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Theorem mplcoe1 20234
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 2824 . . . . . 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 20201 . . . . 5 (𝜑𝑋:𝐷⟶(Base‘𝑅))
76feqmptd 6716 . . . 4 (𝜑𝑋 = (𝑦𝐷 ↦ (𝑋𝑦)))
8 iftrue 4454 . . . . . . 7 (𝑦 ∈ (𝑋 supp 0 ) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
98adantl 485 . . . . . 6 (((𝜑𝑦𝐷) ∧ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
10 eldif 3928 . . . . . . . 8 (𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 )) ↔ (𝑦𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )))
11 ifid 4487 . . . . . . . . 9 if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), (𝑋𝑦)) = (𝑋𝑦)
12 ssidd 3974 . . . . . . . . . . 11 (𝜑 → (𝑋 supp 0 ) ⊆ (𝑋 supp 0 ))
13 ovex 7173 . . . . . . . . . . . . 13 (ℕ0m 𝐼) ∈ V
144, 13rabex2 5220 . . . . . . . . . . . 12 𝐷 ∈ V
1514a1i 11 . . . . . . . . . . 11 (𝜑𝐷 ∈ V)
16 mplcoe1.z . . . . . . . . . . . . 13 0 = (0g𝑅)
1716fvexi 6667 . . . . . . . . . . . 12 0 ∈ V
1817a1i 11 . . . . . . . . . . 11 (𝜑0 ∈ V)
196, 12, 15, 18suppssr 7846 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋𝑦) = 0 )
2019ifeq2d 4467 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), (𝑋𝑦)) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))
2111, 20syl5reqr 2874 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2210, 21sylan2br 597 . . . . . . 7 ((𝜑 ∧ (𝑦𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2322anassrs 471 . . . . . 6 (((𝜑𝑦𝐷) ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
249, 23pm2.61dan 812 . . . . 5 ((𝜑𝑦𝐷) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2524mpteq2dva 5144 . . . 4 (𝜑 → (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ (𝑋𝑦)))
267, 25eqtr4d 2862 . . 3 (𝜑𝑋 = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
27 suppssdm 7828 . . . . 5 (𝑋 supp 0 ) ⊆ dom 𝑋
2827, 6fssdm 6513 . . . 4 (𝜑 → (𝑋 supp 0 ) ⊆ 𝐷)
29 eqid 2824 . . . . . . . . 9 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
30 eqid 2824 . . . . . . . . 9 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
311, 29, 30, 16, 3mplelbas 20198 . . . . . . . 8 (𝑋𝐵 ↔ (𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 finSupp 0 ))
3231simprbi 500 . . . . . . 7 (𝑋𝐵𝑋 finSupp 0 )
335, 32syl 17 . . . . . 6 (𝜑𝑋 finSupp 0 )
3433fsuppimpd 8826 . . . . 5 (𝜑 → (𝑋 supp 0 ) ∈ Fin)
35 sseq1 3976 . . . . . . . 8 (𝑤 = ∅ → (𝑤𝐷 ↔ ∅ ⊆ 𝐷))
36 mpteq1 5137 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ ∅ ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
37 mpt0 6473 . . . . . . . . . . . 12 (𝑘 ∈ ∅ ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅
3836, 37syl6eq 2875 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅)
3938oveq2d 7156 . . . . . . . . . 10 (𝑤 = ∅ → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σg ∅))
40 eqid 2824 . . . . . . . . . . 11 (0g𝑃) = (0g𝑃)
4140gsum0 17885 . . . . . . . . . 10 (𝑃 Σg ∅) = (0g𝑃)
4239, 41syl6eq 2875 . . . . . . . . 9 (𝑤 = ∅ → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (0g𝑃))
43 noel 4278 . . . . . . . . . . . 12 ¬ 𝑦 ∈ ∅
44 eleq2 2904 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑦𝑤𝑦 ∈ ∅))
4543, 44mtbiri 330 . . . . . . . . . . 11 (𝑤 = ∅ → ¬ 𝑦𝑤)
4645iffalsed 4459 . . . . . . . . . 10 (𝑤 = ∅ → if(𝑦𝑤, (𝑋𝑦), 0 ) = 0 )
4746mpteq2dv 5145 . . . . . . . . 9 (𝑤 = ∅ → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷0 ))
4842, 47eqeq12d 2840 . . . . . . . 8 (𝑤 = ∅ → ((𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) ↔ (0g𝑃) = (𝑦𝐷0 )))
4935, 48imbi12d 348 . . . . . . 7 (𝑤 = ∅ → ((𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 ))) ↔ (∅ ⊆ 𝐷 → (0g𝑃) = (𝑦𝐷0 ))))
5049imbi2d 344 . . . . . 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 3976 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝐷𝑥𝐷))
52 mpteq1 5137 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
5352oveq2d 7156 . . . . . . . . 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 2904 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
5554ifbid 4470 . . . . . . . . . 10 (𝑤 = 𝑥 → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
5655mpteq2dv 5145 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))
5753, 56eqeq12d 2840 . . . . . . . 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 348 . . . . . . 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 344 . . . . . 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 3976 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑤𝐷 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐷))
61 mpteq1 5137 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
6261oveq2d 7156 . . . . . . . . 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 2904 . . . . . . . . . . 11 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦𝑤𝑦 ∈ (𝑥 ∪ {𝑧})))
6463ifbid 4470 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))
6564mpteq2dv 5145 . . . . . . . . 9 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )))
6662, 65eqeq12d 2840 . . . . . . . 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 348 . . . . . . 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 344 . . . . . 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 3976 . . . . . . . 8 (𝑤 = (𝑋 supp 0 ) → (𝑤𝐷 ↔ (𝑋 supp 0 ) ⊆ 𝐷))
70 mpteq1 5137 . . . . . . . . . 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 7156 . . . . . . . . 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 2904 . . . . . . . . . . 11 (𝑤 = (𝑋 supp 0 ) → (𝑦𝑤𝑦 ∈ (𝑋 supp 0 )))
7372ifbid 4470 . . . . . . . . . 10 (𝑤 = (𝑋 supp 0 ) → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))
7473mpteq2dv 5145 . . . . . . . . 9 (𝑤 = (𝑋 supp 0 ) → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
7571, 74eqeq12d 2840 . . . . . . . 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 348 . . . . . . 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 344 . . . . . 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 19293 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
8179, 80syl 17 . . . . . . . . 9 (𝜑𝑅 ∈ Grp)
821, 4, 16, 40, 78, 81mpl0 20209 . . . . . . . 8 (𝜑 → (0g𝑃) = (𝐷 × { 0 }))
83 fconstmpt 5597 . . . . . . . 8 (𝐷 × { 0 }) = (𝑦𝐷0 )
8482, 83syl6eq 2875 . . . . . . 7 (𝜑 → (0g𝑃) = (𝑦𝐷0 ))
8584a1d 25 . . . . . 6 (𝜑 → (∅ ⊆ 𝐷 → (0g𝑃) = (𝑦𝐷0 )))
86 ssun1 4132 . . . . . . . . . . 11 𝑥 ⊆ (𝑥 ∪ {𝑧})
87 sstr2 3958 . . . . . . . . . . 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 7147 . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . 14 (+g𝑃) = (+g𝑃)
921mplring 20220 . . . . . . . . . . . . . . . . 17 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
9378, 79, 92syl2anc 587 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ Ring)
94 ringcmn 19322 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
9593, 94syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑃 ∈ CMnd)
9695adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ CMnd)
97 simprll 778 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥 ∈ Fin)
98 simprr 772 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑥 ∪ {𝑧}) ⊆ 𝐷)
9998unssad 4147 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥𝐷)
10099sselda 3951 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝑥) → 𝑘𝐷)
10178adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝐼𝑊)
10279adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑅 ∈ Ring)
1031mpllmod 20219 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ LMod)
104101, 102, 103syl2anc 587 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → 𝑃 ∈ LMod)
1056ffvelrnda 6834 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → (𝑋𝑘) ∈ (Base‘𝑅))
1061, 78, 79mplsca 20213 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑅 = (Scalar‘𝑃))
107106adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝑅 = (Scalar‘𝑃))
108107fveq2d 6657 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
109105, 108eleqtrd 2918 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → (𝑋𝑘) ∈ (Base‘(Scalar‘𝑃)))
110 mplcoe1.o . . . . . . . . . . . . . . . . . 18 1 = (1r𝑅)
111 simpr 488 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑘𝐷)
1121, 3, 16, 110, 4, 101, 102, 111mplmon 20232 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵)
113 eqid 2824 . . . . . . . . . . . . . . . . . 18 (Scalar‘𝑃) = (Scalar‘𝑃)
114 mplcoe1.n . . . . . . . . . . . . . . . . . 18 · = ( ·𝑠𝑃)
115 eqid 2824 . . . . . . . . . . . . . . . . . 18 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
1163, 113, 114, 115lmodvscl 19639 . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐷) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
118117adantlr 714 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝐷) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
119100, 118syldan 594 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝑥) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
120 vex 3482 . . . . . . . . . . . . . . 15 𝑧 ∈ V
121120a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧 ∈ V)
122 simprlr 779 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ¬ 𝑧𝑥)
12378, 79, 103syl2anc 587 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ LMod)
124123adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ LMod)
1256adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑋:𝐷⟶(Base‘𝑅))
12698unssbd 4148 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → {𝑧} ⊆ 𝐷)
127120snss 4701 . . . . . . . . . . . . . . . . . 18 (𝑧𝐷 ↔ {𝑧} ⊆ 𝐷)
128126, 127sylibr 237 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧𝐷)
129125, 128ffvelrnd 6835 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋𝑧) ∈ (Base‘𝑅))
130106adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 = (Scalar‘𝑃))
131130fveq2d 6657 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
132129, 131eleqtrd 2918 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋𝑧) ∈ (Base‘(Scalar‘𝑃)))
13378adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐼𝑊)
13479adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 ∈ Ring)
1351, 3, 16, 110, 4, 133, 134, 128mplmon 20232 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) ∈ 𝐵)
1363, 113, 114, 115lmodvscl 19639 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) ∈ 𝐵)
138 fveq2 6653 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑋𝑘) = (𝑋𝑧))
139 equequ2 2034 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑧 → (𝑦 = 𝑘𝑦 = 𝑧))
140139ifbid 4470 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → if(𝑦 = 𝑘, 1 , 0 ) = if(𝑦 = 𝑧, 1 , 0 ))
141140mpteq2dv 5145 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))
142138, 141oveq12d 7158 . . . . . . . . . . . . . 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 19071 . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . 15 (+g𝑅) = (+g𝑅)
145125ffvelrnda 6834 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (𝑋𝑦) ∈ (Base‘𝑅))
1462, 16ring0cl 19310 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
14779, 146syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑0 ∈ (Base‘𝑅))
148147ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → 0 ∈ (Base‘𝑅))
149145, 148ifcld 4493 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → if(𝑦𝑥, (𝑋𝑦), 0 ) ∈ (Base‘𝑅))
150149fmpttd 6862 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )):𝐷⟶(Base‘𝑅))
151 fvex 6666 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) ∈ V
152151, 14elmap 8420 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ ((Base‘𝑅) ↑m 𝐷) ↔ (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )):𝐷⟶(Base‘𝑅))
153150, 152sylibr 237 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ ((Base‘𝑅) ↑m 𝐷))
15429, 2, 4, 30, 133psrbas 20146 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m 𝐷))
155153, 154eleqtrrd 2919 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)))
15614mptex 6969 . . . . . . . . . . . . . . . . . . 19 (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V
157 funmpt 6376 . . . . . . . . . . . . . . . . . . 19 Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))
158156, 157, 173pm3.2i 1336 . . . . . . . . . . . . . . . . . 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 4088 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝐷𝑥) → ¬ 𝑦𝑥)
161160adantl 485 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷𝑥)) → ¬ 𝑦𝑥)
162161iffalsed 4459 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷𝑥)) → if(𝑦𝑥, (𝑋𝑦), 0 ) = 0 )
16314a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐷 ∈ V)
164162, 163suppss2 7849 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) supp 0 ) ⊆ 𝑥)
165 suppssfifsupp 8834 . . . . . . . . . . . . . . . . 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 835 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) finSupp 0 )
1671, 29, 30, 16, 3mplelbas 20198 . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ 𝐵)
1691, 3, 144, 91, 168, 137mpladd 20210 . . . . . . . . . . . . . 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 7175 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) ∈ V)
171 eqidd 2825 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))
172 eqid 2824 . . . . . . . . . . . . . . . . 17 (.r𝑅) = (.r𝑅)
1731, 114, 2, 3, 172, 4, 129, 135mplvsca 20215 . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (𝑋𝑧) ∈ (Base‘𝑅))
1752, 110ringidcl 19309 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
176175, 146ifcld 4493 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∈ Ring → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
17779, 176syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
178177ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
179 fconstmpt 5597 . . . . . . . . . . . . . . . . . 18 (𝐷 × {(𝑋𝑧)}) = (𝑦𝐷 ↦ (𝑋𝑧))
180179a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝐷 × {(𝑋𝑧)}) = (𝑦𝐷 ↦ (𝑋𝑧)))
181 eqidd 2825 . . . . . . . . . . . . . . . . 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 7411 . . . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . . . . 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 7411 . . . . . . . . . . . . . 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, 16grplid 18124 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Grp ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
187185, 129, 186syl2anc 587 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
188187ad2antrr 725 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
189 simpr 488 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ {𝑧})
190 velsn 4564 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
191189, 190sylib 221 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 = 𝑧)
192191fveq2d 6657 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → (𝑋𝑦) = (𝑋𝑧))
193188, 192eqtr4d 2862 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑦))
194122ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑧𝑥)
195191, 194eqneltrd 2935 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑦𝑥)
196195iffalsed 4459 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦𝑥, (𝑋𝑦), 0 ) = 0 )
197191iftrued 4456 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 1 )
198197oveq2d 7156 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋𝑧)(.r𝑅) 1 ))
1992, 172, 110ringridm 19313 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
200134, 129, 199syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
201200ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
202198, 201eqtrd 2859 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = (𝑋𝑧))
203196, 202oveq12d 7158 . . . . . . . . . . . . . . . . 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 4137 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑧} → 𝑦 ∈ (𝑥 ∪ {𝑧}))
205204adantl 485 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ (𝑥 ∪ {𝑧}))
206205iftrued 4456 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ) = (𝑋𝑦))
207193, 203, 2063eqtr4d 2869 . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → 𝑅 ∈ Grp)
2092, 144, 16grprid 18125 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Grp ∧ if(𝑦𝑥, (𝑋𝑦), 0 ) ∈ (Base‘𝑅)) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
210208, 149, 209syl2anc 587 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
211210adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
212 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 ∈ {𝑧})
213212, 190sylnib 331 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 = 𝑧)
214213iffalsed 4459 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 0 )
215214oveq2d 7156 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋𝑧)(.r𝑅) 0 ))
2162, 172, 16ringrz 19329 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
217134, 129, 216syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
218217ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
219215, 218eqtrd 2859 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = 0 )
220219oveq2d 7156 . . . . . . . . . . . . . . . . 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 biorf 934 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ {𝑧} → (𝑦𝑥 ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥)))
222 elun 4109 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦𝑥𝑦 ∈ {𝑧}))
223 orcom 867 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝑥𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥))
224222, 223bitri 278 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥))
225221, 224syl6rbbr 293 . . . . . . . . . . . . . . . . . . 19 𝑦 ∈ {𝑧} → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦𝑥))
226225adantl 485 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦𝑥))
227226ifbid 4470 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
228211, 220, 2273eqtr4d 2869 . . . . . . . . . . . . . . . 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 812 . . . . . . . . . . . . . . 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 5144 . . . . . . . . . . . . . 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 2864 . . . . . . . . . . . . 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 2840 . . . . . . . . . . . 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 249 . . . . . . . . . . 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 460 . . . . . . . . . 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 417 . . . . . . 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 8745 . . . . 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 2862 . 2 (𝜑𝑋 = (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
24328resmptd 5891 . . . 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 7156 . . 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 6862 . . . 4 (𝜑 → (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))):𝐷𝐵)
2466, 12, 15, 18suppssr 7846 . . . . . . 7 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋𝑘) = 0 )
247246oveq1d 7155 . . . . . 6 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
248 eldifi 4087 . . . . . . 7 (𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 )) → 𝑘𝐷)
249107fveq2d 6657 . . . . . . . . . 10 ((𝜑𝑘𝐷) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
25016, 249syl5eq 2871 . . . . . . . . 9 ((𝜑𝑘𝐷) → 0 = (0g‘(Scalar‘𝑃)))
251250oveq1d 7155 . . . . . . . 8 ((𝜑𝑘𝐷) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
252 eqid 2824 . . . . . . . . . 10 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
2533, 113, 114, 252, 40lmod0vs 19655 . . . . . . . . 9 ((𝑃 ∈ LMod ∧ (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵) → ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
254104, 112, 253syl2anc 587 . . . . . . . 8 ((𝜑𝑘𝐷) → ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
255251, 254eqtrd 2859 . . . . . . 7 ((𝜑𝑘𝐷) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
256248, 255sylan2 595 . . . . . 6 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
257247, 256eqtrd 2859 . . . . 5 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
258257, 15suppss2 7849 . . . 4 (𝜑 → ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) supp (0g𝑃)) ⊆ (𝑋 supp 0 ))
25914mptex 6969 . . . . . . 7 (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V
260 funmpt 6376 . . . . . . 7 Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
261 fvex 6666 . . . . . . 7 (0g𝑃) ∈ V
262259, 260, 2613pm3.2i 1336 . . . . . 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 8834 . . . . 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 835 . . . 4 (𝜑 → (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) finSupp (0g𝑃))
2663, 40, 95, 15, 245, 258, 265gsumres 19024 . . 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 2861 . 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 2859 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 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  {crab 3136  Vcvv 3479  cdif 3915  cun 3916  wss 3918  c0 4274  ifcif 4448  {csn 4548   class class class wbr 5049  cmpt 5129   × cxp 5536  ccnv 5537  cres 5540  cima 5541  Fun wfun 6332  wf 6334  cfv 6338  (class class class)co 7140  f cof 7392   supp csupp 7815  m cmap 8391  Fincfn 8494   finSupp cfsupp 8819  cn 11625  0cn0 11885  Basecbs 16474  +gcplusg 16556  .rcmulr 16557  Scalarcsca 16559   ·𝑠 cvsca 16560  0gc0g 16704   Σg cgsu 16705  Grpcgrp 18094  CMndccmn 18897  1rcur 19242  Ringcrg 19288  LModclmod 19622   mPwSer cmps 20119   mPoly cmpl 20121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-cnex 10580  ax-resscn 10581  ax-1cn 10582  ax-icn 10583  ax-addcl 10584  ax-addrcl 10585  ax-mulcl 10586  ax-mulrcl 10587  ax-mulcom 10588  ax-addass 10589  ax-mulass 10590  ax-distr 10591  ax-i2m1 10592  ax-1ne0 10593  ax-1rid 10594  ax-rnegex 10595  ax-rrecex 10596  ax-cnre 10597  ax-pre-lttri 10598  ax-pre-lttrn 10599  ax-pre-ltadd 10600  ax-pre-mulgt0 10601
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-iin 4905  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-se 5498  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-isom 6347  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-of 7394  df-ofr 7395  df-om 7566  df-1st 7674  df-2nd 7675  df-supp 7816  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-2o 8088  df-oadd 8091  df-er 8274  df-map 8393  df-pm 8394  df-ixp 8447  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-fsupp 8820  df-oi 8960  df-card 9354  df-pnf 10664  df-mnf 10665  df-xr 10666  df-ltxr 10667  df-le 10668  df-sub 10859  df-neg 10860  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-struct 16476  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-mulr 16570  df-sca 16572  df-vsca 16573  df-tset 16575  df-0g 16706  df-gsum 16707  df-mre 16848  df-mrc 16849  df-acs 16851  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-mhm 17947  df-submnd 17948  df-grp 18097  df-minusg 18098  df-sbg 18099  df-mulg 18216  df-subg 18267  df-ghm 18347  df-cntz 18438  df-cmn 18899  df-abl 18900  df-mgp 19231  df-ur 19243  df-ring 19290  df-subrg 19521  df-lmod 19624  df-lss 19692  df-psr 20124  df-mpl 20126
This theorem is referenced by:  mplbas2  20239  mplcoe4  20271  ply1coe  20452
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