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Theorem mplcoe1 22072
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 2734 . . . . . 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 22035 . . . . 5 (𝜑𝑋:𝐷⟶(Base‘𝑅))
76feqmptd 6976 . . . 4 (𝜑𝑋 = (𝑦𝐷 ↦ (𝑋𝑦)))
8 iftrue 4536 . . . . . . 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 3972 . . . . . . . 8 (𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 )) ↔ (𝑦𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )))
11 ssidd 4018 . . . . . . . . . . 11 (𝜑 → (𝑋 supp 0 ) ⊆ (𝑋 supp 0 ))
12 ovex 7463 . . . . . . . . . . . . 13 (ℕ0m 𝐼) ∈ V
134, 12rabex2 5346 . . . . . . . . . . . 12 𝐷 ∈ V
1413a1i 11 . . . . . . . . . . 11 (𝜑𝐷 ∈ V)
15 mplcoe1.z . . . . . . . . . . . . 13 0 = (0g𝑅)
1615fvexi 6920 . . . . . . . . . . . 12 0 ∈ V
1716a1i 11 . . . . . . . . . . 11 (𝜑0 ∈ V)
186, 11, 14, 17suppssr 8218 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋𝑦) = 0 )
1918ifeq2d 4550 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), (𝑋𝑦)) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))
20 ifid 4570 . . . . . . . . 9 if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), (𝑋𝑦)) = (𝑋𝑦)
2119, 20eqtr3di 2789 . . . . . . . 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 467 . . . . . 6 (((𝜑𝑦𝐷) ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
249, 23pm2.61dan 813 . . . . 5 ((𝜑𝑦𝐷) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2524mpteq2dva 5247 . . . 4 (𝜑 → (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ (𝑋𝑦)))
267, 25eqtr4d 2777 . . 3 (𝜑𝑋 = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
27 suppssdm 8200 . . . . 5 (𝑋 supp 0 ) ⊆ dom 𝑋
2827, 6fssdm 6755 . . . 4 (𝜑 → (𝑋 supp 0 ) ⊆ 𝐷)
29 eqid 2734 . . . . . . . . 9 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
30 eqid 2734 . . . . . . . . 9 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
311, 29, 30, 15, 3mplelbas 22028 . . . . . . . 8 (𝑋𝐵 ↔ (𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 finSupp 0 ))
3231simprbi 496 . . . . . . 7 (𝑋𝐵𝑋 finSupp 0 )
335, 32syl 17 . . . . . 6 (𝜑𝑋 finSupp 0 )
3433fsuppimpd 9406 . . . . 5 (𝜑 → (𝑋 supp 0 ) ∈ Fin)
35 sseq1 4020 . . . . . . . 8 (𝑤 = ∅ → (𝑤𝐷 ↔ ∅ ⊆ 𝐷))
36 mpteq1 5240 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ ∅ ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
37 mpt0 6710 . . . . . . . . . . . 12 (𝑘 ∈ ∅ ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅
3836, 37eqtrdi 2790 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅)
3938oveq2d 7446 . . . . . . . . . 10 (𝑤 = ∅ → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σg ∅))
40 eqid 2734 . . . . . . . . . . 11 (0g𝑃) = (0g𝑃)
4140gsum0 18709 . . . . . . . . . 10 (𝑃 Σg ∅) = (0g𝑃)
4239, 41eqtrdi 2790 . . . . . . . . 9 (𝑤 = ∅ → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (0g𝑃))
43 noel 4343 . . . . . . . . . . . 12 ¬ 𝑦 ∈ ∅
44 eleq2 2827 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑦𝑤𝑦 ∈ ∅))
4543, 44mtbiri 327 . . . . . . . . . . 11 (𝑤 = ∅ → ¬ 𝑦𝑤)
4645iffalsed 4541 . . . . . . . . . 10 (𝑤 = ∅ → if(𝑦𝑤, (𝑋𝑦), 0 ) = 0 )
4746mpteq2dv 5249 . . . . . . . . 9 (𝑤 = ∅ → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷0 ))
4842, 47eqeq12d 2750 . . . . . . . 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 4020 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝐷𝑥𝐷))
52 mpteq1 5240 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
5352oveq2d 7446 . . . . . . . . 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 2827 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
5554ifbid 4553 . . . . . . . . . 10 (𝑤 = 𝑥 → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
5655mpteq2dv 5249 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))
5753, 56eqeq12d 2750 . . . . . . . 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 4020 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑤𝐷 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐷))
61 mpteq1 5240 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
6261oveq2d 7446 . . . . . . . . 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 2827 . . . . . . . . . . 11 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦𝑤𝑦 ∈ (𝑥 ∪ {𝑧})))
6463ifbid 4553 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))
6564mpteq2dv 5249 . . . . . . . . 9 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )))
6662, 65eqeq12d 2750 . . . . . . . 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 4020 . . . . . . . 8 (𝑤 = (𝑋 supp 0 ) → (𝑤𝐷 ↔ (𝑋 supp 0 ) ⊆ 𝐷))
70 mpteq1 5240 . . . . . . . . . 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 7446 . . . . . . . . 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 2827 . . . . . . . . . . 11 (𝑤 = (𝑋 supp 0 ) → (𝑦𝑤𝑦 ∈ (𝑋 supp 0 )))
7372ifbid 4553 . . . . . . . . . 10 (𝑤 = (𝑋 supp 0 ) → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))
7473mpteq2dv 5249 . . . . . . . . 9 (𝑤 = (𝑋 supp 0 ) → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
7571, 74eqeq12d 2750 . . . . . . . 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 20255 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
8179, 80syl 17 . . . . . . . . 9 (𝜑𝑅 ∈ Grp)
821, 4, 15, 40, 78, 81mpl0 22043 . . . . . . . 8 (𝜑 → (0g𝑃) = (𝐷 × { 0 }))
83 fconstmpt 5750 . . . . . . . 8 (𝐷 × { 0 }) = (𝑦𝐷0 )
8482, 83eqtrdi 2790 . . . . . . 7 (𝜑 → (0g𝑃) = (𝑦𝐷0 ))
8584a1d 25 . . . . . 6 (𝜑 → (∅ ⊆ 𝐷 → (0g𝑃) = (𝑦𝐷0 )))
86 ssun1 4187 . . . . . . . . . . 11 𝑥 ⊆ (𝑥 ∪ {𝑧})
87 sstr2 4001 . . . . . . . . . . 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 7437 . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . 14 (+g𝑃) = (+g𝑃)
921, 78, 79mplringd 22060 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ Ring)
93 ringcmn 20295 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
9492, 93syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑃 ∈ CMnd)
9594adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ CMnd)
96 simprll 779 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥 ∈ Fin)
97 simprr 773 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑥 ∪ {𝑧}) ⊆ 𝐷)
9897unssad 4202 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥𝐷)
9998sselda 3994 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝑥) → 𝑘𝐷)
10078adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝐼𝑊)
10179adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑅 ∈ Ring)
1021, 100, 101mpllmodd 22061 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → 𝑃 ∈ LMod)
1036ffvelcdmda 7103 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → (𝑋𝑘) ∈ (Base‘𝑅))
1041, 78, 79mplsca 22050 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑅 = (Scalar‘𝑃))
105104adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝑅 = (Scalar‘𝑃))
106105fveq2d 6910 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
107103, 106eleqtrd 2840 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → (𝑋𝑘) ∈ (Base‘(Scalar‘𝑃)))
108 mplcoe1.o . . . . . . . . . . . . . . . . . 18 1 = (1r𝑅)
109 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑘𝐷)
1101, 3, 15, 108, 4, 100, 101, 109mplmon 22070 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵)
111 eqid 2734 . . . . . . . . . . . . . . . . . 18 (Scalar‘𝑃) = (Scalar‘𝑃)
112 mplcoe1.n . . . . . . . . . . . . . . . . . 18 · = ( ·𝑠𝑃)
113 eqid 2734 . . . . . . . . . . . . . . . . . 18 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
1143, 111, 112, 113lmodvscl 20892 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ LMod ∧ (𝑋𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
115102, 107, 110, 114syl3anc 1370 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐷) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
116115adantlr 715 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝐷) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
11799, 116syldan 591 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝑥) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
118 vex 3481 . . . . . . . . . . . . . . 15 𝑧 ∈ V
119118a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧 ∈ V)
120 simprlr 780 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ¬ 𝑧𝑥)
1211, 78, 79mpllmodd 22061 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ LMod)
122121adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ LMod)
1236adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑋:𝐷⟶(Base‘𝑅))
12497unssbd 4203 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → {𝑧} ⊆ 𝐷)
125118snss 4789 . . . . . . . . . . . . . . . . . 18 (𝑧𝐷 ↔ {𝑧} ⊆ 𝐷)
126124, 125sylibr 234 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧𝐷)
127123, 126ffvelcdmd 7104 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋𝑧) ∈ (Base‘𝑅))
128104adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 = (Scalar‘𝑃))
129128fveq2d 6910 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
130127, 129eleqtrd 2840 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋𝑧) ∈ (Base‘(Scalar‘𝑃)))
13178adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐼𝑊)
13279adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 ∈ Ring)
1331, 3, 15, 108, 4, 131, 132, 126mplmon 22070 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) ∈ 𝐵)
1343, 111, 112, 113lmodvscl 20892 . . . . . . . . . . . . . . 15 ((𝑃 ∈ LMod ∧ (𝑋𝑧) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) ∈ 𝐵) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) ∈ 𝐵)
135122, 130, 133, 134syl3anc 1370 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) ∈ 𝐵)
136 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑋𝑘) = (𝑋𝑧))
137 equequ2 2022 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑧 → (𝑦 = 𝑘𝑦 = 𝑧))
138137ifbid 4553 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → if(𝑦 = 𝑘, 1 , 0 ) = if(𝑦 = 𝑧, 1 , 0 ))
139138mpteq2dv 5249 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))
140136, 139oveq12d 7448 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))))
1413, 91, 95, 96, 117, 119, 120, 135, 140gsumunsn 19992 . . . . . . . . . . . . 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 )))))
142 eqid 2734 . . . . . . . . . . . . . . 15 (+g𝑅) = (+g𝑅)
143123ffvelcdmda 7103 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (𝑋𝑦) ∈ (Base‘𝑅))
1442, 15ring0cl 20280 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
14579, 144syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑0 ∈ (Base‘𝑅))
146145ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → 0 ∈ (Base‘𝑅))
147143, 146ifcld 4576 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → if(𝑦𝑥, (𝑋𝑦), 0 ) ∈ (Base‘𝑅))
148147fmpttd 7134 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )):𝐷⟶(Base‘𝑅))
149 fvex 6919 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) ∈ V
150149, 13elmap 8909 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ ((Base‘𝑅) ↑m 𝐷) ↔ (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )):𝐷⟶(Base‘𝑅))
151148, 150sylibr 234 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ ((Base‘𝑅) ↑m 𝐷))
15229, 2, 4, 30, 131psrbas 21970 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m 𝐷))
153151, 152eleqtrrd 2841 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)))
15413mptex 7242 . . . . . . . . . . . . . . . . . . 19 (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V
155 funmpt 6605 . . . . . . . . . . . . . . . . . . 19 Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))
156154, 155, 163pm3.2i 1338 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V ∧ Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∧ 0 ∈ V)
157156a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V ∧ Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∧ 0 ∈ V))
158 eldifn 4141 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝐷𝑥) → ¬ 𝑦𝑥)
159158adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷𝑥)) → ¬ 𝑦𝑥)
160159iffalsed 4541 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷𝑥)) → if(𝑦𝑥, (𝑋𝑦), 0 ) = 0 )
16113a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐷 ∈ V)
162160, 161suppss2 8223 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) supp 0 ) ⊆ 𝑥)
163 suppssfifsupp 9417 . . . . . . . . . . . . . . . . 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 )
164157, 96, 162, 163syl12anc 837 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) finSupp 0 )
1651, 29, 30, 15, 3mplelbas 22028 . . . . . . . . . . . . . . . 16 ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ 𝐵 ↔ ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) finSupp 0 ))
166153, 164, 165sylanbrc 583 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ 𝐵)
1671, 3, 142, 91, 166, 135mpladd 22046 . . . . . . . . . . . . . 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 )))))
168 ovexd 7465 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) ∈ V)
169 eqidd 2735 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))
170 eqid 2734 . . . . . . . . . . . . . . . . 17 (.r𝑅) = (.r𝑅)
1711, 112, 2, 3, 170, 4, 127, 133mplvsca 22052 . . . . . . . . . . . . . . . 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 ))))
172127adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (𝑋𝑧) ∈ (Base‘𝑅))
1732, 108ringidcl 20279 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
174173, 144ifcld 4576 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∈ Ring → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
17579, 174syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
176175ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
177 fconstmpt 5750 . . . . . . . . . . . . . . . . . 18 (𝐷 × {(𝑋𝑧)}) = (𝑦𝐷 ↦ (𝑋𝑧))
178177a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝐷 × {(𝑋𝑧)}) = (𝑦𝐷 ↦ (𝑋𝑧)))
179 eqidd 2735 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))
180161, 172, 176, 178, 179offval2 7716 . . . . . . . . . . . . . . . 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 ))))
181171, 180eqtrd 2774 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) = (𝑦𝐷 ↦ ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 ))))
182161, 147, 168, 169, 181offval2 7716 . . . . . . . . . . . . . 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 )))))
183132, 80syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 ∈ Grp)
1842, 142, 15grplid 18997 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Grp ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
185183, 127, 184syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
186185ad2antrr 726 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
187 simpr 484 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ {𝑧})
188 velsn 4646 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
189187, 188sylib 218 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 = 𝑧)
190189fveq2d 6910 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → (𝑋𝑦) = (𝑋𝑧))
191186, 190eqtr4d 2777 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑦))
192120ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑧𝑥)
193189, 192eqneltrd 2858 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑦𝑥)
194193iffalsed 4541 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦𝑥, (𝑋𝑦), 0 ) = 0 )
195189iftrued 4538 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 1 )
196195oveq2d 7446 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋𝑧)(.r𝑅) 1 ))
1972, 170, 108ringridm 20283 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
198132, 127, 197syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
199198ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
200196, 199eqtrd 2774 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = (𝑋𝑧))
201194, 200oveq12d 7448 . . . . . . . . . . . . . . . . 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)))
202 elun2 4192 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑧} → 𝑦 ∈ (𝑥 ∪ {𝑧}))
203202adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ (𝑥 ∪ {𝑧}))
204203iftrued 4538 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ) = (𝑋𝑦))
205191, 201, 2043eqtr4d 2784 . . . . . . . . . . . . . . . 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 ))
20681ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → 𝑅 ∈ Grp)
2072, 142, 15grprid 18998 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Grp ∧ if(𝑦𝑥, (𝑋𝑦), 0 ) ∈ (Base‘𝑅)) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
208206, 147, 207syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
209208adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
210 simpr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 ∈ {𝑧})
211210, 188sylnib 328 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 = 𝑧)
212211iffalsed 4541 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 0 )
213212oveq2d 7446 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋𝑧)(.r𝑅) 0 ))
2142, 170, 15ringrz 20307 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
215132, 127, 214syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
216215ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
217213, 216eqtrd 2774 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = 0 )
218217oveq2d 7446 . . . . . . . . . . . . . . . . 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 ))
219 elun 4162 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦𝑥𝑦 ∈ {𝑧}))
220 orcom 870 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝑥𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥))
221219, 220bitri 275 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥))
222 biorf 936 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ {𝑧} → (𝑦𝑥 ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥)))
223221, 222bitr4id 290 . . . . . . . . . . . . . . . . . . 19 𝑦 ∈ {𝑧} → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦𝑥))
224223adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦𝑥))
225224ifbid 4553 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
226209, 218, 2253eqtr4d 2784 . . . . . . . . . . . . . . . 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 ))
227205, 226pm2.61dan 813 . . . . . . . . . . . . . . 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 ))
228227mpteq2dva 5247 . . . . . . . . . . . . . 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 )))
229167, 182, 2283eqtrrd 2779 . . . . . . . . . . . . 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 )))))
230141, 229eqeq12d 2750 . . . . . . . . . . . 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 ))))))
23190, 230imbitrrid 246 . . . . . . . . . . 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 ))))
232231expr 456 . . . . . . . . . 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 )))))
233232a2d 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 )))))
23489, 233syl5 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 )))))
235234expcom 413 . . . . . . 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 ))))))
236235a2d 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 ))))))
23750, 59, 68, 77, 85, 236findcard2s 9203 . . . . 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 )))))
23834, 237mpcom 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 ))))
23928, 238mpd 15 . . 3 (𝜑 → (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
24026, 239eqtr4d 2777 . 2 (𝜑𝑋 = (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
24128resmptd 6059 . . . 4 (𝜑 → ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ↾ (𝑋 supp 0 )) = (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
242241oveq2d 7446 . . 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 ))))))
243115fmpttd 7134 . . . 4 (𝜑 → (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))):𝐷𝐵)
2446, 11, 14, 17suppssr 8218 . . . . . . 7 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋𝑘) = 0 )
245244oveq1d 7445 . . . . . 6 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
246 eldifi 4140 . . . . . . 7 (𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 )) → 𝑘𝐷)
247105fveq2d 6910 . . . . . . . . . 10 ((𝜑𝑘𝐷) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
24815, 247eqtrid 2786 . . . . . . . . 9 ((𝜑𝑘𝐷) → 0 = (0g‘(Scalar‘𝑃)))
249248oveq1d 7445 . . . . . . . 8 ((𝜑𝑘𝐷) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
250 eqid 2734 . . . . . . . . . 10 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
2513, 111, 112, 250, 40lmod0vs 20909 . . . . . . . . 9 ((𝑃 ∈ LMod ∧ (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵) → ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
252102, 110, 251syl2anc 584 . . . . . . . 8 ((𝜑𝑘𝐷) → ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
253249, 252eqtrd 2774 . . . . . . 7 ((𝜑𝑘𝐷) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
254246, 253sylan2 593 . . . . . 6 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
255245, 254eqtrd 2774 . . . . 5 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
256255, 14suppss2 8223 . . . 4 (𝜑 → ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) supp (0g𝑃)) ⊆ (𝑋 supp 0 ))
25713mptex 7242 . . . . . . 7 (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V
258 funmpt 6605 . . . . . . 7 Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
259 fvex 6919 . . . . . . 7 (0g𝑃) ∈ V
260257, 258, 2593pm3.2i 1338 . . . . . 6 ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V ∧ Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∧ (0g𝑃) ∈ V)
261260a1i 11 . . . . 5 (𝜑 → ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V ∧ Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∧ (0g𝑃) ∈ V))
262 suppssfifsupp 9417 . . . . 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))
263261, 34, 256, 262syl12anc 837 . . . 4 (𝜑 → (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) finSupp (0g𝑃))
2643, 40, 94, 14, 243, 256, 263gsumres 19945 . . 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 ))))))
265242, 264eqtr3d 2776 . 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 ))))))
266240, 265eqtrd 2774 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 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  {crab 3432  Vcvv 3477  cdif 3959  cun 3960  wss 3962  c0 4338  ifcif 4530  {csn 4630   class class class wbr 5147  cmpt 5230   × cxp 5686  ccnv 5687  cres 5690  cima 5691  Fun wfun 6556  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694   supp csupp 8183  m cmap 8864  Fincfn 8983   finSupp cfsupp 9398  cn 12263  0cn0 12523  Basecbs 17244  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17485   Σg cgsu 17486  Grpcgrp 18963  CMndccmn 19812  1rcur 20198  Ringcrg 20250  LModclmod 20874   mPwSer cmps 21941   mPoly cmpl 21943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-subrng 20562  df-subrg 20586  df-lmod 20876  df-lss 20947  df-psr 21946  df-mpl 21948
This theorem is referenced by:  mplbas2  22077  mplcoe4  22112  ply1coe  22317
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