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Theorem pmatcollpw3fi1lem2 20871
Description: Lemma 2 for pmatcollpw3fi1 20872. (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 6374 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝑛) = (𝑔𝑛))
21fveq2d 6379 . . . . . . 7 (𝑓 = 𝑔 → (𝑇‘(𝑓𝑛)) = (𝑇‘(𝑔𝑛)))
32oveq2d 6858 . . . . . 6 (𝑓 = 𝑔 → ((𝑛 𝑋) (𝑇‘(𝑓𝑛))) = ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))
43mpteq2dv 4904 . . . . 5 (𝑓 = 𝑔 → (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))
54oveq2d 6858 . . . 4 (𝑓 = 𝑔 → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛))))))
65eqeq2d 2775 . . 3 (𝑓 = 𝑔 → (𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))))
76cbvrexv 3320 . 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 18825 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
98anim2i 610 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
1093adant3 1162 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
1110ad2antrr 717 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
12 simplr 785 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑔𝑛)))))) → 𝑔 ∈ (𝐷𝑚 {0}))
13 simpr 477 . . . . . 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 2765 . . . . . . 7 (0g𝐴) = (0g𝐴)
24 eqid 2765 . . . . . . 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 20870 . . . . . 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 1490 . . . . 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 11287 . . . . . . 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 6850 . . . . . . . . 9 (𝑠 = 1 → (0...𝑠) = (0...1))
3029oveq2d 6858 . . . . . . . 8 (𝑠 = 1 → (𝐷𝑚 (0...𝑠)) = (𝐷𝑚 (0...1)))
3129mpteq1d 4897 . . . . . . . . . 10 (𝑠 = 1 → (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))
3231oveq2d 6858 . . . . . . . . 9 (𝑠 = 1 → (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))
3332eqeq2d 2775 . . . . . . . 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 3301 . . . . . . 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 8082 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝐷𝑚 {0}) → 𝑔:{0}⟶𝐷)
37 c0ex 10287 . . . . . . . . . . . . . . . . . . 19 0 ∈ V
3837snid 4366 . . . . . . . . . . . . . . . . . 18 0 ∈ {0}
3938a1i 11 . . . . . . . . . . . . . . . . 17 (𝑙 ∈ (0...1) → 0 ∈ {0})
40 ffvelrn 6547 . . . . . . . . . . . . . . . . 17 ((𝑔:{0}⟶𝐷 ∧ 0 ∈ {0}) → (𝑔‘0) ∈ 𝐷)
4139, 40sylan2 586 . . . . . . . . . . . . . . . 16 ((𝑔:{0}⟶𝐷𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
4241ex 401 . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
4621matring 20525 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
478, 46sylan2 586 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
48473adant3 1162 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐴 ∈ Ring)
4922, 23ring0cl 18836 . . . . . . . . . . . . . 14 (𝐴 ∈ Ring → (0g𝐴) ∈ 𝐷)
5048, 49syl 17 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (0g𝐴) ∈ 𝐷)
5150ad2antrr 717 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑙 ∈ (0...1)) → (0g𝐴) ∈ 𝐷)
5245, 51ifcld 4288 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝑔‘0), (0g𝐴)) ∈ 𝐷)
5352fmpttd 6575 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))):(0...1)⟶𝐷)
5422fvexi 6389 . . . . . . . . . . . 12 𝐷 ∈ V
55 ovex 6874 . . . . . . . . . . . 12 (0...1) ∈ V
5654, 55pm3.2i 462 . . . . . . . . . . 11 (𝐷 ∈ V ∧ (0...1) ∈ V)
57 elmapg 8073 . . . . . . . . . . 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))
5856, 57mp1i 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))
5953, 58mpbird 248 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑔 ∈ (𝐷𝑚 {0})) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) ∈ (𝐷𝑚 (0...1)))
6059adantr 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)))
61 fveq1 6374 . . . . . . . . . . . . . 14 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → (𝑓𝑛) = ((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛))
6261fveq2d 6379 . . . . . . . . . . . . 13 (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴))) → (𝑇‘(𝑓𝑛)) = (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g𝐴)))‘𝑛)))
6362oveq2d 6858 . . . . . . . . . . . 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))))
6463mpteq2dv 4904 . . . . . . . . . . 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)))))
6564oveq2d 6858 . . . . . . . . . 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))))))
6665eqeq2d 2775 . . . . . . . . 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)))))))
6766adantl 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)))))))
6860, 67rspcedv 3465 . . . . . . 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)))))))
6968imp 395 . . . . . 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))))))
7028, 35, 69rspcedvd 3468 . . . . 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))))))
7126, 70mpdan 678 . . . 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))))))
7271ex 401 . . 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)))))))
7372rexlimdva 3178 . 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)))))))
747, 73syl5bi 233 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wrex 3056  Vcvv 3350  ifcif 4243  {csn 4334  cmpt 4888  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  Fincfn 8160  0cc0 10189  1c1 10190  cn 11274  ...cfz 12533  Basecbs 16130   ·𝑠 cvsca 16218  0gc0g 16366   Σg cgsu 16367  .gcmg 17807  mulGrpcmgp 18756  Ringcrg 18814  CRingccrg 18815  var1cv1 19819  Poly1cpl1 19820   Mat cmat 20489   matToPolyMat cmat2pmat 20788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-ot 4343  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-sca 16230  df-vsca 16231  df-ip 16232  df-tset 16233  df-ple 16234  df-ds 16236  df-hom 16238  df-cco 16239  df-0g 16368  df-gsum 16369  df-prds 16374  df-pws 16376  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-mhm 17601  df-submnd 17602  df-grp 17692  df-minusg 17693  df-sbg 17694  df-mulg 17808  df-subg 17855  df-ghm 17922  df-cntz 18013  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-cring 18817  df-subrg 19047  df-lmod 19134  df-lss 19202  df-sra 19446  df-rgmod 19447  df-ascl 19588  df-psr 19630  df-mvr 19631  df-mpl 19632  df-opsr 19634  df-psr1 19823  df-vr1 19824  df-ply1 19825  df-dsmm 20352  df-frlm 20367  df-mamu 20466  df-mat 20490  df-mat2pmat 20791
This theorem is referenced by:  pmatcollpw3fi1  20872
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