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Theorem mplcoe1 19513
Description: Decompose a polynomial into a finite sum of monomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
Hypotheses
Ref Expression
mplcoe1.p 𝑃 = (𝐼 mPoly 𝑅)
mplcoe1.d 𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
mplcoe1.z 0 = (0g𝑅)
mplcoe1.o 1 = (1r𝑅)
mplcoe1.i (𝜑𝐼𝑊)
mplcoe1.b 𝐵 = (Base‘𝑃)
mplcoe1.n · = ( ·𝑠𝑃)
mplcoe1.r (𝜑𝑅 ∈ Ring)
mplcoe1.x (𝜑𝑋𝐵)
Assertion
Ref Expression
mplcoe1 (𝜑𝑋 = (𝑃 Σg (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
Distinct variable groups:   𝑦,𝑘, 1   𝐵,𝑘   𝑓,𝑘,𝑦,𝐼   𝜑,𝑘,𝑦   𝑅,𝑓,𝑦   𝐷,𝑘,𝑦   𝑃,𝑘   0 ,𝑓,𝑘,𝑦   𝑓,𝑋,𝑘,𝑦   𝑘,𝑊,𝑦   · ,𝑘
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑦,𝑓)   𝐷(𝑓)   𝑃(𝑦,𝑓)   𝑅(𝑘)   · (𝑦,𝑓)   1 (𝑓)   𝑊(𝑓)

Proof of Theorem mplcoe1
Dummy variables 𝑤 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mplcoe1.p . . . . . 6 𝑃 = (𝐼 mPoly 𝑅)
2 eqid 2651 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
3 mplcoe1.b . . . . . 6 𝐵 = (Base‘𝑃)
4 mplcoe1.d . . . . . 6 𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
5 mplcoe1.x . . . . . 6 (𝜑𝑋𝐵)
61, 2, 3, 4, 5mplelf 19481 . . . . 5 (𝜑𝑋:𝐷⟶(Base‘𝑅))
76feqmptd 6288 . . . 4 (𝜑𝑋 = (𝑦𝐷 ↦ (𝑋𝑦)))
8 iftrue 4125 . . . . . . 7 (𝑦 ∈ (𝑋 supp 0 ) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
98adantl 481 . . . . . 6 (((𝜑𝑦𝐷) ∧ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
10 eldif 3617 . . . . . . . 8 (𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 )) ↔ (𝑦𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )))
11 ifid 4158 . . . . . . . . 9 if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), (𝑋𝑦)) = (𝑋𝑦)
12 ssid 3657 . . . . . . . . . . . 12 (𝑋 supp 0 ) ⊆ (𝑋 supp 0 )
1312a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑋 supp 0 ) ⊆ (𝑋 supp 0 ))
14 ovex 6718 . . . . . . . . . . . . 13 (ℕ0𝑚 𝐼) ∈ V
154, 14rabex2 4847 . . . . . . . . . . . 12 𝐷 ∈ V
1615a1i 11 . . . . . . . . . . 11 (𝜑𝐷 ∈ V)
17 mplcoe1.z . . . . . . . . . . . . 13 0 = (0g𝑅)
18 fvex 6239 . . . . . . . . . . . . 13 (0g𝑅) ∈ V
1917, 18eqeltri 2726 . . . . . . . . . . . 12 0 ∈ V
2019a1i 11 . . . . . . . . . . 11 (𝜑0 ∈ V)
216, 13, 16, 20suppssr 7371 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋𝑦) = 0 )
2221ifeq2d 4138 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), (𝑋𝑦)) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))
2311, 22syl5reqr 2700 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2410, 23sylan2br 492 . . . . . . 7 ((𝜑 ∧ (𝑦𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2524anassrs 681 . . . . . 6 (((𝜑𝑦𝐷) ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
269, 25pm2.61dan 849 . . . . 5 ((𝜑𝑦𝐷) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2726mpteq2dva 4777 . . . 4 (𝜑 → (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ (𝑋𝑦)))
287, 27eqtr4d 2688 . . 3 (𝜑𝑋 = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
29 suppssdm 7353 . . . . 5 (𝑋 supp 0 ) ⊆ dom 𝑋
30 fdm 6089 . . . . . 6 (𝑋:𝐷⟶(Base‘𝑅) → dom 𝑋 = 𝐷)
316, 30syl 17 . . . . 5 (𝜑 → dom 𝑋 = 𝐷)
3229, 31syl5sseq 3686 . . . 4 (𝜑 → (𝑋 supp 0 ) ⊆ 𝐷)
33 eqid 2651 . . . . . . . . 9 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
34 eqid 2651 . . . . . . . . 9 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
351, 33, 34, 17, 3mplelbas 19478 . . . . . . . 8 (𝑋𝐵 ↔ (𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 finSupp 0 ))
3635simprbi 479 . . . . . . 7 (𝑋𝐵𝑋 finSupp 0 )
375, 36syl 17 . . . . . 6 (𝜑𝑋 finSupp 0 )
3837fsuppimpd 8323 . . . . 5 (𝜑 → (𝑋 supp 0 ) ∈ Fin)
39 sseq1 3659 . . . . . . . 8 (𝑤 = ∅ → (𝑤𝐷 ↔ ∅ ⊆ 𝐷))
40 mpteq1 4770 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ ∅ ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
41 mpt0 6059 . . . . . . . . . . . 12 (𝑘 ∈ ∅ ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅
4240, 41syl6eq 2701 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅)
4342oveq2d 6706 . . . . . . . . . 10 (𝑤 = ∅ → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σg ∅))
44 eqid 2651 . . . . . . . . . . 11 (0g𝑃) = (0g𝑃)
4544gsum0 17325 . . . . . . . . . 10 (𝑃 Σg ∅) = (0g𝑃)
4643, 45syl6eq 2701 . . . . . . . . 9 (𝑤 = ∅ → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (0g𝑃))
47 noel 3952 . . . . . . . . . . . 12 ¬ 𝑦 ∈ ∅
48 eleq2 2719 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑦𝑤𝑦 ∈ ∅))
4947, 48mtbiri 316 . . . . . . . . . . 11 (𝑤 = ∅ → ¬ 𝑦𝑤)
5049iffalsed 4130 . . . . . . . . . 10 (𝑤 = ∅ → if(𝑦𝑤, (𝑋𝑦), 0 ) = 0 )
5150mpteq2dv 4778 . . . . . . . . 9 (𝑤 = ∅ → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷0 ))
5246, 51eqeq12d 2666 . . . . . . . 8 (𝑤 = ∅ → ((𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) ↔ (0g𝑃) = (𝑦𝐷0 )))
5339, 52imbi12d 333 . . . . . . 7 (𝑤 = ∅ → ((𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 ))) ↔ (∅ ⊆ 𝐷 → (0g𝑃) = (𝑦𝐷0 ))))
5453imbi2d 329 . . . . . 6 (𝑤 = ∅ → ((𝜑 → (𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )))) ↔ (𝜑 → (∅ ⊆ 𝐷 → (0g𝑃) = (𝑦𝐷0 )))))
55 sseq1 3659 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝐷𝑥𝐷))
56 mpteq1 4770 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
5756oveq2d 6706 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
58 eleq2 2719 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
5958ifbid 4141 . . . . . . . . . 10 (𝑤 = 𝑥 → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
6059mpteq2dv 4778 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))
6157, 60eqeq12d 2666 . . . . . . . 8 (𝑤 = 𝑥 → ((𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) ↔ (𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))))
6255, 61imbi12d 333 . . . . . . 7 (𝑤 = 𝑥 → ((𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 ))) ↔ (𝑥𝐷 → (𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))))
6362imbi2d 329 . . . . . 6 (𝑤 = 𝑥 → ((𝜑 → (𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )))) ↔ (𝜑 → (𝑥𝐷 → (𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))))))
64 sseq1 3659 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑤𝐷 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐷))
65 mpteq1 4770 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
6665oveq2d 6706 . . . . . . . . 9 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
67 eleq2 2719 . . . . . . . . . . 11 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦𝑤𝑦 ∈ (𝑥 ∪ {𝑧})))
6867ifbid 4141 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))
6968mpteq2dv 4778 . . . . . . . . 9 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )))
7066, 69eqeq12d 2666 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑧}) → ((𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) ↔ (𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))))
7164, 70imbi12d 333 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑧}) → ((𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 ))) ↔ ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )))))
7271imbi2d 329 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑧}) → ((𝜑 → (𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )))) ↔ (𝜑 → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))))))
73 sseq1 3659 . . . . . . . 8 (𝑤 = (𝑋 supp 0 ) → (𝑤𝐷 ↔ (𝑋 supp 0 ) ⊆ 𝐷))
74 mpteq1 4770 . . . . . . . . . 10 (𝑤 = (𝑋 supp 0 ) → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
7574oveq2d 6706 . . . . . . . . 9 (𝑤 = (𝑋 supp 0 ) → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
76 eleq2 2719 . . . . . . . . . . 11 (𝑤 = (𝑋 supp 0 ) → (𝑦𝑤𝑦 ∈ (𝑋 supp 0 )))
7776ifbid 4141 . . . . . . . . . 10 (𝑤 = (𝑋 supp 0 ) → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))
7877mpteq2dv 4778 . . . . . . . . 9 (𝑤 = (𝑋 supp 0 ) → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
7975, 78eqeq12d 2666 . . . . . . . 8 (𝑤 = (𝑋 supp 0 ) → ((𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) ↔ (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))))
8073, 79imbi12d 333 . . . . . . 7 (𝑤 = (𝑋 supp 0 ) → ((𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 ))) ↔ ((𝑋 supp 0 ) ⊆ 𝐷 → (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))))
8180imbi2d 329 . . . . . 6 (𝑤 = (𝑋 supp 0 ) → ((𝜑 → (𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )))) ↔ (𝜑 → ((𝑋 supp 0 ) ⊆ 𝐷 → (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))))))
82 mplcoe1.i . . . . . . . . 9 (𝜑𝐼𝑊)
83 mplcoe1.r . . . . . . . . . 10 (𝜑𝑅 ∈ Ring)
84 ringgrp 18598 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
8583, 84syl 17 . . . . . . . . 9 (𝜑𝑅 ∈ Grp)
861, 4, 17, 44, 82, 85mpl0 19489 . . . . . . . 8 (𝜑 → (0g𝑃) = (𝐷 × { 0 }))
87 fconstmpt 5197 . . . . . . . 8 (𝐷 × { 0 }) = (𝑦𝐷0 )
8886, 87syl6eq 2701 . . . . . . 7 (𝜑 → (0g𝑃) = (𝑦𝐷0 ))
8988a1d 25 . . . . . 6 (𝜑 → (∅ ⊆ 𝐷 → (0g𝑃) = (𝑦𝐷0 )))
90 ssun1 3809 . . . . . . . . . . 11 𝑥 ⊆ (𝑥 ∪ {𝑧})
91 sstr2 3643 . . . . . . . . . . 11 (𝑥 ⊆ (𝑥 ∪ {𝑧}) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷𝑥𝐷))
9290, 91ax-mp 5 . . . . . . . . . 10 ((𝑥 ∪ {𝑧}) ⊆ 𝐷𝑥𝐷)
9392imim1i 63 . . . . . . . . 9 ((𝑥𝐷 → (𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))))
94 oveq1 6697 . . . . . . . . . . . 12 ((𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) → ((𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))(+g𝑃)((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))) = ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))(+g𝑃)((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))))
95 eqid 2651 . . . . . . . . . . . . . 14 (+g𝑃) = (+g𝑃)
961mplring 19500 . . . . . . . . . . . . . . . . 17 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
9782, 83, 96syl2anc 694 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ Ring)
98 ringcmn 18627 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
9997, 98syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑃 ∈ CMnd)
10099adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ CMnd)
101 simprll 819 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥 ∈ Fin)
102 simprr 811 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑥 ∪ {𝑧}) ⊆ 𝐷)
103102unssad 3823 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥𝐷)
104103sselda 3636 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝑥) → 𝑘𝐷)
10582adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝐼𝑊)
10683adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑅 ∈ Ring)
1071mpllmod 19499 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ LMod)
108105, 106, 107syl2anc 694 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → 𝑃 ∈ LMod)
1096ffvelrnda 6399 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → (𝑋𝑘) ∈ (Base‘𝑅))
1101, 82, 83mplsca 19493 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑅 = (Scalar‘𝑃))
111110adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝑅 = (Scalar‘𝑃))
112111fveq2d 6233 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
113109, 112eleqtrd 2732 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → (𝑋𝑘) ∈ (Base‘(Scalar‘𝑃)))
114 mplcoe1.o . . . . . . . . . . . . . . . . . 18 1 = (1r𝑅)
115 simpr 476 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑘𝐷)
1161, 3, 17, 114, 4, 105, 106, 115mplmon 19511 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵)
117 eqid 2651 . . . . . . . . . . . . . . . . . 18 (Scalar‘𝑃) = (Scalar‘𝑃)
118 mplcoe1.n . . . . . . . . . . . . . . . . . 18 · = ( ·𝑠𝑃)
119 eqid 2651 . . . . . . . . . . . . . . . . . 18 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
1203, 117, 118, 119lmodvscl 18928 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ LMod ∧ (𝑋𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
121108, 113, 116, 120syl3anc 1366 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐷) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
122121adantlr 751 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝐷) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
123104, 122syldan 486 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝑥) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
124 vex 3234 . . . . . . . . . . . . . . 15 𝑧 ∈ V
125124a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧 ∈ V)
126 simprlr 820 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ¬ 𝑧𝑥)
12782, 83, 107syl2anc 694 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ LMod)
128127adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ LMod)
1296adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑋:𝐷⟶(Base‘𝑅))
130102unssbd 3824 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → {𝑧} ⊆ 𝐷)
131124snss 4348 . . . . . . . . . . . . . . . . . 18 (𝑧𝐷 ↔ {𝑧} ⊆ 𝐷)
132130, 131sylibr 224 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧𝐷)
133129, 132ffvelrnd 6400 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋𝑧) ∈ (Base‘𝑅))
134110adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 = (Scalar‘𝑃))
135134fveq2d 6233 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
136133, 135eleqtrd 2732 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋𝑧) ∈ (Base‘(Scalar‘𝑃)))
13782adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐼𝑊)
13883adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 ∈ Ring)
1391, 3, 17, 114, 4, 137, 138, 132mplmon 19511 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) ∈ 𝐵)
1403, 117, 118, 119lmodvscl 18928 . . . . . . . . . . . . . . 15 ((𝑃 ∈ LMod ∧ (𝑋𝑧) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) ∈ 𝐵) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) ∈ 𝐵)
141128, 136, 139, 140syl3anc 1366 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) ∈ 𝐵)
142 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑋𝑘) = (𝑋𝑧))
143 equequ2 1999 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑧 → (𝑦 = 𝑘𝑦 = 𝑧))
144143ifbid 4141 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → if(𝑦 = 𝑘, 1 , 0 ) = if(𝑦 = 𝑧, 1 , 0 ))
145144mpteq2dv 4778 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))
146142, 145oveq12d 6708 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))))
1473, 95, 100, 101, 123, 125, 126, 141, 146gsumunsn 18405 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = ((𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))(+g𝑃)((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))))
148 eqid 2651 . . . . . . . . . . . . . . 15 (+g𝑅) = (+g𝑅)
149129ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (𝑋𝑦) ∈ (Base‘𝑅))
1502, 17ring0cl 18615 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
15183, 150syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑0 ∈ (Base‘𝑅))
152151ad2antrr 762 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → 0 ∈ (Base‘𝑅))
153149, 152ifcld 4164 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → if(𝑦𝑥, (𝑋𝑦), 0 ) ∈ (Base‘𝑅))
154 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))
155153, 154fmptd 6425 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )):𝐷⟶(Base‘𝑅))
156 fvex 6239 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) ∈ V
157156, 15elmap 7928 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ ((Base‘𝑅) ↑𝑚 𝐷) ↔ (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )):𝐷⟶(Base‘𝑅))
158155, 157sylibr 224 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ ((Base‘𝑅) ↑𝑚 𝐷))
15933, 2, 4, 34, 137psrbas 19426 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑𝑚 𝐷))
160158, 159eleqtrrd 2733 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)))
16115mptex 6527 . . . . . . . . . . . . . . . . . . 19 (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V
162 funmpt 5964 . . . . . . . . . . . . . . . . . . 19 Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))
163161, 162, 193pm3.2i 1259 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V ∧ Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∧ 0 ∈ V)
164163a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V ∧ Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∧ 0 ∈ V))
165 eldifn 3766 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝐷𝑥) → ¬ 𝑦𝑥)
166165adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷𝑥)) → ¬ 𝑦𝑥)
167166iffalsed 4130 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷𝑥)) → if(𝑦𝑥, (𝑋𝑦), 0 ) = 0 )
16815a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐷 ∈ V)
169167, 168suppss2 7374 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) supp 0 ) ⊆ 𝑥)
170 suppssfifsupp 8331 . . . . . . . . . . . . . . . . 17 ((((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V ∧ Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∧ 0 ∈ V) ∧ (𝑥 ∈ Fin ∧ ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) supp 0 ) ⊆ 𝑥)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) finSupp 0 )
171164, 101, 169, 170syl12anc 1364 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) finSupp 0 )
1721, 33, 34, 17, 3mplelbas 19478 . . . . . . . . . . . . . . . 16 ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ 𝐵 ↔ ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) finSupp 0 ))
173160, 171, 172sylanbrc 699 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ 𝐵)
1741, 3, 148, 95, 173, 141mpladd 19490 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))(+g𝑃)((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))) = ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∘𝑓 (+g𝑅)((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))))
175 ovexd 6720 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) ∈ V)
176 eqidd 2652 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))
177 eqid 2651 . . . . . . . . . . . . . . . . 17 (.r𝑅) = (.r𝑅)
1781, 118, 2, 3, 177, 4, 133, 139mplvsca 19495 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) = ((𝐷 × {(𝑋𝑧)}) ∘𝑓 (.r𝑅)(𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))))
179133adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (𝑋𝑧) ∈ (Base‘𝑅))
1802, 114ringidcl 18614 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
181180, 150ifcld 4164 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∈ Ring → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
18283, 181syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
183182ad2antrr 762 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
184 fconstmpt 5197 . . . . . . . . . . . . . . . . . 18 (𝐷 × {(𝑋𝑧)}) = (𝑦𝐷 ↦ (𝑋𝑧))
185184a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝐷 × {(𝑋𝑧)}) = (𝑦𝐷 ↦ (𝑋𝑧)))
186 eqidd 2652 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))
187168, 179, 183, 185, 186offval2 6956 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝐷 × {(𝑋𝑧)}) ∘𝑓 (.r𝑅)(𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) = (𝑦𝐷 ↦ ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 ))))
188178, 187eqtrd 2685 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) = (𝑦𝐷 ↦ ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 ))))
189168, 153, 175, 176, 188offval2 6956 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∘𝑓 (+g𝑅)((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))) = (𝑦𝐷 ↦ (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅)((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )))))
190138, 84syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 ∈ Grp)
1912, 148, 17grplid 17499 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Grp ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
192190, 133, 191syl2anc 694 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
193192ad2antrr 762 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
194 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ {𝑧})
195 velsn 4226 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
196194, 195sylib 208 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 = 𝑧)
197196fveq2d 6233 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → (𝑋𝑦) = (𝑋𝑧))
198193, 197eqtr4d 2688 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑦))
199126ad2antrr 762 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑧𝑥)
200196, 199eqneltrd 2749 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑦𝑥)
201200iffalsed 4130 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦𝑥, (𝑋𝑦), 0 ) = 0 )
202196iftrued 4127 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 1 )
203202oveq2d 6706 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋𝑧)(.r𝑅) 1 ))
2042, 177, 114ringridm 18618 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
205138, 133, 204syl2anc 694 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
206205ad2antrr 762 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
207203, 206eqtrd 2685 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = (𝑋𝑧))
208201, 207oveq12d 6708 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅)((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 ))) = ( 0 (+g𝑅)(𝑋𝑧)))
209 elun2 3814 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑧} → 𝑦 ∈ (𝑥 ∪ {𝑧}))
210209adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ (𝑥 ∪ {𝑧}))
211210iftrued 4127 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ) = (𝑋𝑦))
212198, 208, 2113eqtr4d 2695 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅)((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 ))) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))
21385ad2antrr 762 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → 𝑅 ∈ Grp)
2142, 148, 17grprid 17500 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Grp ∧ if(𝑦𝑥, (𝑋𝑦), 0 ) ∈ (Base‘𝑅)) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
215213, 153, 214syl2anc 694 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
216215adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
217 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 ∈ {𝑧})
218217, 195sylnib 317 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 = 𝑧)
219218iffalsed 4130 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 0 )
220219oveq2d 6706 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋𝑧)(.r𝑅) 0 ))
2212, 177, 17ringrz 18634 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
222138, 133, 221syl2anc 694 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
223222ad2antrr 762 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
224220, 223eqtrd 2685 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = 0 )
225224oveq2d 6706 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅)((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 ))) = (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ))
226 biorf 419 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ {𝑧} → (𝑦𝑥 ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥)))
227 elun 3786 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦𝑥𝑦 ∈ {𝑧}))
228 orcom 401 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝑥𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥))
229227, 228bitri 264 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥))
230226, 229syl6rbbr 279 . . . . . . . . . . . . . . . . . . 19 𝑦 ∈ {𝑧} → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦𝑥))
231230adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦𝑥))
232231ifbid 4141 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
233216, 225, 2323eqtr4d 2695 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅)((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 ))) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))
234212, 233pm2.61dan 849 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅)((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 ))) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))
235234mpteq2dva 4777 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅)((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )))
236174, 189, 2353eqtrrd 2690 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )) = ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))(+g𝑃)((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))))
237147, 236eqeq12d 2666 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )) ↔ ((𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))(+g𝑃)((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))) = ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))(+g𝑃)((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))))))
23894, 237syl5ibr 236 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) → (𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))))
239238expr 642 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧𝑥)) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → ((𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) → (𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )))))
240239a2d 29 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧𝑥)) → (((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )))))
24193, 240syl5 34 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧𝑥)) → ((𝑥𝐷 → (𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )))))
242241expcom 450 . . . . . . 7 ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) → (𝜑 → ((𝑥𝐷 → (𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))))))
243242a2d 29 . . . . . 6 ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) → ((𝜑 → (𝑥𝐷 → (𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))) → (𝜑 → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))))))
24454, 63, 72, 81, 89, 243findcard2s 8242 . . . . 5 ((𝑋 supp 0 ) ∈ Fin → (𝜑 → ((𝑋 supp 0 ) ⊆ 𝐷 → (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))))
24538, 244mpcom 38 . . . 4 (𝜑 → ((𝑋 supp 0 ) ⊆ 𝐷 → (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))))
24632, 245mpd 15 . . 3 (𝜑 → (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
24728, 246eqtr4d 2688 . 2 (𝜑𝑋 = (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
24832resmptd 5487 . . . 4 (𝜑 → ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ↾ (𝑋 supp 0 )) = (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
249248oveq2d 6706 . . 3 (𝜑 → (𝑃 Σg ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ↾ (𝑋 supp 0 ))) = (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
250 eqid 2651 . . . . 5 (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
251121, 250fmptd 6425 . . . 4 (𝜑 → (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))):𝐷𝐵)
2526, 13, 16, 20suppssr 7371 . . . . . . 7 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋𝑘) = 0 )
253252oveq1d 6705 . . . . . 6 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
254 eldifi 3765 . . . . . . 7 (𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 )) → 𝑘𝐷)
255111fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑘𝐷) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
25617, 255syl5eq 2697 . . . . . . . . 9 ((𝜑𝑘𝐷) → 0 = (0g‘(Scalar‘𝑃)))
257256oveq1d 6705 . . . . . . . 8 ((𝜑𝑘𝐷) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
258 eqid 2651 . . . . . . . . . 10 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
2593, 117, 118, 258, 44lmod0vs 18944 . . . . . . . . 9 ((𝑃 ∈ LMod ∧ (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵) → ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
260108, 116, 259syl2anc 694 . . . . . . . 8 ((𝜑𝑘𝐷) → ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
261257, 260eqtrd 2685 . . . . . . 7 ((𝜑𝑘𝐷) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
262254, 261sylan2 490 . . . . . 6 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
263253, 262eqtrd 2685 . . . . 5 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
264263, 16suppss2 7374 . . . 4 (𝜑 → ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) supp (0g𝑃)) ⊆ (𝑋 supp 0 ))
26515mptex 6527 . . . . . . 7 (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V
266 funmpt 5964 . . . . . . 7 Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
267 fvex 6239 . . . . . . 7 (0g𝑃) ∈ V
268265, 266, 2673pm3.2i 1259 . . . . . 6 ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V ∧ Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∧ (0g𝑃) ∈ V)
269268a1i 11 . . . . 5 (𝜑 → ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V ∧ Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∧ (0g𝑃) ∈ V))
270 suppssfifsupp 8331 . . . . 5 ((((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V ∧ Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∧ (0g𝑃) ∈ V) ∧ ((𝑋 supp 0 ) ∈ Fin ∧ ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) supp (0g𝑃)) ⊆ (𝑋 supp 0 ))) → (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) finSupp (0g𝑃))
271269, 38, 264, 270syl12anc 1364 . . . 4 (𝜑 → (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) finSupp (0g𝑃))
2723, 44, 99, 16, 251, 264, 271gsumres 18360 . . 3 (𝜑 → (𝑃 Σg ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ↾ (𝑋 supp 0 ))) = (𝑃 Σg (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
273249, 272eqtr3d 2687 . 2 (𝜑 → (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σg (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
274247, 273eqtrd 2685 1 (𝜑𝑋 = (𝑃 Σg (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  cdif 3604  cun 3605  wss 3607  c0 3948  ifcif 4119  {csn 4210   class class class wbr 4685  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  cres 5145  cima 5146  Fun wfun 5920  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937   supp csupp 7340  𝑚 cmap 7899  Fincfn 7997   finSupp cfsupp 8316  cn 11058  0cn0 11330  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147   Σg cgsu 16148  Grpcgrp 17469  CMndccmn 18239  1rcur 18547  Ringcrg 18593  LModclmod 18911   mPwSer cmps 19399   mPoly cmpl 19401
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-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-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-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-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-tset 16007  df-0g 16149  df-gsum 16150  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-subrg 18826  df-lmod 18913  df-lss 18981  df-psr 19404  df-mpl 19406
This theorem is referenced by:  mplbas2  19518  mplcoe4  19551  ply1coe  19714
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