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Theorem mplcoe5 19231
Description: Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 19232), 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 19127 . . . . . . . . . 10 (𝐼𝑊 → (𝑌𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (𝑌 “ ℕ) ∈ Fin)))
52, 4syl 17 . . . . . . . . 9 (𝜑 → (𝑌𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (𝑌 “ ℕ) ∈ Fin)))
61, 5mpbid 220 . . . . . . . 8 (𝜑 → (𝑌:𝐼⟶ℕ0 ∧ (𝑌 “ ℕ) ∈ Fin))
76simpld 473 . . . . . . 7 (𝜑𝑌:𝐼⟶ℕ0)
87feqmptd 6140 . . . . . 6 (𝜑𝑌 = (𝑖𝐼 ↦ (𝑌𝑖)))
9 iftrue 4037 . . . . . . . . 9 (𝑖 ∈ (𝑌 “ ℕ) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
109adantl 480 . . . . . . . 8 (((𝜑𝑖𝐼) ∧ 𝑖 ∈ (𝑌 “ ℕ)) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
11 eldif 3545 . . . . . . . . . 10 (𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ)) ↔ (𝑖𝐼 ∧ ¬ 𝑖 ∈ (𝑌 “ ℕ)))
12 ifid 4070 . . . . . . . . . . 11 if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), (𝑌𝑖)) = (𝑌𝑖)
13 frnnn0supp 11192 . . . . . . . . . . . . . . 15 ((𝐼𝑊𝑌:𝐼⟶ℕ0) → (𝑌 supp 0) = (𝑌 “ ℕ))
142, 7, 13syl2anc 690 . . . . . . . . . . . . . 14 (𝜑 → (𝑌 supp 0) = (𝑌 “ ℕ))
15 eqimss 3615 . . . . . . . . . . . . . 14 ((𝑌 supp 0) = (𝑌 “ ℕ) → (𝑌 supp 0) ⊆ (𝑌 “ ℕ))
1614, 15syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑌 supp 0) ⊆ (𝑌 “ ℕ))
17 c0ex 9886 . . . . . . . . . . . . . 14 0 ∈ V
1817a1i 11 . . . . . . . . . . . . 13 (𝜑 → 0 ∈ V)
197, 16, 2, 18suppssr 7186 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (𝑌𝑖) = 0)
2019ifeq2d 4050 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), (𝑌𝑖)) = if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))
2112, 20syl5reqr 2654 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2211, 21sylan2br 491 . . . . . . . . 9 ((𝜑 ∧ (𝑖𝐼 ∧ ¬ 𝑖 ∈ (𝑌 “ ℕ))) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2322anassrs 677 . . . . . . . 8 (((𝜑𝑖𝐼) ∧ ¬ 𝑖 ∈ (𝑌 “ ℕ)) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2410, 23pm2.61dan 827 . . . . . . 7 ((𝜑𝑖𝐼) → if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0) = (𝑌𝑖))
2524mpteq2dva 4662 . . . . . 6 (𝜑 → (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)) = (𝑖𝐼 ↦ (𝑌𝑖)))
268, 25eqtr4d 2642 . . . . 5 (𝜑𝑌 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)))
2726eqeq2d 2615 . . . 4 (𝜑 → (𝑦 = 𝑌𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))))
2827ifbid 4053 . . 3 (𝜑 → if(𝑦 = 𝑌, 1 , 0 ) = if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 ))
2928mpteq2dv 4663 . 2 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )))
30 cnvimass 5387 . . . . 5 (𝑌 “ ℕ) ⊆ dom 𝑌
31 fdm 5946 . . . . . 6 (𝑌:𝐼⟶ℕ0 → dom 𝑌 = 𝐼)
327, 31syl 17 . . . . 5 (𝜑 → dom 𝑌 = 𝐼)
3330, 32syl5sseq 3611 . . . 4 (𝜑 → (𝑌 “ ℕ) ⊆ 𝐼)
346simprd 477 . . . . 5 (𝜑 → (𝑌 “ ℕ) ∈ Fin)
35 sseq1 3584 . . . . . . . 8 (𝑤 = ∅ → (𝑤𝐼 ↔ ∅ ⊆ 𝐼))
36 noel 3873 . . . . . . . . . . . . . . . 16 ¬ 𝑖 ∈ ∅
37 eleq2 2672 . . . . . . . . . . . . . . . 16 (𝑤 = ∅ → (𝑖𝑤𝑖 ∈ ∅))
3836, 37mtbiri 315 . . . . . . . . . . . . . . 15 (𝑤 = ∅ → ¬ 𝑖𝑤)
3938iffalsed 4042 . . . . . . . . . . . . . 14 (𝑤 = ∅ → if(𝑖𝑤, (𝑌𝑖), 0) = 0)
4039mpteq2dv 4663 . . . . . . . . . . . . 13 (𝑤 = ∅ → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ 0))
41 fconstmpt 5071 . . . . . . . . . . . . 13 (𝐼 × {0}) = (𝑖𝐼 ↦ 0)
4240, 41syl6eqr 2657 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝐼 × {0}))
4342eqeq2d 2615 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝐼 × {0})))
4443ifbid 4053 . . . . . . . . . 10 (𝑤 = ∅ → if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
4544mpteq2dv 4663 . . . . . . . . 9 (𝑤 = ∅ → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
46 mpteq1 4655 . . . . . . . . . . . 12 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ ∅ ↦ ((𝑌𝑘) (𝑉𝑘))))
47 mpt0 5916 . . . . . . . . . . . 12 (𝑘 ∈ ∅ ↦ ((𝑌𝑘) (𝑉𝑘))) = ∅
4846, 47syl6eq 2655 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = ∅)
4948oveq2d 6539 . . . . . . . . . 10 (𝑤 = ∅ → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg ∅))
50 mplcoe2.g . . . . . . . . . . . 12 𝐺 = (mulGrp‘𝑃)
51 eqid 2605 . . . . . . . . . . . 12 (1r𝑃) = (1r𝑃)
5250, 51ringidval 18268 . . . . . . . . . . 11 (1r𝑃) = (0g𝐺)
5352gsum0 17043 . . . . . . . . . 10 (𝐺 Σg ∅) = (1r𝑃)
5449, 53syl6eq 2655 . . . . . . . . 9 (𝑤 = ∅ → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (1r𝑃))
5545, 54eqeq12d 2620 . . . . . . . 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)))
5635, 55imbi12d 332 . . . . . . 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))))
5756imbi2d 328 . . . . . 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)))))
58 sseq1 3584 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝐼𝑥𝐼))
59 eleq2 2672 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑖𝑤𝑖𝑥))
6059ifbid 4053 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → if(𝑖𝑤, (𝑌𝑖), 0) = if(𝑖𝑥, (𝑌𝑖), 0))
6160mpteq2dv 4663 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)))
6261eqeq2d 2615 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0))))
6362ifbid 4053 . . . . . . . . . 10 (𝑤 = 𝑥 → if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 ))
6463mpteq2dv 4663 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 )) = (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)), 1 , 0 )))
65 mpteq1 4655 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))
6665oveq2d 6539 . . . . . . . . 9 (𝑤 = 𝑥 → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘)))))
6764, 66eqeq12d 2620 . . . . . . . 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))))))
6858, 67imbi12d 332 . . . . . . 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)))))))
6968imbi2d 328 . . . . . 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))))))))
70 sseq1 3584 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑤𝐼 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐼))
71 eleq2 2672 . . . . . . . . . . . . . 14 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑖𝑤𝑖 ∈ (𝑥 ∪ {𝑧})))
7271ifbid 4053 . . . . . . . . . . . . 13 (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑖𝑤, (𝑌𝑖), 0) = if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0))
7372mpteq2dv 4663 . . . . . . . . . . . 12 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)))
7473eqeq2d 2615 . . . . . . . . . . 11 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0))))
7574ifbid 4053 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0)), 1 , 0 ))
7675mpteq2dv 4663 . . . . . . . . 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 )))
77 mpteq1 4655 . . . . . . . . . 10 (𝑤 = (𝑥 ∪ {𝑧}) → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))))
7877oveq2d 6539 . . . . . . . . 9 (𝑤 = (𝑥 ∪ {𝑧}) → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))))
7976, 78eqeq12d 2620 . . . . . . . 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))))))
8070, 79imbi12d 332 . . . . . . 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)))))))
8180imbi2d 328 . . . . . 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))))))))
82 sseq1 3584 . . . . . . . 8 (𝑤 = (𝑌 “ ℕ) → (𝑤𝐼 ↔ (𝑌 “ ℕ) ⊆ 𝐼))
83 eleq2 2672 . . . . . . . . . . . . . 14 (𝑤 = (𝑌 “ ℕ) → (𝑖𝑤𝑖 ∈ (𝑌 “ ℕ)))
8483ifbid 4053 . . . . . . . . . . . . 13 (𝑤 = (𝑌 “ ℕ) → if(𝑖𝑤, (𝑌𝑖), 0) = if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))
8584mpteq2dv 4663 . . . . . . . . . . . 12 (𝑤 = (𝑌 “ ℕ) → (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)))
8685eqeq2d 2615 . . . . . . . . . . 11 (𝑤 = (𝑌 “ ℕ) → (𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)) ↔ 𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0))))
8786ifbid 4053 . . . . . . . . . 10 (𝑤 = (𝑌 “ ℕ) → if(𝑦 = (𝑖𝐼 ↦ if(𝑖𝑤, (𝑌𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 ))
8887mpteq2dv 4663 . . . . . . . . 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 )))
89 mpteq1 4655 . . . . . . . . . 10 (𝑤 = (𝑌 “ ℕ) → (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘))))
9089oveq2d 6539 . . . . . . . . 9 (𝑤 = (𝑌 “ ℕ) → (𝐺 Σg (𝑘𝑤 ↦ ((𝑌𝑘) (𝑉𝑘)))) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘)))))
9188, 90eqeq12d 2620 . . . . . . . 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))))))
9282, 91imbi12d 332 . . . . . . 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)))))))
9392imbi2d 328 . . . . . 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))))))))
94 mplcoe1.p . . . . . . . . 9 𝑃 = (𝐼 mPoly 𝑅)
95 mplcoe1.z . . . . . . . . 9 0 = (0g𝑅)
96 mplcoe1.o . . . . . . . . 9 1 = (1r𝑅)
97 mplcoe5.r . . . . . . . . 9 (𝜑𝑅 ∈ Ring)
9894, 3, 95, 96, 51, 2, 97mpl1 19207 . . . . . . . 8 (𝜑 → (1r𝑃) = (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
9998eqcomd 2611 . . . . . . 7 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r𝑃))
10099a1d 25 . . . . . 6 (𝜑 → (∅ ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r𝑃)))
101 ssun1 3733 . . . . . . . . . . 11 𝑥 ⊆ (𝑥 ∪ {𝑧})
102 sstr2 3570 . . . . . . . . . . 11 (𝑥 ⊆ (𝑥 ∪ {𝑧}) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼𝑥𝐼))
103101, 102ax-mp 5 . . . . . . . . . 10 ((𝑥 ∪ {𝑧}) ⊆ 𝐼𝑥𝐼)
104103imim1i 60 . . . . . . . . 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))))))
105 oveq1 6530 . . . . . . . . . . . 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))))
106 eqid 2605 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
1072adantr 479 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝐼𝑊)
10897adantr 479 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑅 ∈ Ring)
1097adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑌:𝐼⟶ℕ0)
110109ffvelrnda 6248 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → (𝑌𝑖) ∈ ℕ0)
111 0nn0 11150 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
112 ifcl 4075 . . . . . . . . . . . . . . . . . 18 (((𝑌𝑖) ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℕ0)
113110, 111, 112sylancl 692 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℕ0)
114 eqid 2605 . . . . . . . . . . . . . . . . 17 (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0))
115113, 114fmptd 6273 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)):𝐼⟶ℕ0)
116 frnnn0supp 11192 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑊 ∧ (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)):𝐼⟶ℕ0) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) = ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) “ ℕ))
117107, 115, 116syl2anc 690 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) = ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) “ ℕ))
118 simprll 797 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑥 ∈ Fin)
119 eldifn 3690 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (𝐼𝑥) → ¬ 𝑖𝑥)
120119adantl 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ (𝐼𝑥)) → ¬ 𝑖𝑥)
121120iffalsed 4042 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ (𝐼𝑥)) → if(𝑖𝑥, (𝑌𝑖), 0) = 0)
122121, 107suppss2 7189 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) ⊆ 𝑥)
123 ssfi 8038 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ Fin ∧ ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) ⊆ 𝑥) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) ∈ Fin)
124118, 122, 123syl2anc 690 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) supp 0) ∈ Fin)
125117, 124eqeltrrd 2684 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) “ ℕ) ∈ Fin)
1263psrbag 19127 . . . . . . . . . . . . . . . . 17 (𝐼𝑊 → ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∈ 𝐷 ↔ ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)):𝐼⟶ℕ0 ∧ ((𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) “ ℕ) ∈ Fin)))
127107, 126syl 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)))
128115, 125, 127mpbir2and 958 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) ∈ 𝐷)
129 eqid 2605 . . . . . . . . . . . . . . 15 (.r𝑃) = (.r𝑃)
130 ssun2 3734 . . . . . . . . . . . . . . . . . . 19 {𝑧} ⊆ (𝑥 ∪ {𝑧})
131 simprr 791 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑥 ∪ {𝑧}) ⊆ 𝐼)
132130, 131syl5ss 3574 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → {𝑧} ⊆ 𝐼)
133 vex 3171 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ V
134133snss 4254 . . . . . . . . . . . . . . . . . 18 (𝑧𝐼 ↔ {𝑧} ⊆ 𝐼)
135132, 134sylibr 222 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑧𝐼)
136109, 135ffvelrnd 6249 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑌𝑧) ∈ ℕ0)
1373snifpsrbag 19129 . . . . . . . . . . . . . . . 16 ((𝐼𝑊 ∧ (𝑌𝑧) ∈ ℕ0) → (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)) ∈ 𝐷)
138107, 136, 137syl2anc 690 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)) ∈ 𝐷)
13994, 106, 95, 96, 3, 107, 108, 128, 129, 138mplmonmul 19227 . . . . . . . . . . . . . 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 )))
140 mplcoe2.m . . . . . . . . . . . . . . . 16 = (.g𝐺)
141 mplcoe2.v . . . . . . . . . . . . . . . 16 𝑉 = (𝐼 mVar 𝑅)
14294, 3, 95, 96, 107, 50, 140, 141, 108, 135, 136mplcoe3 19229 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)), 1 , 0 )) = ((𝑌𝑧) (𝑉𝑧)))
143142oveq2d 6539 . . . . . . . . . . . . . 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))))
144136adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → (𝑌𝑧) ∈ ℕ0)
145 ifcl 4075 . . . . . . . . . . . . . . . . . . . 20 (((𝑌𝑧) ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑖 = 𝑧, (𝑌𝑧), 0) ∈ ℕ0)
146144, 111, 145sylancl 692 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → if(𝑖 = 𝑧, (𝑌𝑧), 0) ∈ ℕ0)
147 eqidd 2606 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)) = (𝑖𝐼 ↦ if(𝑖𝑥, (𝑌𝑖), 0)))
148 eqidd 2606 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)) = (𝑖𝐼 ↦ if(𝑖 = 𝑧, (𝑌𝑧), 0)))
149107, 113, 146, 147, 148offval2 6785 . . . . . . . . . . . . . . . . . 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))))
150110adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌𝑖) ∈ ℕ0)
151150nn0cnd 11196 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌𝑖) ∈ ℂ)
152151addid2d 10084 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (0 + (𝑌𝑖)) = (𝑌𝑖))
153 elsni 4137 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ {𝑧} → 𝑖 = 𝑧)
154153adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 = 𝑧)
155 simprlr 798 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ¬ 𝑧𝑥)
156155ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → ¬ 𝑧𝑥)
157154, 156eqneltrd 2702 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → ¬ 𝑖𝑥)
158157iffalsed 4042 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖𝑥, (𝑌𝑖), 0) = 0)
159154iftrued 4039 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌𝑧), 0) = (𝑌𝑧))
160154fveq2d 6088 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌𝑖) = (𝑌𝑧))
161159, 160eqtr4d 2642 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌𝑧), 0) = (𝑌𝑖))
162158, 161oveq12d 6541 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → (if(𝑖𝑥, (𝑌𝑖), 0) + if(𝑖 = 𝑧, (𝑌𝑧), 0)) = (0 + (𝑌𝑖)))
163 simpr 475 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 ∈ {𝑧})
164130, 163sseldi 3561 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 ∈ (𝑥 ∪ {𝑧}))
165164iftrued 4039 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0) = (𝑌𝑖))
166152, 162, 1653eqtr4d 2649 . . . . . . . . . . . . . . . . . . . 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))
167113adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℕ0)
168167nn0cnd 11196 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖𝑥, (𝑌𝑖), 0) ∈ ℂ)
169168addid1d 10083 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (if(𝑖𝑥, (𝑌𝑖), 0) + 0) = if(𝑖𝑥, (𝑌𝑖), 0))
170 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → ¬ 𝑖 ∈ {𝑧})
171 velsn 4136 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ {𝑧} ↔ 𝑖 = 𝑧)
172170, 171sylnib 316 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → ¬ 𝑖 = 𝑧)
173172iffalsed 4042 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌𝑧), 0) = 0)
174173oveq2d 6539 . . . . . . . . . . . . . . . . . . . . 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))
175 biorf 418 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑖 ∈ {𝑧} → (𝑖𝑥 ↔ (𝑖 ∈ {𝑧} ∨ 𝑖𝑥)))
176 elun 3710 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑖𝑥𝑖 ∈ {𝑧}))
177 orcom 400 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖𝑥𝑖 ∈ {𝑧}) ↔ (𝑖 ∈ {𝑧} ∨ 𝑖𝑥))
178176, 177bitri 262 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑖 ∈ {𝑧} ∨ 𝑖𝑥))
179175, 178syl6rbbr 277 . . . . . . . . . . . . . . . . . . . . . . 23 𝑖 ∈ {𝑧} → (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑖𝑥))
180179adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑖𝑥))
181180ifbid 4053 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0) = if(𝑖𝑥, (𝑌𝑖), 0))
182169, 174, 1813eqtr4d 2649 . . . . . . . . . . . . . . . . . . . 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))
183166, 182pm2.61dan 827 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖𝐼) → (if(𝑖𝑥, (𝑌𝑖), 0) + if(𝑖 = 𝑧, (𝑌𝑧), 0)) = if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌𝑖), 0))
184183mpteq2dva 4662 . . . . . . . . . . . . . . . . . 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)))
185149, 184eqtrd 2639 . . . . . . . . . . . . . . . . 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)))
186185eqeq2d 2615 . . . . . . . . . . . . . . . 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))))
187186ifbid 4053 . . . . . . . . . . . . . . 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 ))
188187mpteq2dv 4663 . . . . . . . . . . . . . 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 )))
189139, 143, 1883eqtr3rd 2648 . . . . . . . . . . . . 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))))
19050, 106mgpbas 18260 . . . . . . . . . . . . . 14 (Base‘𝑃) = (Base‘𝐺)
19150, 129mgpplusg 18258 . . . . . . . . . . . . . 14 (.r𝑃) = (+g𝐺)
192 eqid 2605 . . . . . . . . . . . . . 14 (Cntz‘𝐺) = (Cntz‘𝐺)
193 eqid 2605 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))
19494mplring 19215 . . . . . . . . . . . . . . . . 17 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
1952, 97, 194syl2anc 690 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ Ring)
19650ringmgp 18318 . . . . . . . . . . . . . . . 16 (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
197195, 196syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺 ∈ Mnd)
198197adantr 479 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝐺 ∈ Mnd)
1991adantr 479 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑌𝐷)
200 mplcoe5.c . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑥𝐼𝑦𝐼 ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)))
201 fveq2 6084 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝑉𝑥) = (𝑉𝑎))
202201oveq2d 6539 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑦)(+g𝐺)(𝑉𝑎)))
203201oveq1d 6538 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑉𝑥)(+g𝐺)(𝑉𝑦)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑦)))
204202, 203eqeq12d 2620 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)) ↔ ((𝑉𝑦)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑦))))
205 fveq2 6084 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑏 → (𝑉𝑦) = (𝑉𝑏))
206205oveq1d 6538 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → ((𝑉𝑦)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑏)(+g𝐺)(𝑉𝑎)))
207205oveq2d 6539 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑏 → ((𝑉𝑎)(+g𝐺)(𝑉𝑦)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
208206, 207eqeq12d 2620 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏 → (((𝑉𝑦)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑦)) ↔ ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏))))
209204, 208cbvral2v 3150 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐼𝑦𝐼 ((𝑉𝑦)(+g𝐺)(𝑉𝑥)) = ((𝑉𝑥)(+g𝐺)(𝑉𝑦)) ↔ ∀𝑎𝐼𝑏𝐼 ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
210200, 209sylib 206 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑎𝐼𝑏𝐼 ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
211210adantr 479 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ∀𝑎𝐼𝑏𝐼 ((𝑉𝑏)(+g𝐺)(𝑉𝑎)) = ((𝑉𝑎)(+g𝐺)(𝑉𝑏)))
21294, 3, 95, 96, 107, 50, 140, 141, 108, 199, 211, 131mplcoe5lem 19230 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ran (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))))
213101, 131syl5ss 3574 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑥𝐼)
214213sselda 3563 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘𝑥) → 𝑘𝐼)
215197adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → 𝐺 ∈ Mnd)
2167ffvelrnda 6248 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → (𝑌𝑘) ∈ ℕ0)
2172adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → 𝐼𝑊)
21897adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → 𝑅 ∈ Ring)
219 simpr 475 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → 𝑘𝐼)
22094, 141, 106, 217, 218, 219mvrcl 19212 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → (𝑉𝑘) ∈ (Base‘𝑃))
221190, 140mulgnn0cl 17323 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ Mnd ∧ (𝑌𝑘) ∈ ℕ0 ∧ (𝑉𝑘) ∈ (Base‘𝑃)) → ((𝑌𝑘) (𝑉𝑘)) ∈ (Base‘𝑃))
222215, 216, 220, 221syl3anc 1317 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐼) → ((𝑌𝑘) (𝑉𝑘)) ∈ (Base‘𝑃))
223222adantlr 746 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘𝐼) → ((𝑌𝑘) (𝑉𝑘)) ∈ (Base‘𝑃))
224214, 223syldan 485 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘𝑥) → ((𝑌𝑘) (𝑉𝑘)) ∈ (Base‘𝑃))
22594, 141, 106, 107, 108, 135mvrcl 19212 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑉𝑧) ∈ (Base‘𝑃))
226190, 140mulgnn0cl 17323 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Mnd ∧ (𝑌𝑧) ∈ ℕ0 ∧ (𝑉𝑧) ∈ (Base‘𝑃)) → ((𝑌𝑧) (𝑉𝑧)) ∈ (Base‘𝑃))
227198, 136, 225, 226syl3anc 1317 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑌𝑧) (𝑉𝑧)) ∈ (Base‘𝑃))
228 fveq2 6084 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (𝑌𝑘) = (𝑌𝑧))
229 fveq2 6084 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (𝑉𝑘) = (𝑉𝑧))
230228, 229oveq12d 6541 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → ((𝑌𝑘) (𝑉𝑘)) = ((𝑌𝑧) (𝑉𝑧)))
231230adantl 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 = 𝑧) → ((𝑌𝑘) (𝑉𝑘)) = ((𝑌𝑧) (𝑉𝑧)))
232190, 191, 192, 193, 198, 118, 212, 224, 135, 155, 227, 231gsumzunsnd 18120 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌𝑘) (𝑉𝑘)))) = ((𝐺 Σg (𝑘𝑥 ↦ ((𝑌𝑘) (𝑉𝑘))))(.r𝑃)((𝑌𝑧) (𝑉𝑧))))
233189, 232eqeq12d 2620 . . . . . . . . . . . 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)))))
234105, 233syl5ibr 234 . . . . . . . . . . 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))))))
235234expr 640 . . . . . . . . . 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)))))))
236235a2d 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)))))))
237104, 236syl5 33 . . . . . . . 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)))))))
238237expcom 449 . . . . . . 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))))))))
239238a2d 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))))))))
24057, 69, 81, 93, 100, 239findcard2s 8059 . . . . 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)))))))
24134, 240mpcom 37 . . . 4 (𝜑 → ((𝑌 “ ℕ) ⊆ 𝐼 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘))))))
24233, 241mpd 15 . . 3 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘)))))
24333resmptd 5354 . . . 4 (𝜑 → ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ↾ (𝑌 “ ℕ)) = (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘))))
244243oveq2d 6539 . . 3 (𝜑 → (𝐺 Σg ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ↾ (𝑌 “ ℕ))) = (𝐺 Σg (𝑘 ∈ (𝑌 “ ℕ) ↦ ((𝑌𝑘) (𝑉𝑘)))))
245 eqid 2605 . . . . 5 (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) = (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))
246222, 245fmptd 6273 . . . 4 (𝜑 → (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))):𝐼⟶(Base‘𝑃))
247 ssid 3582 . . . . . 6 𝐼𝐼
248247a1i 11 . . . . 5 (𝜑𝐼𝐼)
24994, 3, 95, 96, 2, 50, 140, 141, 97, 1, 200, 248mplcoe5lem 19230 . . . 4 (𝜑 → ran (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))
2507, 16, 2, 18suppssr 7186 . . . . . . 7 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (𝑌𝑘) = 0)
251250oveq1d 6538 . . . . . 6 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → ((𝑌𝑘) (𝑉𝑘)) = (0 (𝑉𝑘)))
252 eldifi 3689 . . . . . . . 8 (𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ)) → 𝑘𝐼)
253252, 220sylan2 489 . . . . . . 7 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (𝑉𝑘) ∈ (Base‘𝑃))
254190, 52, 140mulg0 17311 . . . . . . 7 ((𝑉𝑘) ∈ (Base‘𝑃) → (0 (𝑉𝑘)) = (1r𝑃))
255253, 254syl 17 . . . . . 6 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → (0 (𝑉𝑘)) = (1r𝑃))
256251, 255eqtrd 2639 . . . . 5 ((𝜑𝑘 ∈ (𝐼 ∖ (𝑌 “ ℕ))) → ((𝑌𝑘) (𝑉𝑘)) = (1r𝑃))
257256, 2suppss2 7189 . . . 4 (𝜑 → ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) supp (1r𝑃)) ⊆ (𝑌 “ ℕ))
258 mptexg 6363 . . . . . 6 (𝐼𝑊 → (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ∈ V)
2592, 258syl 17 . . . . 5 (𝜑 → (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ∈ V)
260 funmpt 5822 . . . . . 6 Fun (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))
261260a1i 11 . . . . 5 (𝜑 → Fun (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))))
262 fvex 6094 . . . . . 6 (1r𝑃) ∈ V
263262a1i 11 . . . . 5 (𝜑 → (1r𝑃) ∈ V)
264 suppssfifsupp 8146 . . . . 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))
265259, 261, 263, 34, 257, 264syl32anc 1325 . . . 4 (𝜑 → (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) finSupp (1r𝑃))
266190, 52, 192, 197, 2, 246, 249, 257, 265gsumzres 18075 . . 3 (𝜑 → (𝐺 Σg ((𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘))) ↾ (𝑌 “ ℕ))) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))
267242, 244, 2663eqtr2d 2645 . 2 (𝜑 → (𝑦𝐷 ↦ if(𝑦 = (𝑖𝐼 ↦ if(𝑖 ∈ (𝑌 “ ℕ), (𝑌𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘𝐼 ↦ ((𝑌𝑘) (𝑉𝑘)))))
26829, 267eqtrd 2639 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 194  wo 381  wa 382   = wceq 1474  wcel 1975  wral 2891  {crab 2895  Vcvv 3168  cdif 3532  cun 3533  wss 3535  c0 3869  ifcif 4031  {csn 4120   class class class wbr 4573  cmpt 4633   × cxp 5022  ccnv 5023  dom cdm 5024  cres 5026  cima 5027  Fun wfun 5780  wf 5782  cfv 5786  (class class class)co 6523  𝑓 cof 6766   supp csupp 7155  𝑚 cmap 7717  Fincfn 7814   finSupp cfsupp 8131  0cc0 9788   + caddc 9791  cn 10863  0cn0 11135  Basecbs 15637  +gcplusg 15710  .rcmulr 15711  0gc0g 15865   Σg cgsu 15866  Mndcmnd 17059  .gcmg 17305  Cntzccntz 17513  mulGrpcmgp 18254  1rcur 18266  Ringcrg 18312   mVar cmvr 19115   mPoly cmpl 19116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-iin 4448  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-of 6768  df-ofr 6769  df-om 6931  df-1st 7032  df-2nd 7033  df-supp 7156  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-ixp 7768  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-fsupp 8132  df-oi 8271  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-5 10925  df-6 10926  df-7 10927  df-8 10928  df-9 10929  df-n0 11136  df-z 11207  df-uz 11516  df-fz 12149  df-fzo 12286  df-seq 12615  df-hash 12931  df-struct 15639  df-ndx 15640  df-slot 15641  df-base 15642  df-sets 15643  df-ress 15644  df-plusg 15723  df-mulr 15724  df-sca 15726  df-vsca 15727  df-tset 15729  df-0g 15867  df-gsum 15868  df-mre 16011  df-mrc 16012  df-acs 16014  df-mgm 17007  df-sgrp 17049  df-mnd 17060  df-mhm 17100  df-submnd 17101  df-grp 17190  df-minusg 17191  df-mulg 17306  df-subg 17356  df-ghm 17423  df-cntz 17515  df-cmn 17960  df-abl 17961  df-mgp 18255  df-ur 18267  df-srg 18271  df-ring 18314  df-subrg 18543  df-psr 19119  df-mvr 19120  df-mpl 19121
This theorem is referenced by:  mplcoe2  19232  ply1coe  19429
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