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Theorem mplcoe5 22075
Description: Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 22076), it is sufficient that 𝑅 is a ring and all the variables of the multivariate polynomial commute. (Contributed by AV, 7-Oct-2019.)
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)
mplcoe2.g 𝐺 = (mulGrp‘𝑃)
mplcoe2.m = (.g𝐺)
mplcoe2.v 𝑉 = (𝐼 mVar 𝑅)
mplcoe5.r (𝜑𝑅 ∈ Ring)
mplcoe5.y (𝜑𝑌𝐷)
mplcoe5.c (𝜑 → ∀𝑥𝐼𝑦𝐼 ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)))
Assertion
Ref Expression
mplcoe5 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))
Distinct variable groups:   𝑥,𝑘, ,𝑦   1 ,𝑘   𝑥,𝑦, 1   𝑘,𝐺,𝑥   𝑓,𝑘,𝑥,𝑦,𝐼   𝜑,𝑘,𝑥,𝑦   𝑅,𝑓,𝑦   𝐷,𝑘,𝑥,𝑦   𝑃,𝑘,𝑥   𝑘,𝑉,𝑥   0 ,𝑓,𝑘,𝑥,𝑦   𝑓,𝑌,𝑘,𝑥,𝑦   𝑘,𝑊,𝑦   𝑦,𝐺   𝑦,𝑉   𝑦,
Allowed substitution hints:   𝜑(𝑓)   𝐷(𝑓)   𝑃(𝑦,𝑓)   𝑅(𝑥,𝑘)   1 (𝑓)   (𝑓)   𝐺(𝑓)   𝑉(𝑓)   𝑊(𝑥,𝑓)

Proof of Theorem mplcoe5
Dummy variables 𝑖 𝑤 𝑧 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mplcoe5.y . . . . . . . . 9 (𝜑𝑌𝐷)
2 mplcoe1.i . . . . . . . . . 10 (𝜑𝐼𝑊)
3 mplcoe1.d . . . . . . . . . . 11 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
43psrbag 21954 . . . . . . . . . 10 (𝐼𝑊 → (𝑌𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (𝑌 “ ℕ) ∈ Fin)))
52, 4syl 17 . . . . . . . . 9 (𝜑 → (𝑌𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (𝑌 “ ℕ) ∈ Fin)))
61, 5mpbid 232 . . . . . . . 8 (𝜑 → (𝑌:𝐼⟶ℕ0 ∧ (𝑌 “ ℕ) ∈ Fin))
76simpld 494 . . . . . . 7 (𝜑𝑌:𝐼⟶ℕ0)
87feqmptd 6976 . . . . . 6 (𝜑𝑌 = (𝑖𝐼 ↦ (𝑌𝑖)))
9 iftrue 4536 . . . . . . . . 9 (𝑖 ∈ (𝑌 “ ℕ) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
109adantl 481 . . . . . . . 8 (((𝜑𝑖𝐼) ∧ 𝑖 ∈ (𝑌 “ ℕ)) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
11 eldif 3972 . . . . . . . . . 10 (𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ)) ↔ (𝑖𝐼 ∧ ¬ 𝑖 ∈ (𝑌 “ ℕ)))
12 fcdmnn0supp 12580 . . . . . . . . . . . . . . 15 ((𝐼𝑊𝑌:𝐼⟶ℕ0) → (𝑌 supp 0) = (𝑌 “ ℕ))
132, 7, 12syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → (𝑌 supp 0) = (𝑌 “ ℕ))
14 eqimss 4053 . . . . . . . . . . . . . 14 ((𝑌 supp 0) = (𝑌 “ ℕ) → (𝑌 supp 0) ⊆ (𝑌 “ ℕ))
1513, 14syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑌 supp 0) ⊆ (𝑌 “ ℕ))
16 c0ex 11252 . . . . . . . . . . . . . 14 0 ∈ V
1716a1i 11 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ V)
187, 15, 2, 17suppssr 8218 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (𝑌𝑖) = 0)
1918ifeq2d 4550 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), (𝑌𝑖)) = if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))
20 ifid 4570 . . . . . . . . . . 11 if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), (𝑌𝑖)) = (𝑌𝑖)
2119, 20eqtr3di 2789 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2211, 21sylan2br 595 . . . . . . . . 9 ((𝜑 ∧ (𝑖𝐼 ∧ ¬ 𝑖 ∈ (𝑌 “ ℕ))) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2322anassrs 467 . . . . . . . 8 (((𝜑𝑖𝐼) ∧ ¬ 𝑖 ∈ (𝑌 “ ℕ)) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2410, 23pm2.61dan 813 . . . . . . 7 ((𝜑𝑖𝐼) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2524mpteq2dva 5247 . . . . . 6 (𝜑 → (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)) = (𝑖𝐼 ↦ (𝑌𝑖)))
268, 25eqtr4d 2777 . . . . 5 (𝜑𝑌 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)))
2726eqeq2d 2745 . . . 4 (𝜑 → (𝑦 = 𝑌𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))))
2827ifbid 4553 . . 3 (𝜑 → if(𝑦 = 𝑌, 1 , 0 ) = if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 ))
2928mpteq2dv 5249 . 2 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )))
30 cnvimass 6101 . . . . 5 (𝑌 “ ℕ) ⊆ dom 𝑌
3130, 7fssdm 6755 . . . 4 (𝜑 → (𝑌 “ ℕ) ⊆ 𝐼)
326simprd 495 . . . . 5 (𝜑 → (𝑌 “ ℕ) ∈ Fin)
33 sseq1 4020 . . . . . . . 8 (𝑤 = ∅ → (𝑤𝐼 ↔ ∅ ⊆ 𝐼))
34 noel 4343 . . . . . . . . . . . . . . . 16 ¬ 𝑖 ∈ ∅
35 eleq2 2827 . . . . . . . . . . . . . . . 16 (𝑤 = ∅ → (𝑖𝑤𝑖 ∈ ∅))
3634, 35mtbiri 327 . . . . . . . . . . . . . . 15 (𝑤 = ∅ → ¬ 𝑖𝑤)
3736iffalsed 4541 . . . . . . . . . . . . . 14 (𝑤 = ∅ → if(𝑖𝑤, (𝑌𝑖), 0) = 0)
3837mpteq2dv 5249 . . . . . . . . . . . . 13 (𝑤 = ∅ → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ 0))
39 fconstmpt 5750 . . . . . . . . . . . . 13 (𝐼 × {0}) = (𝑖𝐼 ↦ 0)
4038, 39eqtr4di 2792 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝐼 × {0}))
4140eqeq2d 2745 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝐼 × {0})))
4241ifbid 4553 . . . . . . . . . 10 (𝑤 = ∅ → if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
4342mpteq2dv 5249 . . . . . . . . 9 (𝑤 = ∅ → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
44 mpteq1 5240 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ ∅ ↦ ((𝑌𝑘) (𝑉𝑘))))
45 mpt0 6710 . . . . . . . . . . . 12 (𝑘 ∈ ∅ ↦ ((𝑌𝑘) (𝑉𝑘))) = ∅
4644, 45eqtrdi 2790 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = ∅)
4746oveq2d 7446 . . . . . . . . . 10 (𝑤 = ∅ → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg ∅))
48 mplcoe2.g . . . . . . . . . . . 12 𝐺 = (mulGrp‘𝑃)
49 eqid 2734 . . . . . . . . . . . 12 (1r𝑃) = (1r𝑃)
5048, 49ringidval 20200 . . . . . . . . . . 11 (1r𝑃) = (0g𝐺)
5150gsum0 18709 . . . . . . . . . 10 (𝐺 Σg ∅) = (1r𝑃)
5247, 51eqtrdi 2790 . . . . . . . . 9 (𝑤 = ∅ → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (1r𝑃))
5343, 52eqeq12d 2750 . . . . . . . 8 (𝑤 = ∅ → ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) ↔ (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r𝑃)))
5433, 53imbi12d 344 . . . . . . 7 (𝑤 = ∅ → ((𝑤𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))))) ↔ (∅ ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r𝑃))))
5554imbi2d 340 . . . . . 6 (𝑤 = ∅ → ((𝜑 → (𝑤𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))))) ↔ (𝜑 → (∅ ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r𝑃)))))
56 sseq1 4020 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝐼𝑥𝐼))
57 eleq2 2827 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑖𝑤𝑖𝑥))
5857ifbid 4553 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → if(𝑖𝑤, (𝑌𝑖), 0) = if(𝑖𝑥, (𝑌𝑖), 0))
5958mpteq2dv 5249 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)))
6059eqeq2d 2745 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0))))
6160ifbid 4553 . . . . . . . . . 10 (𝑤 = 𝑥 → if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 ))
6261mpteq2dv 5249 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )))
63 mpteq1 5240 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))
6463oveq2d 7446 . . . . . . . . 9 (𝑤 = 𝑥 → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘)))))
6562, 64eqeq12d 2750 . . . . . . . 8 (𝑤 = 𝑥 → ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) ↔ (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))))
6656, 65imbi12d 344 . . . . . . 7 (𝑤 = 𝑥 → ((𝑤𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))))) ↔ (𝑥𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘)))))))
6766imbi2d 340 . . . . . 6 (𝑤 = 𝑥 → ((𝜑 → (𝑤𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))))) ↔ (𝜑 → (𝑥𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))))))
68 sseq1 4020 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑤𝐼 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐼))
69 eleq2 2827 . . . . . . . . . . . . . 14 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑖𝑤𝑖 ∈ (𝑥 ∪ {𝑧})))
7069ifbid 4553 . . . . . . . . . . . . 13 (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑖𝑤, (𝑌𝑖), 0) = if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0))
7170mpteq2dv 5249 . . . . . . . . . . . 12 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)))
7271eqeq2d 2745 . . . . . . . . . . 11 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0))))
7372ifbid 4553 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 ))
7473mpteq2dv 5249 . . . . . . . . 9 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )))
75 mpteq1 5240 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))))
7675oveq2d 7446 . . . . . . . . 9 (𝑤 = (𝑥 ∪ {𝑧}) → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))))
7774, 76eqeq12d 2750 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑧}) → ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) ↔ (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))))))
7868, 77imbi12d 344 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑧}) → ((𝑤𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))))) ↔ ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))))))
7978imbi2d 340 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑧}) → ((𝜑 → (𝑤𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))))) ↔ (𝜑 → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))))))))
80 sseq1 4020 . . . . . . . 8 (𝑤 = (𝑌 “ ℕ) → (𝑤𝐼 ↔ (𝑌 “ ℕ) ⊆ 𝐼))
81 eleq2 2827 . . . . . . . . . . . . . 14 (𝑤 = (𝑌 “ ℕ) → (𝑖𝑤𝑖 ∈ (𝑌 “ ℕ)))
8281ifbid 4553 . . . . . . . . . . . . 13 (𝑤 = (𝑌 “ ℕ) → if(𝑖𝑤, (𝑌𝑖), 0) = if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))
8382mpteq2dv 5249 . . . . . . . . . . . 12 (𝑤 = (𝑌 “ ℕ) → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)))
8483eqeq2d 2745 . . . . . . . . . . 11 (𝑤 = (𝑌 “ ℕ) → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))))
8584ifbid 4553 . . . . . . . . . 10 (𝑤 = (𝑌 “ ℕ) → if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 ))
8685mpteq2dv 5249 . . . . . . . . 9 (𝑤 = (𝑌 “ ℕ) → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )))
87 mpteq1 5240 . . . . . . . . . 10 (𝑤 = (𝑌 “ ℕ) → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘))))
8887oveq2d 7446 . . . . . . . . 9 (𝑤 = (𝑌 “ ℕ) → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘)))))
8986, 88eqeq12d 2750 . . . . . . . 8 (𝑤 = (𝑌 “ ℕ) → ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) ↔ (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘))))))
9080, 89imbi12d 344 . . . . . . 7 (𝑤 = (𝑌 “ ℕ) → ((𝑤𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))))) ↔ ((𝑌 “ ℕ) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘)))))))
9190imbi2d 340 . . . . . 6 (𝑤 = (𝑌 “ ℕ) → ((𝜑 → (𝑤𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))))) ↔ (𝜑 → ((𝑌 “ ℕ) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘))))))))
92 mplcoe1.p . . . . . . . . 9 𝑃 = (𝐼 mPoly 𝑅)
93 mplcoe1.z . . . . . . . . 9 0 = (0g𝑅)
94 mplcoe1.o . . . . . . . . 9 1 = (1r𝑅)
95 mplcoe5.r . . . . . . . . 9 (𝜑𝑅 ∈ Ring)
9692, 3, 93, 94, 49, 2, 95mpl1 22049 . . . . . . . 8 (𝜑 → (1r𝑃) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
9796, 49eqtr3di 2789 . . . . . . 7 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r𝑃))
9897a1d 25 . . . . . 6 (𝜑 → (∅ ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r𝑃)))
99 ssun1 4187 . . . . . . . . . . 11 𝑥 ⊆ (𝑥 ∪ {𝑧})
100 sstr2 4001 . . . . . . . . . . 11 (𝑥 ⊆ (𝑥 ∪ {𝑧}) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼𝑥𝐼))
10199, 100ax-mp 5 . . . . . . . . . 10 ((𝑥 ∪ {𝑧}) ⊆ 𝐼𝑥𝐼)
102101imim1i 63 . . . . . . . . 9 ((𝑥𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))))
103 oveq1 7437 . . . . . . . . . . . 12 ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘)))) → ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 ))(.r𝑃)((𝑌𝑧) (𝑉𝑧))) = ((𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))(.r𝑃)((𝑌𝑧) (𝑉𝑧))))
104 eqid 2734 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
1052adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝐼𝑊)
10695adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑅 ∈ Ring)
1077adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑌:𝐼⟶ℕ0)
108107ffvelcdmda 7103 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → (𝑌𝑖) ∈ ℕ0)
109 0nn0 12538 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
110 ifcl 4575 . . . . . . . . . . . . . . . . . 18 (((𝑌𝑖) ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℕ0)
111108, 109, 110sylancl 586 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℕ0)
112111fmpttd 7134 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)):𝐼⟶ℕ0)
113 fcdmnn0supp 12580 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑊 ∧ (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)):𝐼⟶ℕ0) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) = ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) “ ℕ))
114105, 112, 113syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) = ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) “ ℕ))
115 simprll 779 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑥 ∈ Fin)
116 eldifn 4141 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (𝐼𝑥) → ¬ 𝑖𝑥)
117116adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ (𝐼𝑥)) → ¬ 𝑖𝑥)
118117iffalsed 4541 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ (𝐼𝑥)) → if(𝑖𝑥, (𝑌𝑖), 0) = 0)
119118, 105suppss2 8223 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) ⊆ 𝑥)
120115, 119ssfid 9298 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) ∈ Fin)
121114, 120eqeltrrd 2839 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) “ ℕ) ∈ Fin)
1223psrbag 21954 . . . . . . . . . . . . . . . . 17 (𝐼𝑊 → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∈ 𝐷 ↔ ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)):𝐼⟶ℕ0 ∧ ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) “ ℕ) ∈ Fin)))
123105, 122syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∈ 𝐷 ↔ ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)):𝐼⟶ℕ0 ∧ ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) “ ℕ) ∈ Fin)))
124112, 121, 123mpbir2and 713 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∈ 𝐷)
125 eqid 2734 . . . . . . . . . . . . . . 15 (.r𝑃) = (.r𝑃)
126 ssun2 4188 . . . . . . . . . . . . . . . . . . 19 {𝑧} ⊆ (𝑥 ∪ {𝑧})
127 simprr 773 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑥 ∪ {𝑧}) ⊆ 𝐼)
128126, 127sstrid 4006 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → {𝑧} ⊆ 𝐼)
129 vex 3481 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ V
130129snss 4789 . . . . . . . . . . . . . . . . . 18 (𝑧𝐼 ↔ {𝑧} ⊆ 𝐼)
131128, 130sylibr 234 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑧𝐼)
132107, 131ffvelcdmd 7104 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑌𝑧) ∈ ℕ0)
1333snifpsrbag 21957 . . . . . . . . . . . . . . . 16 ((𝐼𝑊 ∧ (𝑌𝑧) ∈ ℕ0) → (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)) ∈ 𝐷)
134105, 132, 133syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)) ∈ 𝐷)
13592, 104, 93, 94, 3, 105, 106, 124, 125, 134mplmonmul 22071 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 ))(.r𝑃)(𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)), 1 , 0 ))) = (𝑦𝐷 ↦ if(𝑦 = ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∘f + (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0))), 1 , 0 )))
136 mplcoe2.m . . . . . . . . . . . . . . . 16 = (.g𝐺)
137 mplcoe2.v . . . . . . . . . . . . . . . 16 𝑉 = (𝐼 mVar 𝑅)
13892, 3, 93, 94, 105, 48, 136, 137, 106, 131, 132mplcoe3 22073 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)), 1 , 0 )) = ((𝑌𝑧) (𝑉𝑧)))
139138oveq2d 7446 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 ))(.r𝑃)(𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)), 1 , 0 ))) = ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 ))(.r𝑃)((𝑌𝑧) (𝑉𝑧))))
140132adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → (𝑌𝑧) ∈ ℕ0)
141 ifcl 4575 . . . . . . . . . . . . . . . . . . . 20 (((𝑌𝑧) ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑖 = 𝑧, (𝑌𝑧), 0) ∈ ℕ0)
142140, 109, 141sylancl 586 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → if(𝑖 = 𝑧, (𝑌𝑧), 0) ∈ ℕ0)
143 eqidd 2735 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)))
144 eqidd 2735 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)) = (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)))
145105, 111, 142, 143, 144offval2 7716 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∘f + (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0))) = (𝑖𝐼 ↦ (if(𝑖𝑥, (𝑌𝑖), 0) + if(𝑖 = 𝑧, (𝑌𝑧), 0))))
146108adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌𝑖) ∈ ℕ0)
147146nn0cnd 12586 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌𝑖) ∈ ℂ)
148147addlidd 11459 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (0 + (𝑌𝑖)) = (𝑌𝑖))
149 elsni 4647 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ {𝑧} → 𝑖 = 𝑧)
150149adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 = 𝑧)
151 simprlr 780 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ¬ 𝑧𝑥)
152151ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → ¬ 𝑧𝑥)
153150, 152eqneltrd 2858 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → ¬ 𝑖𝑥)
154153iffalsed 4541 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖𝑥, (𝑌𝑖), 0) = 0)
155150iftrued 4538 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌𝑧), 0) = (𝑌𝑧))
156150fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌𝑖) = (𝑌𝑧))
157155, 156eqtr4d 2777 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌𝑧), 0) = (𝑌𝑖))
158154, 157oveq12d 7448 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (if(𝑖𝑥, (𝑌𝑖), 0) + if(𝑖 = 𝑧, (𝑌𝑧), 0)) = (0 + (𝑌𝑖)))
159 simpr 484 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 ∈ {𝑧})
160126, 159sselid 3992 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 ∈ (𝑥 ∪ {𝑧}))
161160iftrued 4538 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0) = (𝑌𝑖))
162148, 158, 1613eqtr4d 2784 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (if(𝑖𝑥, (𝑌𝑖), 0) + if(𝑖 = 𝑧, (𝑌𝑧), 0)) = if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0))
163111adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℕ0)
164163nn0cnd 12586 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℂ)
165164addridd 11458 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (if(𝑖𝑥, (𝑌𝑖), 0) + 0) = if(𝑖𝑥, (𝑌𝑖), 0))
166 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → ¬ 𝑖 ∈ {𝑧})
167 velsn 4646 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ {𝑧} ↔ 𝑖 = 𝑧)
168166, 167sylnib 328 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → ¬ 𝑖 = 𝑧)
169168iffalsed 4541 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌𝑧), 0) = 0)
170169oveq2d 7446 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (if(𝑖𝑥, (𝑌𝑖), 0) + if(𝑖 = 𝑧, (𝑌𝑧), 0)) = (if(𝑖𝑥, (𝑌𝑖), 0) + 0))
171 elun 4162 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑖𝑥𝑖 ∈ {𝑧}))
172 orcom 870 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖𝑥𝑖 ∈ {𝑧}) ↔ (𝑖 ∈ {𝑧} ∨ 𝑖𝑥))
173171, 172bitri 275 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑖 ∈ {𝑧} ∨ 𝑖𝑥))
174 biorf 936 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑖 ∈ {𝑧} → (𝑖𝑥 ↔ (𝑖 ∈ {𝑧} ∨ 𝑖𝑥)))
175173, 174bitr4id 290 . . . . . . . . . . . . . . . . . . . . . . 23 𝑖 ∈ {𝑧} → (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑖𝑥))
176175adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑖𝑥))
177176ifbid 4553 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0) = if(𝑖𝑥, (𝑌𝑖), 0))
178165, 170, 1773eqtr4d 2784 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (if(𝑖𝑥, (𝑌𝑖), 0) + if(𝑖 = 𝑧, (𝑌𝑧), 0)) = if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0))
179162, 178pm2.61dan 813 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → (if(𝑖𝑥, (𝑌𝑖), 0) + if(𝑖 = 𝑧, (𝑌𝑧), 0)) = if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0))
180179mpteq2dva 5247 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ (if(𝑖𝑥, (𝑌𝑖), 0) + if(𝑖 = 𝑧, (𝑌𝑧), 0))) = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)))
181145, 180eqtrd 2774 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∘f + (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0))) = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)))
182181eqeq2d 2745 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑦 = ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∘f + (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0))) ↔ 𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0))))
183182ifbid 4553 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → if(𝑦 = ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∘f + (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0))), 1 , 0 ) = if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 ))
184183mpteq2dv 5249 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑦𝐷 ↦ if(𝑦 = ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∘f + (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0))), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )))
185135, 139, 1843eqtr3rd 2783 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )) = ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 ))(.r𝑃)((𝑌𝑧) (𝑉𝑧))))
18648, 104mgpbas 20157 . . . . . . . . . . . . . 14 (Base‘𝑃) = (Base‘𝐺)
18748, 125mgpplusg 20155 . . . . . . . . . . . . . 14 (.r𝑃) = (+g𝐺)
188 eqid 2734 . . . . . . . . . . . . . 14 (Cntz‘𝐺) = (Cntz‘𝐺)
189 eqid 2734 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))
19092, 2, 95mplringd 22060 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ Ring)
19148ringmgp 20256 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
192190, 191syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺 ∈ Mnd)
193192adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝐺 ∈ Mnd)
1941adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑌𝐷)
195 mplcoe5.c . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑥𝐼𝑦𝐼 ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)))
196 fveq2 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝑉𝑥) = (𝑉𝑎))
197196oveq2d 7446 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑦)(+g𝐺)(𝑉𝑎)))
198196oveq1d 7445 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑉𝑥)(+g𝐺)(𝑉𝑦)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑦)))
199197, 198eqeq12d 2750 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)) ↔ ((𝑉𝑦)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑦))))
200 fveq2 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑏 → (𝑉𝑦) = (𝑉𝑏))
201200oveq1d 7445 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → ((𝑉𝑦)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑏)(+g𝐺)(𝑉𝑎)))
202200oveq2d 7446 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → ((𝑉𝑎)(+g𝐺)(𝑉𝑦)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
203201, 202eqeq12d 2750 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → (((𝑉𝑦)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑦)) ↔ ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏))))
204199, 203cbvral2vw 3238 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐼𝑦𝐼 ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)) ↔ ∀𝑎𝐼𝑏𝐼 ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
205195, 204sylib 218 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑎𝐼𝑏𝐼 ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
206205adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ∀𝑎𝐼𝑏𝐼 ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
20792, 3, 93, 94, 105, 48, 136, 137, 106, 194, 206, 127mplcoe5lem 22074 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ran (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))))
20899, 127sstrid 4006 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑥𝐼)
209208sselda 3994 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘𝑥) → 𝑘𝐼)
210192adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → 𝐺 ∈ Mnd)
2117ffvelcdmda 7103 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → (𝑌𝑘) ∈ ℕ0)
2122adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → 𝐼𝑊)
21395adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → 𝑅 ∈ Ring)
214 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → 𝑘𝐼)
21592, 137, 104, 212, 213, 214mvrcl 22029 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → (𝑉𝑘) ∈ (Base‘𝑃))
216186, 136, 210, 211, 215mulgnn0cld 19125 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐼) → ((𝑌𝑘) (𝑉𝑘)) ∈ (Base‘𝑃))
217216adantlr 715 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘𝐼) → ((𝑌𝑘) (𝑉𝑘)) ∈ (Base‘𝑃))
218209, 217syldan 591 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘𝑥) → ((𝑌𝑘) (𝑉𝑘)) ∈ (Base‘𝑃))
21992, 137, 104, 105, 106, 131mvrcl 22029 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑉𝑧) ∈ (Base‘𝑃))
220186, 136, 193, 132, 219mulgnn0cld 19125 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑌𝑧) (𝑉𝑧)) ∈ (Base‘𝑃))
221 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (𝑌𝑘) = (𝑌𝑧))
222 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (𝑉𝑘) = (𝑉𝑧))
223221, 222oveq12d 7448 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → ((𝑌𝑘) (𝑉𝑘)) = ((𝑌𝑧) (𝑉𝑧)))
224223adantl 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 = 𝑧) → ((𝑌𝑘) (𝑉𝑘)) = ((𝑌𝑧) (𝑉𝑧)))
225186, 187, 188, 189, 193, 115, 207, 218, 131, 151, 220, 224gsumzunsnd 19988 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))) = ((𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))(.r𝑃)((𝑌𝑧) (𝑉𝑧))))
226185, 225eqeq12d 2750 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))) ↔ ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 ))(.r𝑃)((𝑌𝑧) (𝑉𝑧))) = ((𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))(.r𝑃)((𝑌𝑧) (𝑉𝑧)))))
227103, 226imbitrrid 246 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘)))) → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))))))
228227expr 456 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧𝑥)) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → ((𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘)))) → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))))))
229228a2d 29 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧𝑥)) → (((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))))))
230102, 229syl5 34 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧𝑥)) → ((𝑥𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))))))
231230expcom 413 . . . . . . 7 ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) → (𝜑 → ((𝑥𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))))))))
232231a2d 29 . . . . . 6 ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) → ((𝜑 → (𝑥𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘)))))) → (𝜑 → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))))))))
23355, 67, 79, 91, 98, 232findcard2s 9203 . . . . 5 ((𝑌 “ ℕ) ∈ Fin → (𝜑 → ((𝑌 “ ℕ) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘)))))))
23432, 233mpcom 38 . . . 4 (𝜑 → ((𝑌 “ ℕ) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘))))))
23531, 234mpd 15 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘)))))
23631resmptd 6059 . . . 4 (𝜑 → ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ↾ (𝑌 “ ℕ)) = (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘))))
237236oveq2d 7446 . . 3 (𝜑 → (𝐺 Σg ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ↾ (𝑌 “ ℕ))) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘)))))
238216fmpttd 7134 . . . 4 (𝜑 → (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))):𝐼⟶(Base‘𝑃))
239 ssidd 4018 . . . . 5 (𝜑𝐼𝐼)
24092, 3, 93, 94, 2, 48, 136, 137, 95, 1, 195, 239mplcoe5lem 22074 . . . 4 (𝜑 → ran (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))
2417, 15, 2, 17suppssr 8218 . . . . . . 7 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (𝑌𝑘) = 0)
242241oveq1d 7445 . . . . . 6 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → ((𝑌𝑘) (𝑉𝑘)) = (0 (𝑉𝑘)))
243 eldifi 4140 . . . . . . . 8 (𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ)) → 𝑘𝐼)
244243, 215sylan2 593 . . . . . . 7 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (𝑉𝑘) ∈ (Base‘𝑃))
245186, 50, 136mulg0 19104 . . . . . . 7 ((𝑉𝑘) ∈ (Base‘𝑃) → (0 (𝑉𝑘)) = (1r𝑃))
246244, 245syl 17 . . . . . 6 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (0 (𝑉𝑘)) = (1r𝑃))
247242, 246eqtrd 2774 . . . . 5 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → ((𝑌𝑘) (𝑉𝑘)) = (1r𝑃))
248247, 2suppss2 8223 . . . 4 (𝜑 → ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) supp (1r𝑃)) ⊆ (𝑌 “ ℕ))
2492mptexd 7243 . . . . 5 (𝜑 → (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ∈ V)
250 funmpt 6605 . . . . . 6 Fun (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))
251250a1i 11 . . . . 5 (𝜑 → Fun (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))))
252 fvexd 6921 . . . . 5 (𝜑 → (1r𝑃) ∈ V)
253 suppssfifsupp 9417 . . . . 5 ((((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ∈ V ∧ Fun (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ∧ (1r𝑃) ∈ V) ∧ ((𝑌 “ ℕ) ∈ Fin ∧ ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) supp (1r𝑃)) ⊆ (𝑌 “ ℕ))) → (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) finSupp (1r𝑃))
254249, 251, 252, 32, 248, 253syl32anc 1377 . . . 4 (𝜑 → (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) finSupp (1r𝑃))
255186, 50, 188, 192, 2, 238, 240, 248, 254gsumzres 19941 . . 3 (𝜑 → (𝐺 Σg ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ↾ (𝑌 “ ℕ))) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))
256235, 237, 2553eqtr2d 2780 . 2 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))
25729, 256eqtrd 2774 1 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  wral 3058  {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  0cc0 11152   + caddc 11155  cn 12263  0cn0 12523  Basecbs 17244  +gcplusg 17297  .rcmulr 17298  0gc0g 17485   Σg cgsu 17486  Mndcmnd 18759  .gcmg 19097  Cntzccntz 19345  mulGrpcmgp 20151  1rcur 20198  Ringcrg 20250   mVar cmvr 21942   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-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-srg 20204  df-ring 20252  df-subrng 20562  df-subrg 20586  df-psr 21946  df-mvr 21947  df-mpl 21948
This theorem is referenced by:  mplcoe2  22076  ply1coe  22317
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