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Theorem mplcoe1 19862
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 2778 . . . . . 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 19830 . . . . 5 (𝜑𝑋:𝐷⟶(Base‘𝑅))
76feqmptd 6509 . . . 4 (𝜑𝑋 = (𝑦𝐷 ↦ (𝑋𝑦)))
8 iftrue 4313 . . . . . . 7 (𝑦 ∈ (𝑋 supp 0 ) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
98adantl 475 . . . . . 6 (((𝜑𝑦𝐷) ∧ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
10 eldif 3802 . . . . . . . 8 (𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 )) ↔ (𝑦𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )))
11 ifid 4346 . . . . . . . . 9 if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), (𝑋𝑦)) = (𝑋𝑦)
12 ssidd 3843 . . . . . . . . . . 11 (𝜑 → (𝑋 supp 0 ) ⊆ (𝑋 supp 0 ))
13 ovex 6954 . . . . . . . . . . . . 13 (ℕ0𝑚 𝐼) ∈ V
144, 13rabex2 5051 . . . . . . . . . . . 12 𝐷 ∈ V
1514a1i 11 . . . . . . . . . . 11 (𝜑𝐷 ∈ V)
16 mplcoe1.z . . . . . . . . . . . . 13 0 = (0g𝑅)
1716fvexi 6460 . . . . . . . . . . . 12 0 ∈ V
1817a1i 11 . . . . . . . . . . 11 (𝜑0 ∈ V)
196, 12, 15, 18suppssr 7608 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋𝑦) = 0 )
2019ifeq2d 4326 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), (𝑋𝑦)) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))
2111, 20syl5reqr 2829 . . . . . . . 8 ((𝜑𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2210, 21sylan2br 588 . . . . . . 7 ((𝜑 ∧ (𝑦𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2322anassrs 461 . . . . . 6 (((𝜑𝑦𝐷) ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
249, 23pm2.61dan 803 . . . . 5 ((𝜑𝑦𝐷) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ) = (𝑋𝑦))
2524mpteq2dva 4979 . . . 4 (𝜑 → (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ (𝑋𝑦)))
267, 25eqtr4d 2817 . . 3 (𝜑𝑋 = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
27 suppssdm 7589 . . . . 5 (𝑋 supp 0 ) ⊆ dom 𝑋
2827, 6fssdm 6307 . . . 4 (𝜑 → (𝑋 supp 0 ) ⊆ 𝐷)
29 eqid 2778 . . . . . . . . 9 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
30 eqid 2778 . . . . . . . . 9 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
311, 29, 30, 16, 3mplelbas 19827 . . . . . . . 8 (𝑋𝐵 ↔ (𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 finSupp 0 ))
3231simprbi 492 . . . . . . 7 (𝑋𝐵𝑋 finSupp 0 )
335, 32syl 17 . . . . . 6 (𝜑𝑋 finSupp 0 )
3433fsuppimpd 8570 . . . . 5 (𝜑 → (𝑋 supp 0 ) ∈ Fin)
35 sseq1 3845 . . . . . . . 8 (𝑤 = ∅ → (𝑤𝐷 ↔ ∅ ⊆ 𝐷))
36 mpteq1 4972 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ ∅ ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
37 mpt0 6267 . . . . . . . . . . . 12 (𝑘 ∈ ∅ ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅
3836, 37syl6eq 2830 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅)
3938oveq2d 6938 . . . . . . . . . 10 (𝑤 = ∅ → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σg ∅))
40 eqid 2778 . . . . . . . . . . 11 (0g𝑃) = (0g𝑃)
4140gsum0 17664 . . . . . . . . . 10 (𝑃 Σg ∅) = (0g𝑃)
4239, 41syl6eq 2830 . . . . . . . . 9 (𝑤 = ∅ → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (0g𝑃))
43 noel 4146 . . . . . . . . . . . 12 ¬ 𝑦 ∈ ∅
44 eleq2 2848 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑦𝑤𝑦 ∈ ∅))
4543, 44mtbiri 319 . . . . . . . . . . 11 (𝑤 = ∅ → ¬ 𝑦𝑤)
4645iffalsed 4318 . . . . . . . . . 10 (𝑤 = ∅ → if(𝑦𝑤, (𝑋𝑦), 0 ) = 0 )
4746mpteq2dv 4980 . . . . . . . . 9 (𝑤 = ∅ → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷0 ))
4842, 47eqeq12d 2793 . . . . . . . 8 (𝑤 = ∅ → ((𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) ↔ (0g𝑃) = (𝑦𝐷0 )))
4935, 48imbi12d 336 . . . . . . 7 (𝑤 = ∅ → ((𝑤𝐷 → (𝑃 Σg (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 ))) ↔ (∅ ⊆ 𝐷 → (0g𝑃) = (𝑦𝐷0 ))))
5049imbi2d 332 . . . . . 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 3845 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝐷𝑥𝐷))
52 mpteq1 4972 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘𝑥 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
5352oveq2d 6938 . . . . . . . . 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 2848 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
5554ifbid 4329 . . . . . . . . . 10 (𝑤 = 𝑥 → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
5655mpteq2dv 4980 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))
5753, 56eqeq12d 2793 . . . . . . . 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 336 . . . . . . 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 332 . . . . . 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 3845 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑤𝐷 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐷))
61 mpteq1 4972 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑘𝑤 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
6261oveq2d 6938 . . . . . . . . 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 2848 . . . . . . . . . . 11 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦𝑤𝑦 ∈ (𝑥 ∪ {𝑧})))
6463ifbid 4329 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ))
6564mpteq2dv 4980 . . . . . . . . 9 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 )))
6662, 65eqeq12d 2793 . . . . . . . 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 336 . . . . . . 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 332 . . . . . 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 3845 . . . . . . . 8 (𝑤 = (𝑋 supp 0 ) → (𝑤𝐷 ↔ (𝑋 supp 0 ) ⊆ 𝐷))
70 mpteq1 4972 . . . . . . . . . 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 6938 . . . . . . . . 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 2848 . . . . . . . . . . 11 (𝑤 = (𝑋 supp 0 ) → (𝑦𝑤𝑦 ∈ (𝑋 supp 0 )))
7372ifbid 4329 . . . . . . . . . 10 (𝑤 = (𝑋 supp 0 ) → if(𝑦𝑤, (𝑋𝑦), 0 ) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 ))
7473mpteq2dv 4980 . . . . . . . . 9 (𝑤 = (𝑋 supp 0 ) → (𝑦𝐷 ↦ if(𝑦𝑤, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋𝑦), 0 )))
7571, 74eqeq12d 2793 . . . . . . . 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 336 . . . . . . 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 332 . . . . . 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 18939 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
8179, 80syl 17 . . . . . . . . 9 (𝜑𝑅 ∈ Grp)
821, 4, 16, 40, 78, 81mpl0 19838 . . . . . . . 8 (𝜑 → (0g𝑃) = (𝐷 × { 0 }))
83 fconstmpt 5411 . . . . . . . 8 (𝐷 × { 0 }) = (𝑦𝐷0 )
8482, 83syl6eq 2830 . . . . . . 7 (𝜑 → (0g𝑃) = (𝑦𝐷0 ))
8584a1d 25 . . . . . 6 (𝜑 → (∅ ⊆ 𝐷 → (0g𝑃) = (𝑦𝐷0 )))
86 ssun1 3999 . . . . . . . . . . 11 𝑥 ⊆ (𝑥 ∪ {𝑧})
87 sstr2 3828 . . . . . . . . . . 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 6929 . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . 14 (+g𝑃) = (+g𝑃)
921mplring 19849 . . . . . . . . . . . . . . . . 17 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
9378, 79, 92syl2anc 579 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ Ring)
94 ringcmn 18968 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
9593, 94syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑃 ∈ CMnd)
9695adantr 474 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ CMnd)
97 simprll 769 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥 ∈ Fin)
98 simprr 763 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑥 ∪ {𝑧}) ⊆ 𝐷)
9998unssad 4013 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥𝐷)
10099sselda 3821 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝑥) → 𝑘𝐷)
10178adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝐼𝑊)
10279adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑅 ∈ Ring)
1031mpllmod 19848 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ LMod)
104101, 102, 103syl2anc 579 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → 𝑃 ∈ LMod)
1056ffvelrnda 6623 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → (𝑋𝑘) ∈ (Base‘𝑅))
1061, 78, 79mplsca 19842 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑅 = (Scalar‘𝑃))
107106adantr 474 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐷) → 𝑅 = (Scalar‘𝑃))
108107fveq2d 6450 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
109105, 108eleqtrd 2861 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → (𝑋𝑘) ∈ (Base‘(Scalar‘𝑃)))
110 mplcoe1.o . . . . . . . . . . . . . . . . . 18 1 = (1r𝑅)
111 simpr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐷) → 𝑘𝐷)
1121, 3, 16, 110, 4, 101, 102, 111mplmon 19860 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐷) → (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵)
113 eqid 2778 . . . . . . . . . . . . . . . . . 18 (Scalar‘𝑃) = (Scalar‘𝑃)
114 mplcoe1.n . . . . . . . . . . . . . . . . . 18 · = ( ·𝑠𝑃)
115 eqid 2778 . . . . . . . . . . . . . . . . . 18 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
1163, 113, 114, 115lmodvscl 19272 . . . . . . . . . . . . . . . . 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 1439 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐷) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
118117adantlr 705 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝐷) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
119100, 118syldan 585 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘𝑥) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
120 vex 3401 . . . . . . . . . . . . . . 15 𝑧 ∈ V
121120a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧 ∈ V)
122 simprlr 770 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ¬ 𝑧𝑥)
12378, 79, 103syl2anc 579 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ LMod)
124123adantr 474 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ LMod)
1256adantr 474 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑋:𝐷⟶(Base‘𝑅))
12698unssbd 4014 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → {𝑧} ⊆ 𝐷)
127120snss 4549 . . . . . . . . . . . . . . . . . 18 (𝑧𝐷 ↔ {𝑧} ⊆ 𝐷)
128126, 127sylibr 226 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧𝐷)
129125, 128ffvelrnd 6624 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋𝑧) ∈ (Base‘𝑅))
130106adantr 474 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 = (Scalar‘𝑃))
131130fveq2d 6450 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
132129, 131eleqtrd 2861 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋𝑧) ∈ (Base‘(Scalar‘𝑃)))
13378adantr 474 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐼𝑊)
13479adantr 474 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 ∈ Ring)
1351, 3, 16, 110, 4, 133, 134, 128mplmon 19860 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) ∈ 𝐵)
1363, 113, 114, 115lmodvscl 19272 . . . . . . . . . . . . . . 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 1439 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧) · (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) ∈ 𝐵)
138 fveq2 6446 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑋𝑘) = (𝑋𝑧))
139 equequ2 2073 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑧 → (𝑦 = 𝑘𝑦 = 𝑧))
140139ifbid 4329 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → if(𝑦 = 𝑘, 1 , 0 ) = if(𝑦 = 𝑧, 1 , 0 ))
141140mpteq2dv 4980 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))
142138, 141oveq12d 6940 . . . . . . . . . . . . . 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 18745 . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . 15 (+g𝑅) = (+g𝑅)
145125ffvelrnda 6623 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (𝑋𝑦) ∈ (Base‘𝑅))
1462, 16ring0cl 18956 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
14779, 146syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑0 ∈ (Base‘𝑅))
148147ad2antrr 716 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → 0 ∈ (Base‘𝑅))
149145, 148ifcld 4352 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → if(𝑦𝑥, (𝑋𝑦), 0 ) ∈ (Base‘𝑅))
150149fmpttd 6649 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )):𝐷⟶(Base‘𝑅))
151 fvex 6459 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) ∈ V
152151, 14elmap 8169 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ ((Base‘𝑅) ↑𝑚 𝐷) ↔ (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )):𝐷⟶(Base‘𝑅))
153150, 152sylibr 226 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ ((Base‘𝑅) ↑𝑚 𝐷))
15429, 2, 4, 30, 133psrbas 19775 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑𝑚 𝐷))
155153, 154eleqtrrd 2862 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)))
15614mptex 6758 . . . . . . . . . . . . . . . . . . 19 (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ V
157 funmpt 6173 . . . . . . . . . . . . . . . . . . 19 Fun (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 ))
158156, 157, 173pm3.2i 1395 . . . . . . . . . . . . . . . . . 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 3956 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝐷𝑥) → ¬ 𝑦𝑥)
161160adantl 475 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷𝑥)) → ¬ 𝑦𝑥)
162161iffalsed 4318 . . . . . . . . . . . . . . . . . 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 7611 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) supp 0 ) ⊆ 𝑥)
165 suppssfifsupp 8578 . . . . . . . . . . . . . . . . 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 827 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) finSupp 0 )
1671, 29, 30, 16, 3mplelbas 19827 . . . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) ∈ 𝐵)
1691, 3, 144, 91, 168, 137mpladd 19839 . . . . . . . . . . . . . 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 6956 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) ∈ V)
171 eqidd 2779 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )) = (𝑦𝐷 ↦ if(𝑦𝑥, (𝑋𝑦), 0 )))
172 eqid 2778 . . . . . . . . . . . . . . . . 17 (.r𝑅) = (.r𝑅)
1731, 114, 2, 3, 172, 4, 129, 135mplvsca 19844 . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (𝑋𝑧) ∈ (Base‘𝑅))
1752, 110ringidcl 18955 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
176175, 146ifcld 4352 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∈ Ring → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
17779, 176syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
178177ad2antrr 716 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
179 fconstmpt 5411 . . . . . . . . . . . . . . . . . 18 (𝐷 × {(𝑋𝑧)}) = (𝑦𝐷 ↦ (𝑋𝑧))
180179a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝐷 × {(𝑋𝑧)}) = (𝑦𝐷 ↦ (𝑋𝑧)))
181 eqidd 2779 . . . . . . . . . . . . . . . . 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 7191 . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . 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 7191 . . . . . . . . . . . . . 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 17839 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Grp ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
187185, 129, 186syl2anc 579 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
188187ad2antrr 716 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑧))
189 simpr 479 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ {𝑧})
190 velsn 4414 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
191189, 190sylib 210 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 = 𝑧)
192191fveq2d 6450 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → (𝑋𝑦) = (𝑋𝑧))
193188, 192eqtr4d 2817 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+g𝑅)(𝑋𝑧)) = (𝑋𝑦))
194122ad2antrr 716 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑧𝑥)
195191, 194eqneltrd 2878 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑦𝑥)
196195iffalsed 4318 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦𝑥, (𝑋𝑦), 0 ) = 0 )
197191iftrued 4315 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 1 )
198197oveq2d 6938 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋𝑧)(.r𝑅) 1 ))
1992, 172, 110ringridm 18959 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
200134, 129, 199syl2anc 579 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
201200ad2antrr 716 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅) 1 ) = (𝑋𝑧))
202198, 201eqtrd 2814 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = (𝑋𝑧))
203196, 202oveq12d 6940 . . . . . . . . . . . . . . . . 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 4004 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑧} → 𝑦 ∈ (𝑥 ∪ {𝑧}))
205204adantl 475 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ (𝑥 ∪ {𝑧}))
206205iftrued 4315 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ) = (𝑋𝑦))
207193, 203, 2063eqtr4d 2824 . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → 𝑅 ∈ Grp)
2092, 144, 16grprid 17840 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Grp ∧ if(𝑦𝑥, (𝑋𝑦), 0 ) ∈ (Base‘𝑅)) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
210208, 149, 209syl2anc 579 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
211210adantr 474 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (if(𝑦𝑥, (𝑋𝑦), 0 )(+g𝑅) 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
212 simpr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 ∈ {𝑧})
213212, 190sylnib 320 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 = 𝑧)
214213iffalsed 4318 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 0 )
215214oveq2d 6938 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋𝑧)(.r𝑅) 0 ))
2162, 172, 16ringrz 18975 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑋𝑧) ∈ (Base‘𝑅)) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
217134, 129, 216syl2anc 579 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
218217ad2antrr 716 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅) 0 ) = 0 )
219215, 218eqtrd 2814 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋𝑧)(.r𝑅)if(𝑦 = 𝑧, 1 , 0 )) = 0 )
220219oveq2d 6938 . . . . . . . . . . . . . . . . 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 923 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ {𝑧} → (𝑦𝑥 ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥)))
222 elun 3976 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦𝑥𝑦 ∈ {𝑧}))
223 orcom 859 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝑥𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥))
224222, 223bitri 267 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦𝑥))
225221, 224syl6rbbr 282 . . . . . . . . . . . . . . . . . . 19 𝑦 ∈ {𝑧} → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦𝑥))
226225adantl 475 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦𝑥))
227226ifbid 4329 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋𝑦), 0 ) = if(𝑦𝑥, (𝑋𝑦), 0 ))
228211, 220, 2273eqtr4d 2824 . . . . . . . . . . . . . . . 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 803 . . . . . . . . . . . . . . 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 4979 . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . 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 2793 . . . . . . . . . . . 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 238 . . . . . . . . . . 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 450 . . . . . . . . . 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 404 . . . . . . 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 8489 . . . . 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 2817 . 2 (𝜑𝑋 = (𝑃 Σg (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
24328resmptd 5702 . . . 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 6938 . . 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 6649 . . . 4 (𝜑 → (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))):𝐷𝐵)
2466, 12, 15, 18suppssr 7608 . . . . . . 7 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋𝑘) = 0 )
247246oveq1d 6937 . . . . . 6 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
248 eldifi 3955 . . . . . . 7 (𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 )) → 𝑘𝐷)
249107fveq2d 6450 . . . . . . . . . 10 ((𝜑𝑘𝐷) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
25016, 249syl5eq 2826 . . . . . . . . 9 ((𝜑𝑘𝐷) → 0 = (0g‘(Scalar‘𝑃)))
251250oveq1d 6937 . . . . . . . 8 ((𝜑𝑘𝐷) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
252 eqid 2778 . . . . . . . . . 10 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
2533, 113, 114, 252, 40lmod0vs 19288 . . . . . . . . 9 ((𝑃 ∈ LMod ∧ (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵) → ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
254104, 112, 253syl2anc 579 . . . . . . . 8 ((𝜑𝑘𝐷) → ((0g‘(Scalar‘𝑃)) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
255251, 254eqtrd 2814 . . . . . . 7 ((𝜑𝑘𝐷) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
256248, 255sylan2 586 . . . . . 6 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ( 0 · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
257247, 256eqtrd 2814 . . . . 5 ((𝜑𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0g𝑃))
258257, 15suppss2 7611 . . . 4 (𝜑 → ((𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) supp (0g𝑃)) ⊆ (𝑋 supp 0 ))
25914mptex 6758 . . . . . . 7 (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V
260 funmpt 6173 . . . . . . 7 Fun (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
261 fvex 6459 . . . . . . 7 (0g𝑃) ∈ V
262259, 260, 2613pm3.2i 1395 . . . . . 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 8578 . . . . 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 827 . . . 4 (𝜑 → (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) finSupp (0g𝑃))
2663, 40, 95, 15, 245, 258, 265gsumres 18700 . . 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 2816 . 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 2814 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 198  wa 386  wo 836  w3a 1071   = wceq 1601  wcel 2107  {crab 3094  Vcvv 3398  cdif 3789  cun 3790  wss 3792  c0 4141  ifcif 4307  {csn 4398   class class class wbr 4886  cmpt 4965   × cxp 5353  ccnv 5354  cres 5357  cima 5358  Fun wfun 6129  wf 6131  cfv 6135  (class class class)co 6922  𝑓 cof 7172   supp csupp 7576  𝑚 cmap 8140  Fincfn 8241   finSupp cfsupp 8563  cn 11374  0cn0 11642  Basecbs 16255  +gcplusg 16338  .rcmulr 16339  Scalarcsca 16341   ·𝑠 cvsca 16342  0gc0g 16486   Σg cgsu 16487  Grpcgrp 17809  CMndccmn 18579  1rcur 18888  Ringcrg 18934  LModclmod 19255   mPwSer cmps 19748   mPoly cmpl 19750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-ofr 7175  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-seq 13120  df-hash 13436  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-mulr 16352  df-sca 16354  df-vsca 16355  df-tset 16357  df-0g 16488  df-gsum 16489  df-mre 16632  df-mrc 16633  df-acs 16635  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-mhm 17721  df-submnd 17722  df-grp 17812  df-minusg 17813  df-sbg 17814  df-mulg 17928  df-subg 17975  df-ghm 18042  df-cntz 18133  df-cmn 18581  df-abl 18582  df-mgp 18877  df-ur 18889  df-ring 18936  df-subrg 19170  df-lmod 19257  df-lss 19325  df-psr 19753  df-mpl 19755
This theorem is referenced by:  mplbas2  19867  mplcoe4  19899  ply1coe  20062
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