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Theorem mplcoe5 20249
Description: Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 20250), 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 20144 . . . . . . . . . 10 (𝐼𝑊 → (𝑌𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (𝑌 “ ℕ) ∈ Fin)))
52, 4syl 17 . . . . . . . . 9 (𝜑 → (𝑌𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (𝑌 “ ℕ) ∈ Fin)))
61, 5mpbid 235 . . . . . . . 8 (𝜑 → (𝑌:𝐼⟶ℕ0 ∧ (𝑌 “ ℕ) ∈ Fin))
76simpld 498 . . . . . . 7 (𝜑𝑌:𝐼⟶ℕ0)
87feqmptd 6724 . . . . . 6 (𝜑𝑌 = (𝑖𝐼 ↦ (𝑌𝑖)))
9 iftrue 4456 . . . . . . . . 9 (𝑖 ∈ (𝑌 “ ℕ) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
109adantl 485 . . . . . . . 8 (((𝜑𝑖𝐼) ∧ 𝑖 ∈ (𝑌 “ ℕ)) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
11 eldif 3929 . . . . . . . . . 10 (𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ)) ↔ (𝑖𝐼 ∧ ¬ 𝑖 ∈ (𝑌 “ ℕ)))
12 ifid 4489 . . . . . . . . . . 11 if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), (𝑌𝑖)) = (𝑌𝑖)
13 frnnn0supp 11950 . . . . . . . . . . . . . . 15 ((𝐼𝑊𝑌:𝐼⟶ℕ0) → (𝑌 supp 0) = (𝑌 “ ℕ))
142, 7, 13syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → (𝑌 supp 0) = (𝑌 “ ℕ))
15 eqimss 4009 . . . . . . . . . . . . . 14 ((𝑌 supp 0) = (𝑌 “ ℕ) → (𝑌 supp 0) ⊆ (𝑌 “ ℕ))
1614, 15syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑌 supp 0) ⊆ (𝑌 “ ℕ))
17 c0ex 10633 . . . . . . . . . . . . . 14 0 ∈ V
1817a1i 11 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ V)
197, 16, 2, 18suppssr 7857 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (𝑌𝑖) = 0)
2019ifeq2d 4469 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), (𝑌𝑖)) = if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))
2112, 20syl5reqr 2874 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2211, 21sylan2br 597 . . . . . . . . 9 ((𝜑 ∧ (𝑖𝐼 ∧ ¬ 𝑖 ∈ (𝑌 “ ℕ))) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2322anassrs 471 . . . . . . . 8 (((𝜑𝑖𝐼) ∧ ¬ 𝑖 ∈ (𝑌 “ ℕ)) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2410, 23pm2.61dan 812 . . . . . . 7 ((𝜑𝑖𝐼) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2524mpteq2dva 5147 . . . . . 6 (𝜑 → (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)) = (𝑖𝐼 ↦ (𝑌𝑖)))
268, 25eqtr4d 2862 . . . . 5 (𝜑𝑌 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)))
2726eqeq2d 2835 . . . 4 (𝜑 → (𝑦 = 𝑌𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))))
2827ifbid 4472 . . 3 (𝜑 → if(𝑦 = 𝑌, 1 , 0 ) = if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 ))
2928mpteq2dv 5148 . 2 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )))
30 cnvimass 5936 . . . . 5 (𝑌 “ ℕ) ⊆ dom 𝑌
3130, 7fssdm 6520 . . . 4 (𝜑 → (𝑌 “ ℕ) ⊆ 𝐼)
326simprd 499 . . . . 5 (𝜑 → (𝑌 “ ℕ) ∈ Fin)
33 sseq1 3978 . . . . . . . 8 (𝑤 = ∅ → (𝑤𝐼 ↔ ∅ ⊆ 𝐼))
34 noel 4280 . . . . . . . . . . . . . . . 16 ¬ 𝑖 ∈ ∅
35 eleq2 2904 . . . . . . . . . . . . . . . 16 (𝑤 = ∅ → (𝑖𝑤𝑖 ∈ ∅))
3634, 35mtbiri 330 . . . . . . . . . . . . . . 15 (𝑤 = ∅ → ¬ 𝑖𝑤)
3736iffalsed 4461 . . . . . . . . . . . . . 14 (𝑤 = ∅ → if(𝑖𝑤, (𝑌𝑖), 0) = 0)
3837mpteq2dv 5148 . . . . . . . . . . . . 13 (𝑤 = ∅ → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ 0))
39 fconstmpt 5601 . . . . . . . . . . . . 13 (𝐼 × {0}) = (𝑖𝐼 ↦ 0)
4038, 39syl6eqr 2877 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝐼 × {0}))
4140eqeq2d 2835 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝐼 × {0})))
4241ifbid 4472 . . . . . . . . . 10 (𝑤 = ∅ → if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
4342mpteq2dv 5148 . . . . . . . . 9 (𝑤 = ∅ → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
44 mpteq1 5140 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ ∅ ↦ ((𝑌𝑘) (𝑉𝑘))))
45 mpt0 6479 . . . . . . . . . . . 12 (𝑘 ∈ ∅ ↦ ((𝑌𝑘) (𝑉𝑘))) = ∅
4644, 45syl6eq 2875 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = ∅)
4746oveq2d 7165 . . . . . . . . . 10 (𝑤 = ∅ → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg ∅))
48 mplcoe2.g . . . . . . . . . . . 12 𝐺 = (mulGrp‘𝑃)
49 eqid 2824 . . . . . . . . . . . 12 (1r𝑃) = (1r𝑃)
5048, 49ringidval 19253 . . . . . . . . . . 11 (1r𝑃) = (0g𝐺)
5150gsum0 17894 . . . . . . . . . 10 (𝐺 Σg ∅) = (1r𝑃)
5247, 51syl6eq 2875 . . . . . . . . 9 (𝑤 = ∅ → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (1r𝑃))
5343, 52eqeq12d 2840 . . . . . . . 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 348 . . . . . . 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 344 . . . . . 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 3978 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝐼𝑥𝐼))
57 eleq2 2904 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑖𝑤𝑖𝑥))
5857ifbid 4472 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → if(𝑖𝑤, (𝑌𝑖), 0) = if(𝑖𝑥, (𝑌𝑖), 0))
5958mpteq2dv 5148 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)))
6059eqeq2d 2835 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0))))
6160ifbid 4472 . . . . . . . . . 10 (𝑤 = 𝑥 → if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 ))
6261mpteq2dv 5148 . . . . . . . . 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 5140 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))
6463oveq2d 7165 . . . . . . . . 9 (𝑤 = 𝑥 → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘)))))
6562, 64eqeq12d 2840 . . . . . . . 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 348 . . . . . . 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 344 . . . . . 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 3978 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑤𝐼 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐼))
69 eleq2 2904 . . . . . . . . . . . . . 14 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑖𝑤𝑖 ∈ (𝑥 ∪ {𝑧})))
7069ifbid 4472 . . . . . . . . . . . . 13 (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑖𝑤, (𝑌𝑖), 0) = if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0))
7170mpteq2dv 5148 . . . . . . . . . . . 12 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)))
7271eqeq2d 2835 . . . . . . . . . . 11 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0))))
7372ifbid 4472 . . . . . . . . . 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 5148 . . . . . . . . 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 5140 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))))
7675oveq2d 7165 . . . . . . . . 9 (𝑤 = (𝑥 ∪ {𝑧}) → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))))
7774, 76eqeq12d 2840 . . . . . . . 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 348 . . . . . . 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 344 . . . . . 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 3978 . . . . . . . 8 (𝑤 = (𝑌 “ ℕ) → (𝑤𝐼 ↔ (𝑌 “ ℕ) ⊆ 𝐼))
81 eleq2 2904 . . . . . . . . . . . . . 14 (𝑤 = (𝑌 “ ℕ) → (𝑖𝑤𝑖 ∈ (𝑌 “ ℕ)))
8281ifbid 4472 . . . . . . . . . . . . 13 (𝑤 = (𝑌 “ ℕ) → if(𝑖𝑤, (𝑌𝑖), 0) = if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))
8382mpteq2dv 5148 . . . . . . . . . . . 12 (𝑤 = (𝑌 “ ℕ) → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)))
8483eqeq2d 2835 . . . . . . . . . . 11 (𝑤 = (𝑌 “ ℕ) → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))))
8584ifbid 4472 . . . . . . . . . 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 5148 . . . . . . . . 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 5140 . . . . . . . . . 10 (𝑤 = (𝑌 “ ℕ) → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘))))
8887oveq2d 7165 . . . . . . . . 9 (𝑤 = (𝑌 “ ℕ) → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘)))))
8986, 88eqeq12d 2840 . . . . . . . 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 348 . . . . . . 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 344 . . . . . 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 20224 . . . . . . . 8 (𝜑 → (1r𝑃) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
9796eqcomd 2830 . . . . . . 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 4134 . . . . . . . . . . 11 𝑥 ⊆ (𝑥 ∪ {𝑧})
100 sstr2 3960 . . . . . . . . . . 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 7156 . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
1052adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝐼𝑊)
10695adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑅 ∈ Ring)
1077adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑌:𝐼⟶ℕ0)
108107ffvelrnda 6842 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → (𝑌𝑖) ∈ ℕ0)
109 0nn0 11909 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
110 ifcl 4494 . . . . . . . . . . . . . . . . . 18 (((𝑌𝑖) ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℕ0)
111108, 109, 110sylancl 589 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℕ0)
112111fmpttd 6870 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)):𝐼⟶ℕ0)
113 frnnn0supp 11950 . . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) = ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) “ ℕ))
115 simprll 778 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑥 ∈ Fin)
116 eldifn 4090 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (𝐼𝑥) → ¬ 𝑖𝑥)
117116adantl 485 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ (𝐼𝑥)) → ¬ 𝑖𝑥)
118117iffalsed 4461 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ (𝐼𝑥)) → if(𝑖𝑥, (𝑌𝑖), 0) = 0)
119118, 105suppss2 7860 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) ⊆ 𝑥)
120115, 119ssfid 8738 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) ∈ Fin)
121114, 120eqeltrrd 2917 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) “ ℕ) ∈ Fin)
1223psrbag 20144 . . . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∈ 𝐷)
125 eqid 2824 . . . . . . . . . . . . . . 15 (.r𝑃) = (.r𝑃)
126 ssun2 4135 . . . . . . . . . . . . . . . . . . 19 {𝑧} ⊆ (𝑥 ∪ {𝑧})
127 simprr 772 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑥 ∪ {𝑧}) ⊆ 𝐼)
128126, 127sstrid 3964 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → {𝑧} ⊆ 𝐼)
129 vex 3483 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ V
130129snss 4703 . . . . . . . . . . . . . . . . . 18 (𝑧𝐼 ↔ {𝑧} ⊆ 𝐼)
131128, 130sylibr 237 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑧𝐼)
132107, 131ffvelrnd 6843 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑌𝑧) ∈ ℕ0)
1333snifpsrbag 20146 . . . . . . . . . . . . . . . 16 ((𝐼𝑊 ∧ (𝑌𝑧) ∈ ℕ0) → (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)) ∈ 𝐷)
134105, 132, 133syl2anc 587 . . . . . . . . . . . . . . 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 20245 . . . . . . . . . . . . . 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 20247 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)), 1 , 0 )) = ((𝑌𝑧) (𝑉𝑧)))
139138oveq2d 7165 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → (𝑌𝑧) ∈ ℕ0)
141 ifcl 4494 . . . . . . . . . . . . . . . . . . . 20 (((𝑌𝑧) ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑖 = 𝑧, (𝑌𝑧), 0) ∈ ℕ0)
142140, 109, 141sylancl 589 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → if(𝑖 = 𝑧, (𝑌𝑧), 0) ∈ ℕ0)
143 eqidd 2825 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)))
144 eqidd 2825 . . . . . . . . . . . . . . . . . . 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 7420 . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌𝑖) ∈ ℕ0)
147146nn0cnd 11954 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌𝑖) ∈ ℂ)
148147addid2d 10839 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (0 + (𝑌𝑖)) = (𝑌𝑖))
149 elsni 4567 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ {𝑧} → 𝑖 = 𝑧)
150149adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 = 𝑧)
151 simprlr 779 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ¬ 𝑧𝑥)
152151ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → ¬ 𝑧𝑥)
153150, 152eqneltrd 2935 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → ¬ 𝑖𝑥)
154153iffalsed 4461 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖𝑥, (𝑌𝑖), 0) = 0)
155150iftrued 4458 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌𝑧), 0) = (𝑌𝑧))
156150fveq2d 6665 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌𝑖) = (𝑌𝑧))
157155, 156eqtr4d 2862 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌𝑧), 0) = (𝑌𝑖))
158154, 157oveq12d 7167 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (if(𝑖𝑥, (𝑌𝑖), 0) + if(𝑖 = 𝑧, (𝑌𝑧), 0)) = (0 + (𝑌𝑖)))
159 simpr 488 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 ∈ {𝑧})
160126, 159sseldi 3951 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 ∈ (𝑥 ∪ {𝑧}))
161160iftrued 4458 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0) = (𝑌𝑖))
162148, 158, 1613eqtr4d 2869 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℕ0)
164163nn0cnd 11954 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℂ)
165164addid1d 10838 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (if(𝑖𝑥, (𝑌𝑖), 0) + 0) = if(𝑖𝑥, (𝑌𝑖), 0))
166 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → ¬ 𝑖 ∈ {𝑧})
167 velsn 4566 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ {𝑧} ↔ 𝑖 = 𝑧)
168166, 167sylnib 331 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → ¬ 𝑖 = 𝑧)
169168iffalsed 4461 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌𝑧), 0) = 0)
170169oveq2d 7165 . . . . . . . . . . . . . . . . . . . . 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 biorf 934 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑖 ∈ {𝑧} → (𝑖𝑥 ↔ (𝑖 ∈ {𝑧} ∨ 𝑖𝑥)))
172 elun 4111 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑖𝑥𝑖 ∈ {𝑧}))
173 orcom 867 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖𝑥𝑖 ∈ {𝑧}) ↔ (𝑖 ∈ {𝑧} ∨ 𝑖𝑥))
174172, 173bitri 278 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑖 ∈ {𝑧} ∨ 𝑖𝑥))
175171, 174syl6rbbr 293 . . . . . . . . . . . . . . . . . . . . . . 23 𝑖 ∈ {𝑧} → (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑖𝑥))
176175adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑖𝑥))
177176ifbid 4472 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0) = if(𝑖𝑥, (𝑌𝑖), 0))
178165, 170, 1773eqtr4d 2869 . . . . . . . . . . . . . . . . . . . 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 812 . . . . . . . . . . . . . . . . . . 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 5147 . . . . . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . . . . . 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 4472 . . . . . . . . . . . . . . 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 5148 . . . . . . . . . . . . . 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 2868 . . . . . . . . . . . . 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 19245 . . . . . . . . . . . . . 14 (Base‘𝑃) = (Base‘𝐺)
18748, 125mgpplusg 19243 . . . . . . . . . . . . . 14 (.r𝑃) = (+g𝐺)
188 eqid 2824 . . . . . . . . . . . . . 14 (Cntz‘𝐺) = (Cntz‘𝐺)
189 eqid 2824 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))
19092mplring 20232 . . . . . . . . . . . . . . . . 17 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
1912, 95, 190syl2anc 587 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ Ring)
19248ringmgp 19303 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
193191, 192syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺 ∈ Mnd)
194193adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝐺 ∈ Mnd)
1951adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑌𝐷)
196 mplcoe5.c . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑥𝐼𝑦𝐼 ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)))
197 fveq2 6661 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝑉𝑥) = (𝑉𝑎))
198197oveq2d 7165 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑦)(+g𝐺)(𝑉𝑎)))
199197oveq1d 7164 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑉𝑥)(+g𝐺)(𝑉𝑦)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑦)))
200198, 199eqeq12d 2840 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)) ↔ ((𝑉𝑦)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑦))))
201 fveq2 6661 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑏 → (𝑉𝑦) = (𝑉𝑏))
202201oveq1d 7164 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → ((𝑉𝑦)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑏)(+g𝐺)(𝑉𝑎)))
203201oveq2d 7165 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → ((𝑉𝑎)(+g𝐺)(𝑉𝑦)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
204202, 203eqeq12d 2840 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → (((𝑉𝑦)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑦)) ↔ ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏))))
205200, 204cbvral2vw 3446 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐼𝑦𝐼 ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)) ↔ ∀𝑎𝐼𝑏𝐼 ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
206196, 205sylib 221 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑎𝐼𝑏𝐼 ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
207206adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ∀𝑎𝐼𝑏𝐼 ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
20892, 3, 93, 94, 105, 48, 136, 137, 106, 195, 207, 127mplcoe5lem 20248 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ran (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))))
20999, 127sstrid 3964 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑥𝐼)
210209sselda 3953 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘𝑥) → 𝑘𝐼)
211193adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → 𝐺 ∈ Mnd)
2127ffvelrnda 6842 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → (𝑌𝑘) ∈ ℕ0)
2132adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → 𝐼𝑊)
21495adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → 𝑅 ∈ Ring)
215 simpr 488 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → 𝑘𝐼)
21692, 137, 104, 213, 214, 215mvrcl 20229 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → (𝑉𝑘) ∈ (Base‘𝑃))
217186, 136mulgnn0cl 18244 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Mnd ∧ (𝑌𝑘) ∈ ℕ0 ∧ (𝑉𝑘) ∈ (Base‘𝑃)) → ((𝑌𝑘) (𝑉𝑘)) ∈ (Base‘𝑃))
218211, 212, 216, 217syl3anc 1368 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐼) → ((𝑌𝑘) (𝑉𝑘)) ∈ (Base‘𝑃))
219218adantlr 714 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘𝐼) → ((𝑌𝑘) (𝑉𝑘)) ∈ (Base‘𝑃))
220210, 219syldan 594 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘𝑥) → ((𝑌𝑘) (𝑉𝑘)) ∈ (Base‘𝑃))
22192, 137, 104, 105, 106, 131mvrcl 20229 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑉𝑧) ∈ (Base‘𝑃))
222186, 136mulgnn0cl 18244 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Mnd ∧ (𝑌𝑧) ∈ ℕ0 ∧ (𝑉𝑧) ∈ (Base‘𝑃)) → ((𝑌𝑧) (𝑉𝑧)) ∈ (Base‘𝑃))
223194, 132, 221, 222syl3anc 1368 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑌𝑧) (𝑉𝑧)) ∈ (Base‘𝑃))
224 fveq2 6661 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (𝑌𝑘) = (𝑌𝑧))
225 fveq2 6661 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (𝑉𝑘) = (𝑉𝑧))
226224, 225oveq12d 7167 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → ((𝑌𝑘) (𝑉𝑘)) = ((𝑌𝑧) (𝑉𝑧)))
227226adantl 485 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 = 𝑧) → ((𝑌𝑘) (𝑉𝑘)) = ((𝑌𝑧) (𝑉𝑧)))
228186, 187, 188, 189, 194, 115, 208, 220, 131, 151, 223, 227gsumzunsnd 19076 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))) = ((𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))(.r𝑃)((𝑌𝑧) (𝑉𝑧))))
229185, 228eqeq12d 2840 . . . . . . . . . . . 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)))))
230103, 229syl5ibr 249 . . . . . . . . . . 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))))))
231230expr 460 . . . . . . . . . 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)))))))
232231a2d 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)))))))
233102, 232syl5 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)))))))
234233expcom 417 . . . . . . 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))))))))
235234a2d 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))))))))
23655, 67, 79, 91, 98, 235findcard2s 8756 . . . . 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)))))))
23732, 236mpcom 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))))))
23831, 237mpd 15 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘)))))
23931resmptd 5895 . . . 4 (𝜑 → ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ↾ (𝑌 “ ℕ)) = (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘))))
240239oveq2d 7165 . . 3 (𝜑 → (𝐺 Σg ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ↾ (𝑌 “ ℕ))) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘)))))
241218fmpttd 6870 . . . 4 (𝜑 → (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))):𝐼⟶(Base‘𝑃))
242 ssidd 3976 . . . . 5 (𝜑𝐼𝐼)
24392, 3, 93, 94, 2, 48, 136, 137, 95, 1, 196, 242mplcoe5lem 20248 . . . 4 (𝜑 → ran (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))
2447, 16, 2, 18suppssr 7857 . . . . . . 7 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (𝑌𝑘) = 0)
245244oveq1d 7164 . . . . . 6 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → ((𝑌𝑘) (𝑉𝑘)) = (0 (𝑉𝑘)))
246 eldifi 4089 . . . . . . . 8 (𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ)) → 𝑘𝐼)
247246, 216sylan2 595 . . . . . . 7 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (𝑉𝑘) ∈ (Base‘𝑃))
248186, 50, 136mulg0 18231 . . . . . . 7 ((𝑉𝑘) ∈ (Base‘𝑃) → (0 (𝑉𝑘)) = (1r𝑃))
249247, 248syl 17 . . . . . 6 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (0 (𝑉𝑘)) = (1r𝑃))
250245, 249eqtrd 2859 . . . . 5 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → ((𝑌𝑘) (𝑉𝑘)) = (1r𝑃))
251250, 2suppss2 7860 . . . 4 (𝜑 → ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) supp (1r𝑃)) ⊆ (𝑌 “ ℕ))
2522mptexd 6978 . . . . 5 (𝜑 → (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ∈ V)
253 funmpt 6381 . . . . . 6 Fun (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))
254253a1i 11 . . . . 5 (𝜑 → Fun (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))))
255 fvexd 6676 . . . . 5 (𝜑 → (1r𝑃) ∈ V)
256 suppssfifsupp 8845 . . . . 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))
257252, 254, 255, 32, 251, 256syl32anc 1375 . . . 4 (𝜑 → (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) finSupp (1r𝑃))
258186, 50, 188, 193, 2, 241, 243, 251, 257gsumzres 19029 . . 3 (𝜑 → (𝐺 Σg ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ↾ (𝑌 “ ℕ))) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))
259238, 240, 2583eqtr2d 2865 . 2 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))
26029, 259eqtrd 2859 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 209  wa 399  wo 844   = wceq 1538  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  cdif 3916  cun 3917  wss 3919  c0 4276  ifcif 4450  {csn 4550   class class class wbr 5052  cmpt 5132   × cxp 5540  ccnv 5541  cres 5544  cima 5545  Fun wfun 6337  wf 6339  cfv 6343  (class class class)co 7149  f cof 7401   supp csupp 7826  m cmap 8402  Fincfn 8505   finSupp cfsupp 8830  0cc0 10535   + caddc 10538  cn 11634  0cn0 11894  Basecbs 16483  +gcplusg 16565  .rcmulr 16566  0gc0g 16713   Σg cgsu 16714  Mndcmnd 17911  .gcmg 18224  Cntzccntz 18445  mulGrpcmgp 19239  1rcur 19251  Ringcrg 19297   mVar cmvr 20132   mPoly cmpl 20133
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 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
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 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-seq 13374  df-hash 13696  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-tset 16584  df-0g 16715  df-gsum 16716  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-mulg 18225  df-subg 18276  df-ghm 18356  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-srg 19256  df-ring 19299  df-subrg 19533  df-psr 20136  df-mvr 20137  df-mpl 20138
This theorem is referenced by:  mplcoe2  20250  ply1coe  20464
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