𝑙 ∈ (0...1) → 0 ∈ {0})
40 ffvelrn 6397 . . . . . . . . . . . . . . . . 17 ((𝑔:{0}⟶𝐷 ∧ 0 ∈ {0}) → (𝑔‘0) ∈ 𝐷)
4139, 40sylan2 490 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 20297 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
478, 46sylan2 490 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
48473adant3 1101 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
4922, 23ring0cl 18615 . . . . . . . . . . . . . 14 (𝐴 ∈ Ring → (0g𝐴) ∈ 𝐷)
5048, 49syl 17 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (0g𝐴) ∈ 𝐷)
5150ad2antrr 762 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑙 ∈ (0...1)) → (0g𝐴) ∈ 𝐷)
5245, 51ifcld 4164 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝑔‘0), (0g𝐴)) ∈ 𝐷)
5352, 24fmptd 6425 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))):(0...1)⟶𝐷)
54 fvex 6239 . . . . . . . . . . . . 13 (Base‘𝐴) ∈ V
5522, 54eqeltri 2726 . . . . . . . . . . . 12 𝐷 ∈ V
56 ovex 6718 . . . . . . . . . . . 12 (0...1) ∈ V
5755, 56pm3.2i 470 . . . . . . . . . . 11 (𝐷 ∈ V ∧ (0...1) ∈ V)
58 elmapg 7912 . . . . . . . . . . 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 480 . . . . . . . 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 6228 . . . . . . . . . . . . . 14 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → (𝑓𝑛) = ((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛))
6362fveq2d 6233 . . . . . . . . . . . . 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 6706 . . . . . . . . . . . 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 4778 . . . . . . . . . . 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 6706 . . . . . . . . . 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 2661 . . . . . . . . 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 481 . . . . . . . 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 3344 . . . . . . 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 3348 . . . . 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 703 . . . 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 3060 . 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 1054   = wceq 1523  wcel 2030  wrex 2942  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  cn 11058  ...cfz 12364  Basecbs 15904   ·𝑠 cvsca 15992  0gc0g 16147   Σg cgsu 16148  .gcmg 17587  mulGrpcmgp 18535  Ringcrg 18593  CRingccrg 18594  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-cring 18596  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:  pmatcollpw3fi1  20641
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