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Theorem mplcoe1 21318
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 2736 . . . . . 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 21284 . . . . 5 (𝜑𝑋:𝐷⟶(Base‘𝑅))
76feqmptd 6876 . . . 4 (𝜑𝑋 = (𝑦𝐷 ↦ (𝑋𝑦)))
8 iftrue 4476 . . . . . . 7 (𝑦 ∈ (𝑋 supp 0 ) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
98adantl 482 . . . . . 6 (((𝜑𝑦𝐷) ∧ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
10 eldif 3906 . . . . . . . 8 (𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 )) ↔ (𝑦𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )))
11 ssidd 3953 . . . . . . . . . . 11 (𝜑 → (𝑋 supp 0 ) ⊆ (𝑋 supp 0 ))
12 ovex 7349 . . . . . . . . . . . . 13 (ℕ0m 𝐼) ∈ V
134, 12rabex2 5272 . . . . . . . . . . . 12 𝐷 ∈ V
1413a1i 11 . . . . . . . . . . 11 (𝜑𝐷 ∈ V)
15 mplcoe1.z . . . . . . . . . . . . 13 0 = (0g𝑅)
1615fvexi 6825 . . . . . . . . . . . 12 0 ∈ V
1716a1i 11 . . . . . . . . . . 11 (𝜑0 ∈ V)
186, 11, 14, 17suppssr 8060 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋𝑦) = 0 )
1918ifeq2d 4490 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), (𝑋𝑦)) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))
20 ifid 4510 . . . . . . . . 9 if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), (𝑋𝑦)) = (𝑋𝑦)
2119, 20eqtr3di 2791 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2210, 21sylan2br 595 . . . . . . 7 ((𝜑 ∧ (𝑦𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2322anassrs 468 . . . . . 6 (((𝜑𝑦𝐷) ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
249, 23pm2.61dan 810 . . . . 5 ((𝜑𝑦𝐷) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2524mpteq2dva 5186 . . . 4 (𝜑 → (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ (𝑋𝑦)))
267, 25eqtr4d 2779 . . 3 (𝜑𝑋 = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
27 suppssdm 8041 . . . . 5 (𝑋 supp 0 ) ⊆ dom 𝑋
2827, 6fssdm 6657 . . . 4 (𝜑 → (𝑋 supp 0 ) ⊆ 𝐷)
29 eqid 2736 . . . . . . . . 9 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
30 eqid 2736 . . . . . . . . 9 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
311, 29, 30, 15, 3mplelbas 21279 . . . . . . . 8 (𝑋𝐵 ↔ (𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 finSupp 0 ))
3231simprbi 497 . . . . . . 7 (𝑋𝐵𝑋 finSupp 0 )
335, 32syl 17 . . . . . 6 (𝜑𝑋 finSupp 0 )
3433fsuppimpd 9211 . . . . 5 (𝜑 → (𝑋 supp 0 ) ∈ Fin)
35 sseq1 3955 . . . . . . . 8 (𝑤 = ∅ → (𝑤𝐷 ↔ ∅ ⊆ 𝐷))
36 mpteq1 5179 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ ∅ ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
37 mpt0 6612 . . . . . . . . . . . 12 (𝑘 ∈ ∅ ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅
3836, 37eqtrdi 2792 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅)
3938oveq2d 7332 . . . . . . . . . 10 (𝑤 = ∅ → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σg ∅))
40 eqid 2736 . . . . . . . . . . 11 (0g𝑃) = (0g𝑃)
4140gsum0 18442 . . . . . . . . . 10 (𝑃 Σg ∅) = (0g𝑃)
4239, 41eqtrdi 2792 . . . . . . . . 9 (𝑤 = ∅ → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (0g𝑃))
43 noel 4274 . . . . . . . . . . . 12 ¬ 𝑦 ∈ ∅
44 eleq2 2825 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑦𝑤𝑦 ∈ ∅))
4543, 44mtbiri 326 . . . . . . . . . . 11 (𝑤 = ∅ → ¬ 𝑦𝑤)
4645iffalsed 4481 . . . . . . . . . 10 (𝑤 = ∅ → if(𝑦𝑤, (𝑋𝑦), 0 ) = 0 )
4746mpteq2dv 5188 . . . . . . . . 9 (𝑤 = ∅ → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷0 ))
4842, 47eqeq12d 2752 . . . . . . . 8 (𝑤 = ∅ → ((𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) ↔ (0g𝑃) = (𝑦𝐷0 )))
4935, 48imbi12d 344 . . . . . . 7 (𝑤 = ∅ → ((𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 ))) ↔ (∅ ⊆ 𝐷 → (0g𝑃) = (𝑦𝐷0 ))))
5049imbi2d 340 . . . . . 6 (𝑤 = ∅ → ((𝜑 → (𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )))) ↔ (𝜑 → (∅ ⊆ 𝐷 → (0g𝑃) = (𝑦𝐷0 )))))
51 sseq1 3955 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝐷𝑥𝐷))
52 mpteq1 5179 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
5352oveq2d 7332 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σg (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
54 eleq2 2825 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
5554ifbid 4493 . . . . . . . . . 10 (𝑤 = 𝑥 → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
5655mpteq2dv 5188 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))
5753, 56eqeq12d 2752 . . . . . . . 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 ))))
5851, 57imbi12d 344 . . . . . . 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 )))))
5958imbi2d 340 . . . . . 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 ))))))
60 sseq1 3955 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑤𝐷 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐷))
61 mpteq1 5179 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
6261oveq2d 7332 . . . . . . . . 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 ))))))
63 eleq2 2825 . . . . . . . . . . 11 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦𝑤𝑦 ∈ (𝑥 ∪ {𝑧})))
6463ifbid 4493 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))
6564mpteq2dv 5188 . . . . . . . . 9 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )))
6662, 65eqeq12d 2752 . . . . . . . 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 ))))
6760, 66imbi12d 344 . . . . . . 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 )))))
6867imbi2d 340 . . . . . 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 ))))))
69 sseq1 3955 . . . . . . . 8 (𝑤 = (𝑋 supp 0 ) → (𝑤𝐷 ↔ (𝑋 supp 0 ) ⊆ 𝐷))
70 mpteq1 5179 . . . . . . . . . 10 (𝑤 = (𝑋 supp 0 ) → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
7170oveq2d 7332 . . . . . . . . 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 ))))))
72 eleq2 2825 . . . . . . . . . . 11 (𝑤 = (𝑋 supp 0 ) → (𝑦𝑤𝑦 ∈ (𝑋 supp 0 )))
7372ifbid 4493 . . . . . . . . . 10 (𝑤 = (𝑋 supp 0 ) → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))
7473mpteq2dv 5188 . . . . . . . . 9 (𝑤 = (𝑋 supp 0 ) → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
7571, 74eqeq12d 2752 . . . . . . . 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 ))))
7669, 75imbi12d 344 . . . . . . 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 )))))
7776imbi2d 340 . . . . . 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 ))))))
78 mplcoe1.i . . . . . . . . 9 (𝜑𝐼𝑊)
79 mplcoe1.r . . . . . . . . . 10 (𝜑𝑅 ∈ Ring)
80 ringgrp 19860 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
8179, 80syl 17 . . . . . . . . 9 (𝜑𝑅 ∈ Grp)
821, 4, 15, 40, 78, 81mpl0 21292 . . . . . . . 8 (𝜑 → (0g𝑃) = (𝐷 × { 0 }))
83 fconstmpt 5667 . . . . . . . 8 (𝐷 × { 0 }) = (𝑦𝐷0 )
8482, 83eqtrdi 2792 . . . . . . 7 (𝜑 → (0g𝑃) = (𝑦𝐷0 ))
8584a1d 25 . . . . . 6 (𝜑 → (∅ ⊆ 𝐷 → (0g𝑃) = (𝑦𝐷0 )))
86 ssun1 4116 . . . . . . . . . . 11 𝑥 ⊆ (𝑥 ∪ {𝑧})
87 sstr2 3937 . . . . . . . . . . 11 (𝑥 ⊆ (𝑥 ∪ {𝑧}) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷𝑥𝐷))
8886, 87ax-mp 5 . . . . . . . . . 10 ((𝑥 ∪ {𝑧}) ⊆ 𝐷𝑥𝐷)
8988imim1i 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 ))))
90 oveq1 7323 . . . . . . . . . . . 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 )))))
91 eqid 2736 . . . . . . . . . . . . . 14 (+g𝑃) = (+g𝑃)
921mplring 21304 . . . . . . . . . . . . . . . . 17 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
9378, 79, 92syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ Ring)
94 ringcmn 19892 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
9593, 94syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑃 ∈ CMnd)
9695adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ CMnd)
97 simprll 776 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥 ∈ Fin)
98 simprr 770 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑥 ∪ {𝑧}) ⊆ 𝐷)
9998unssad 4131 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥𝐷)
10099sselda 3930 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝑥) → 𝑘𝐷)
10178adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝐼𝑊)
10279adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑅 ∈ Ring)
1031mpllmod 21303 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ LMod)
104101, 102, 103syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → 𝑃 ∈ LMod)
1056ffvelcdmda 7000 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → (𝑋𝑘) ∈ (Base‘𝑅))
1061, 78, 79mplsca 21297 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑅 = (Scalar‘𝑃))
107106adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝑅 = (Scalar‘𝑃))
108107fveq2d 6815 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
109105, 108eleqtrd 2839 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → (𝑋𝑘) ∈ (Base‘(Scalar‘𝑃)))
110 mplcoe1.o . . . . . . . . . . . . . . . . . 18 1 = (1r𝑅)
111 simpr 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑘𝐷)
1121, 3, 15, 110, 4, 101, 102, 111mplmon 21316 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵)
113 eqid 2736 . . . . . . . . . . . . . . . . . 18 (Scalar‘𝑃) = (Scalar‘𝑃)
114 mplcoe1.n . . . . . . . . . . . . . . . . . 18 · = ( ·𝑠𝑃)
115 eqid 2736 . . . . . . . . . . . . . . . . . 18 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
1163, 113, 114, 115lmodvscl 20220 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ LMod ∧ (𝑋𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
117104, 109, 112, 116syl3anc 1370 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐷) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
118117adantlr 712 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝐷) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
119100, 118syldan 591 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝑥) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
120 vex 3444 . . . . . . . . . . . . . . 15 𝑧 ∈ V
121120a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧 ∈ V)
122 simprlr 777 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ¬ 𝑧𝑥)
12378, 79, 103syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ LMod)
124123adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ LMod)
1256adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑋:𝐷⟶(Base‘𝑅))
12698unssbd 4132 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → {𝑧} ⊆ 𝐷)
127120snss 4730 . . . . . . . . . . . . . . . . . 18 (𝑧𝐷 ↔ {𝑧} ⊆ 𝐷)
128126, 127sylibr 233 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧𝐷)
129125, 128ffvelcdmd 7001 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋𝑧) ∈ (Base‘𝑅))
130106adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 = (Scalar‘𝑃))
131130fveq2d 6815 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
132129, 131eleqtrd 2839 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋𝑧) ∈ (Base‘(Scalar‘𝑃)))
13378adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐼𝑊)
13479adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 ∈ Ring)
1351, 3, 15, 110, 4, 133, 134, 128mplmon 21316 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) ∈ 𝐵)
1363, 113, 114, 115lmodvscl 20220 . . . . . . . . . . . . . . 15 ((𝑃 ∈ LMod ∧ (𝑋𝑧) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) ∈ 𝐵) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) ∈ 𝐵)
137124, 132, 135, 136syl3anc 1370 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) ∈ 𝐵)
138 fveq2 6811 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑋𝑘) = (𝑋𝑧))
139 equequ2 2028 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑧 → (𝑦 = 𝑘𝑦 = 𝑧))
140139ifbid 4493 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → if(𝑦 = 𝑘, 1 , 0 ) = if(𝑦 = 𝑧, 1 , 0 ))
141140mpteq2dv 5188 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))
142138, 141oveq12d 7334 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))))
1433, 91, 96, 97, 119, 121, 122, 137, 142gsumunsn 19633 . . . . . . . . . . . . 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 )))))
144 eqid 2736 . . . . . . . . . . . . . . 15 (+g𝑅) = (+g𝑅)
145125ffvelcdmda 7000 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (𝑋𝑦) ∈ (Base‘𝑅))
1462, 15ring0cl 19880 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
14779, 146syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑0 ∈ (Base‘𝑅))
148147ad2antrr 723 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → 0 ∈ (Base‘𝑅))
149145, 148ifcld 4516 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → if(𝑦𝑥, (𝑋𝑦), 0 ) ∈ (Base‘𝑅))
150149fmpttd 7028 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )):𝐷⟶(Base‘𝑅))
151 fvex 6824 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) ∈ V
152151, 13elmap 8708 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ ((Base‘𝑅) ↑m 𝐷) ↔ (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )):𝐷⟶(Base‘𝑅))
153150, 152sylibr 233 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ ((Base‘𝑅) ↑m 𝐷))
15429, 2, 4, 30, 133psrbas 21227 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m 𝐷))
155153, 154eleqtrrd 2840 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)))
15613mptex 7138 . . . . . . . . . . . . . . . . . . 19 (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V
157 funmpt 6508 . . . . . . . . . . . . . . . . . . 19 Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))
158156, 157, 163pm3.2i 1338 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V ∧ Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∧ 0 ∈ V)
159158a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V ∧ Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∧ 0 ∈ V))
160 eldifn 4072 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝐷𝑥) → ¬ 𝑦𝑥)
161160adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷𝑥)) → ¬ 𝑦𝑥)
162161iffalsed 4481 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷𝑥)) → if(𝑦𝑥, (𝑋𝑦), 0 ) = 0 )
16313a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐷 ∈ V)
164162, 163suppss2 8064 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) supp 0 ) ⊆ 𝑥)
165 suppssfifsupp 9219 . . . . . . . . . . . . . . . . 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 )
166159, 97, 164, 165syl12anc 834 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) finSupp 0 )
1671, 29, 30, 15, 3mplelbas 21279 . . . . . . . . . . . . . . . 16 ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ 𝐵 ↔ ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) finSupp 0 ))
168155, 166, 167sylanbrc 583 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ 𝐵)
1691, 3, 144, 91, 168, 137mpladd 21293 . . . . . . . . . . . . . 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 )))))
170 ovexd 7351 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) ∈ V)
171 eqidd 2737 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))
172 eqid 2736 . . . . . . . . . . . . . . . . 17 (.r𝑅) = (.r𝑅)
1731, 114, 2, 3, 172, 4, 129, 135mplvsca 21299 . . . . . . . . . . . . . . . 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 ))))
174129adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (𝑋𝑧) ∈ (Base‘𝑅))
1752, 110ringidcl 19879 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
176175, 146ifcld 4516 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∈ Ring → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
17779, 176syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
178177ad2antrr 723 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
179 fconstmpt 5667 . . . . . . . . . . . . . . . . . 18 (𝐷 × {(𝑋𝑧)}) = (𝑦𝐷 ↦ (𝑋𝑧))
180179a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝐷 × {(𝑋𝑧)}) = (𝑦𝐷 ↦ (𝑋𝑧)))
181 eqidd 2737 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))
182163, 174, 178, 180, 181offval2 7594 . . . . . . . . . . . . . . . 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 ))))
183173, 182eqtrd 2776 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) = (𝑦𝐷 ↦ ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 ))))
184163, 149, 170, 171, 183offval2 7594 . . . . . . . . . . . . . 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 )))))
185134, 80syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 ∈ Grp)
1862, 144, 15grplid 18682 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Grp ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
187185, 129, 186syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
188187ad2antrr 723 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
189 simpr 485 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ {𝑧})
190 velsn 4586 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
191189, 190sylib 217 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 = 𝑧)
192191fveq2d 6815 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → (𝑋𝑦) = (𝑋𝑧))
193188, 192eqtr4d 2779 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑦))
194122ad2antrr 723 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑧𝑥)
195191, 194eqneltrd 2856 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑦𝑥)
196195iffalsed 4481 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦𝑥, (𝑋𝑦), 0 ) = 0 )
197191iftrued 4478 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 1 )
198197oveq2d 7332 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋𝑧)(.r𝑅) 1 ))
1992, 172, 110ringridm 19883 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
200134, 129, 199syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
201200ad2antrr 723 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
202198, 201eqtrd 2776 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = (𝑋𝑧))
203196, 202oveq12d 7334 . . . . . . . . . . . . . . . . 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)))
204 elun2 4121 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑧} → 𝑦 ∈ (𝑥 ∪ {𝑧}))
205204adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ (𝑥 ∪ {𝑧}))
206205iftrued 4478 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ) = (𝑋𝑦))
207193, 203, 2063eqtr4d 2786 . . . . . . . . . . . . . . . 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 ))
20881ad2antrr 723 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → 𝑅 ∈ Grp)
2092, 144, 15grprid 18683 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Grp ∧ if(𝑦𝑥, (𝑋𝑦), 0 ) ∈ (Base‘𝑅)) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
210208, 149, 209syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
211210adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
212 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 ∈ {𝑧})
213212, 190sylnib 327 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 = 𝑧)
214213iffalsed 4481 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 0 )
215214oveq2d 7332 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋𝑧)(.r𝑅) 0 ))
2162, 172, 15ringrz 19899 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
217134, 129, 216syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
218217ad2antrr 723 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
219215, 218eqtrd 2776 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = 0 )
220219oveq2d 7332 . . . . . . . . . . . . . . . . 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 ))
221 elun 4093 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦𝑥𝑦 ∈ {𝑧}))
222 orcom 867 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝑥𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥))
223221, 222bitri 274 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥))
224 biorf 934 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ {𝑧} → (𝑦𝑥 ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥)))
225223, 224bitr4id 289 . . . . . . . . . . . . . . . . . . 19 𝑦 ∈ {𝑧} → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦𝑥))
226225adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦𝑥))
227226ifbid 4493 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
228211, 220, 2273eqtr4d 2786 . . . . . . . . . . . . . . . 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 ))
229207, 228pm2.61dan 810 . . . . . . . . . . . . . . 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 ))
230229mpteq2dva 5186 . . . . . . . . . . . . . 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 )))
231169, 184, 2303eqtrrd 2781 . . . . . . . . . . . . 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 )))))
232143, 231eqeq12d 2752 . . . . . . . . . . . 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 ))))))
23390, 232syl5ibr 245 . . . . . . . . . . 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 ))))
234233expr 457 . . . . . . . . . 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 )))))
235234a2d 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 )))))
23689, 235syl5 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 )))))
237236expcom 414 . . . . . . 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 ))))))
238237a2d 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 ))))))
23950, 59, 68, 77, 85, 238findcard2s 9008 . . . . 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 )))))
24034, 239mpcom 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 ))))
24128, 240mpd 15 . . 3 (𝜑 → (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
24226, 241eqtr4d 2779 . 2 (𝜑𝑋 = (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
24328resmptd 5967 . . . 4 (𝜑 → ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ↾ (𝑋 supp 0 )) = (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
244243oveq2d 7332 . . 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 ))))))
245117fmpttd 7028 . . . 4 (𝜑 → (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))):𝐷𝐵)
2466, 11, 14, 17suppssr 8060 . . . . . . 7 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋𝑘) = 0 )
247246oveq1d 7331 . . . . . 6 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
248 eldifi 4071 . . . . . . 7 (𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 )) → 𝑘𝐷)
249107fveq2d 6815 . . . . . . . . . 10 ((𝜑𝑘𝐷) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
25015, 249eqtrid 2788 . . . . . . . . 9 ((𝜑𝑘𝐷) → 0 = (0g‘(Scalar‘𝑃)))
251250oveq1d 7331 . . . . . . . 8 ((𝜑𝑘𝐷) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
252 eqid 2736 . . . . . . . . . 10 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
2533, 113, 114, 252, 40lmod0vs 20236 . . . . . . . . 9 ((𝑃 ∈ LMod ∧ (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵) → ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
254104, 112, 253syl2anc 584 . . . . . . . 8 ((𝜑𝑘𝐷) → ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
255251, 254eqtrd 2776 . . . . . . 7 ((𝜑𝑘𝐷) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
256248, 255sylan2 593 . . . . . 6 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
257247, 256eqtrd 2776 . . . . 5 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
258257, 14suppss2 8064 . . . 4 (𝜑 → ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) supp (0g𝑃)) ⊆ (𝑋 supp 0 ))
25913mptex 7138 . . . . . . 7 (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V
260 funmpt 6508 . . . . . . 7 Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
261 fvex 6824 . . . . . . 7 (0g𝑃) ∈ V
262259, 260, 2613pm3.2i 1338 . . . . . 6 ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V ∧ Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∧ (0g𝑃) ∈ V)
263262a1i 11 . . . . 5 (𝜑 → ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V ∧ Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∧ (0g𝑃) ∈ V))
264 suppssfifsupp 9219 . . . . 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))
265263, 34, 258, 264syl12anc 834 . . . 4 (𝜑 → (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) finSupp (0g𝑃))
2663, 40, 95, 14, 245, 258, 265gsumres 19586 . . 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 ))))))
267244, 266eqtr3d 2778 . 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 ))))))
268242, 267eqtrd 2776 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 205  wa 396  wo 844  w3a 1086   = wceq 1540  wcel 2105  {crab 3403  Vcvv 3440  cdif 3893  cun 3894  wss 3896  c0 4266  ifcif 4470  {csn 4570   class class class wbr 5086  cmpt 5169   × cxp 5605  ccnv 5606  cres 5609  cima 5610  Fun wfun 6459  wf 6461  cfv 6465  (class class class)co 7316  f cof 7572   supp csupp 8025  m cmap 8664  Fincfn 8782   finSupp cfsupp 9204  cn 12052  0cn0 12312  Basecbs 16986  +gcplusg 17036  .rcmulr 17037  Scalarcsca 17039   ·𝑠 cvsca 17040  0gc0g 17224   Σg cgsu 17225  Grpcgrp 18650  CMndccmn 19458  1rcur 19809  Ringcrg 19855  LModclmod 20203   mPwSer cmps 21187   mPoly cmpl 21189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629  ax-cnex 11006  ax-resscn 11007  ax-1cn 11008  ax-icn 11009  ax-addcl 11010  ax-addrcl 11011  ax-mulcl 11012  ax-mulrcl 11013  ax-mulcom 11014  ax-addass 11015  ax-mulass 11016  ax-distr 11017  ax-i2m1 11018  ax-1ne0 11019  ax-1rid 11020  ax-rnegex 11021  ax-rrecex 11022  ax-cnre 11023  ax-pre-lttri 11024  ax-pre-lttrn 11025  ax-pre-ltadd 11026  ax-pre-mulgt0 11027
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4850  df-int 4892  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-se 5563  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-isom 6474  df-riota 7273  df-ov 7319  df-oprab 7320  df-mpo 7321  df-of 7574  df-ofr 7575  df-om 7759  df-1st 7877  df-2nd 7878  df-supp 8026  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-rdg 8289  df-1o 8345  df-er 8547  df-map 8666  df-pm 8667  df-ixp 8735  df-en 8783  df-dom 8784  df-sdom 8785  df-fin 8786  df-fsupp 9205  df-oi 9345  df-card 9774  df-pnf 11090  df-mnf 11091  df-xr 11092  df-ltxr 11093  df-le 11094  df-sub 11286  df-neg 11287  df-nn 12053  df-2 12115  df-3 12116  df-4 12117  df-5 12118  df-6 12119  df-7 12120  df-8 12121  df-9 12122  df-n0 12313  df-z 12399  df-uz 12662  df-fz 13319  df-fzo 13462  df-seq 13801  df-hash 14124  df-struct 16922  df-sets 16939  df-slot 16957  df-ndx 16969  df-base 16987  df-ress 17016  df-plusg 17049  df-mulr 17050  df-sca 17052  df-vsca 17053  df-tset 17055  df-0g 17226  df-gsum 17227  df-mre 17369  df-mrc 17370  df-acs 17372  df-mgm 18400  df-sgrp 18449  df-mnd 18460  df-mhm 18504  df-submnd 18505  df-grp 18653  df-minusg 18654  df-sbg 18655  df-mulg 18774  df-subg 18825  df-ghm 18905  df-cntz 18996  df-cmn 19460  df-abl 19461  df-mgp 19793  df-ur 19810  df-ring 19857  df-subrg 20101  df-lmod 20205  df-lss 20274  df-psr 21192  df-mpl 21194
This theorem is referenced by:  mplbas2  21323  mplcoe4  21359  ply1coe  21547
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