Theorem mplcoe5 22000

Description: Decompose a monomial into a finite product of powers of variables. Instead of assuming that 𝑅 is a commutative ring (as in mplcoe2 22001), 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 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
mplcoe1.z	⊢ 0 = (0g‘𝑅)
mplcoe1.o	⊢ 1 = (1r‘𝑅)
mplcoe1.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplcoe2.g	⊢ 𝐺 = (mulGrp‘𝑃)
mplcoe2.m	⊢ ↑ = (.g‘𝐺)
mplcoe2.v	⊢ 𝑉 = (𝐼 mVar 𝑅)
mplcoe5.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mplcoe5.y	⊢ (𝜑 → 𝑌 ∈ 𝐷)
mplcoe5.c	⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)))

Assertion

Ref	Expression
mplcoe5	⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))

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.

Step	Hyp	Ref	Expression
1		mplcoe5.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
2		mplcoe1.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3		mplcoe1.d	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
4	3	psrbag 21867	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑊 → (𝑌 ∈ 𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈ Fin)))
5	2, 4	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ 𝐷 ↔ (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈ Fin)))
6	1, 5	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (𝑌:𝐼⟶ℕ0 ∧ (◡𝑌 “ ℕ) ∈ Fin))
7	6	simpld 493	. . . . . . 7 ⊢ (𝜑 → 𝑌:𝐼⟶ℕ0)
8	7	feqmptd 6966	. . . . . 6 ⊢ (𝜑 → 𝑌 = (𝑖 ∈ 𝐼 ↦ (𝑌‘𝑖)))
9		iftrue 4536	. . . . . . . . 9 ⊢ (𝑖 ∈ (◡𝑌 “ ℕ) → if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0) = (𝑌‘𝑖))
10	9	adantl 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ (◡𝑌 “ ℕ)) → if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0) = (𝑌‘𝑖))
11		eldif 3954	. . . . . . . . . 10 ⊢ (𝑖 ∈ (𝐼 ∖ (◡𝑌 “ ℕ)) ↔ (𝑖 ∈ 𝐼 ∧ ¬ 𝑖 ∈ (◡𝑌 “ ℕ)))
12		fcdmnn0supp 12561	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑌:𝐼⟶ℕ0) → (𝑌 supp 0) = (◡𝑌 “ ℕ))
13	2, 7, 12	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑌 supp 0) = (◡𝑌 “ ℕ))
14		eqimss 4035	. . . . . . . . . . . . . 14 ⊢ ((𝑌 supp 0) = (◡𝑌 “ ℕ) → (𝑌 supp 0) ⊆ (◡𝑌 “ ℕ))
15	13, 14	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌 supp 0) ⊆ (◡𝑌 “ ℕ))
16		c0ex 11240	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
17	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ V)
18	7, 15, 2, 17	suppssr 8201	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐼 ∖ (◡𝑌 “ ℕ))) → (𝑌‘𝑖) = 0)
19	18	ifeq2d 4550	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐼 ∖ (◡𝑌 “ ℕ))) → if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), (𝑌‘𝑖)) = if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0))
20		ifid 4570	. . . . . . . . . . 11 ⊢ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), (𝑌‘𝑖)) = (𝑌‘𝑖)
21	19, 20	eqtr3di 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐼 ∖ (◡𝑌 “ ℕ))) → if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0) = (𝑌‘𝑖))
22	11, 21	sylan2br 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ ¬ 𝑖 ∈ (◡𝑌 “ ℕ))) → if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0) = (𝑌‘𝑖))
23	22	anassrs 466	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 ∈ (◡𝑌 “ ℕ)) → if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0) = (𝑌‘𝑖))
24	10, 23	pm2.61dan 811	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0) = (𝑌‘𝑖))
25	24	mpteq2dva 5249	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)) = (𝑖 ∈ 𝐼 ↦ (𝑌‘𝑖)))
26	8, 25	eqtr4d 2768	. . . . 5 ⊢ (𝜑 → 𝑌 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)))
27	26	eqeq2d 2736	. . . 4 ⊢ (𝜑 → (𝑦 = 𝑌 ↔ 𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0))))
28	27	ifbid 4553	. . 3 ⊢ (𝜑 → if(𝑦 = 𝑌, 1 , 0 ) = if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)), 1 , 0 ))
29	28	mpteq2dv 5251	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)), 1 , 0 )))
30		cnvimass 6086	. . . . 5 ⊢ (◡𝑌 “ ℕ) ⊆ dom 𝑌
31	30, 7	fssdm 6742	. . . 4 ⊢ (𝜑 → (◡𝑌 “ ℕ) ⊆ 𝐼)
32	6	simprd 494	. . . . 5 ⊢ (𝜑 → (◡𝑌 “ ℕ) ∈ Fin)
33		sseq1 4002	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐼 ↔ ∅ ⊆ 𝐼))
34		noel 4330	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑖 ∈ ∅
35		eleq2 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ∅ → (𝑖 ∈ 𝑤 ↔ 𝑖 ∈ ∅))
36	34, 35	mtbiri 326	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ∅ → ¬ 𝑖 ∈ 𝑤)
37	36	iffalsed 4541	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∅ → if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0) = 0)
38	37	mpteq2dv 5251	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∅ → (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)) = (𝑖 ∈ 𝐼 ↦ 0))
39		fconstmpt 5740	. . . . . . . . . . . . 13 ⊢ (𝐼 × {0}) = (𝑖 ∈ 𝐼 ↦ 0)
40	38, 39	eqtr4di 2783	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)) = (𝐼 × {0}))
41	40	eqeq2d 2736	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)) ↔ 𝑦 = (𝐼 × {0})))
42	41	ifbid 4553	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
43	42	mpteq2dv 5251	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
44		mpteq1 5242	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) = (𝑘 ∈ ∅ ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))
45		mpt0 6698	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ∅ ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) = ∅
46	44, 45	eqtrdi 2781	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) = ∅)
47	46	oveq2d 7435	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) = (𝐺 Σg ∅))
48		mplcoe2.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (mulGrp‘𝑃)
49		eqid 2725	. . . . . . . . . . . 12 ⊢ (1r‘𝑃) = (1r‘𝑃)
50	48, 49	ringidval 20135	. . . . . . . . . . 11 ⊢ (1r‘𝑃) = (0g‘𝐺)
51	50	gsum0 18647	. . . . . . . . . 10 ⊢ (𝐺 Σg ∅) = (1r‘𝑃)
52	47, 51	eqtrdi 2781	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) = (1r‘𝑃))
53	43, 52	eqeq12d 2741	. . . . . . . 8 ⊢ (𝑤 = ∅ → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r‘𝑃)))
54	33, 53	imbi12d 343	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) ↔ (∅ ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r‘𝑃))))
55	54	imbi2d 339	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))) ↔ (𝜑 → (∅ ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r‘𝑃)))))
56		sseq1 4002	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐼 ↔ 𝑥 ⊆ 𝐼))
57		eleq2 2814	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑖 ∈ 𝑤 ↔ 𝑖 ∈ 𝑥))
58	57	ifbid 4553	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0) = if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0))
59	58	mpteq2dv 5251	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)) = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)))
60	59	eqeq2d 2736	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)) ↔ 𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0))))
61	60	ifbid 4553	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 ))
62	61	mpteq2dv 5251	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )))
63		mpteq1 5242	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) = (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))
64	63	oveq2d 7435	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
65	62, 64	eqeq12d 2741	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))))
66	56, 65	imbi12d 343	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((𝑤 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) ↔ (𝑥 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))))
67	66	imbi2d 339	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝜑 → (𝑤 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))) ↔ (𝜑 → (𝑥 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))))))
68		sseq1 4002	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑤 ⊆ 𝐼 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐼))
69		eleq2 2814	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑖 ∈ 𝑤 ↔ 𝑖 ∈ (𝑥 ∪ {𝑧})))
70	69	ifbid 4553	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0) = if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0))
71	70	mpteq2dv 5251	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)) = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)))
72	71	eqeq2d 2736	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)) ↔ 𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0))))
73	72	ifbid 4553	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 ))
74	73	mpteq2dv 5251	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )))
75		mpteq1 5242	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))
76	75	oveq2d 7435	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
77	74, 76	eqeq12d 2741	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))))
78	68, 77	imbi12d 343	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → ((𝑤 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) ↔ ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))))
79	78	imbi2d 339	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → ((𝜑 → (𝑤 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))) ↔ (𝜑 → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))))))
80		sseq1 4002	. . . . . . . 8 ⊢ (𝑤 = (◡𝑌 “ ℕ) → (𝑤 ⊆ 𝐼 ↔ (◡𝑌 “ ℕ) ⊆ 𝐼))
81		eleq2 2814	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (◡𝑌 “ ℕ) → (𝑖 ∈ 𝑤 ↔ 𝑖 ∈ (◡𝑌 “ ℕ)))
82	81	ifbid 4553	. . . . . . . . . . . . 13 ⊢ (𝑤 = (◡𝑌 “ ℕ) → if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0) = if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0))
83	82	mpteq2dv 5251	. . . . . . . . . . . 12 ⊢ (𝑤 = (◡𝑌 “ ℕ) → (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)) = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)))
84	83	eqeq2d 2736	. . . . . . . . . . 11 ⊢ (𝑤 = (◡𝑌 “ ℕ) → (𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)) ↔ 𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0))))
85	84	ifbid 4553	. . . . . . . . . 10 ⊢ (𝑤 = (◡𝑌 “ ℕ) → if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 ) = if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)), 1 , 0 ))
86	85	mpteq2dv 5251	. . . . . . . . 9 ⊢ (𝑤 = (◡𝑌 “ ℕ) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)), 1 , 0 )))
87		mpteq1 5242	. . . . . . . . . 10 ⊢ (𝑤 = (◡𝑌 “ ℕ) → (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) = (𝑘 ∈ (◡𝑌 “ ℕ) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))
88	87	oveq2d 7435	. . . . . . . . 9 ⊢ (𝑤 = (◡𝑌 “ ℕ) → (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) = (𝐺 Σg (𝑘 ∈ (◡𝑌 “ ℕ) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
89	86, 88	eqeq12d 2741	. . . . . . . 8 ⊢ (𝑤 = (◡𝑌 “ ℕ) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (◡𝑌 “ ℕ) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))))
90	80, 89	imbi12d 343	. . . . . . 7 ⊢ (𝑤 = (◡𝑌 “ ℕ) → ((𝑤 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) ↔ ((◡𝑌 “ ℕ) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (◡𝑌 “ ℕ) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))))
91	90	imbi2d 339	. . . . . 6 ⊢ (𝑤 = (◡𝑌 “ ℕ) → ((𝜑 → (𝑤 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑤, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑤 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))) ↔ (𝜑 → ((◡𝑌 “ ℕ) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (◡𝑌 “ ℕ) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))))))
92		mplcoe1.p	. . . . . . . . 9 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
93		mplcoe1.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
94		mplcoe1.o	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
95		mplcoe5.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
96	92, 3, 93, 94, 49, 2, 95	mpl1 21974	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑃) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
97	96, 49	eqtr3di 2780	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r‘𝑃))
98	97	a1d 25	. . . . . 6 ⊢ (𝜑 → (∅ ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (1r‘𝑃)))
99		ssun1 4170	. . . . . . . . . . 11 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑧})
100		sstr2 3983	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ (𝑥 ∪ {𝑧}) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → 𝑥 ⊆ 𝐼))
101	99, 100	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → 𝑥 ⊆ 𝐼)
102	101	imim1i 63	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))))
103		oveq1 7426	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 ))(.r‘𝑃)((𝑌‘𝑧) ↑ (𝑉‘𝑧))) = ((𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))(.r‘𝑃)((𝑌‘𝑧) ↑ (𝑉‘𝑧))))
104		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
105	2	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝐼 ∈ 𝑊)
106	95	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑅 ∈ Ring)
107	7	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑌:𝐼⟶ℕ0)
108	107	ffvelcdmda 7093	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) → (𝑌‘𝑖) ∈ ℕ0)
109		0nn0 12520	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ0
110		ifcl 4575	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑌‘𝑖) ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) ∈ ℕ0)
111	108, 109, 110	sylancl 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) → if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) ∈ ℕ0)
112	111	fmpttd 7124	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)):𝐼⟶ℕ0)
113		fcdmnn0supp 12561	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑊 ∧ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)):𝐼⟶ℕ0) → ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) supp 0) = (◡(𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) “ ℕ))
114	105, 112, 113	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) supp 0) = (◡(𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) “ ℕ))
115		simprll 777	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑥 ∈ Fin)
116		eldifn 4124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (𝐼 ∖ 𝑥) → ¬ 𝑖 ∈ 𝑥)
117	116	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ (𝐼 ∖ 𝑥)) → ¬ 𝑖 ∈ 𝑥)
118	117	iffalsed 4541	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ (𝐼 ∖ 𝑥)) → if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) = 0)
119	118, 105	suppss2 8206	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) supp 0) ⊆ 𝑥)
120	115, 119	ssfid 9292	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) supp 0) ∈ Fin)
121	114, 120	eqeltrrd 2826	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (◡(𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) “ ℕ) ∈ Fin)
122	3	psrbag 21867	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∈ 𝑊 → ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) ∈ 𝐷 ↔ ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)):𝐼⟶ℕ0 ∧ (◡(𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) “ ℕ) ∈ Fin)))
123	105, 122	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) ∈ 𝐷 ↔ ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)):𝐼⟶ℕ0 ∧ (◡(𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) “ ℕ) ∈ Fin)))
124	112, 121, 123	mpbir2and 711	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) ∈ 𝐷)
125		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (.r‘𝑃)
126		ssun2 4171	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑧} ⊆ (𝑥 ∪ {𝑧})
127		simprr 771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑥 ∪ {𝑧}) ⊆ 𝐼)
128	126, 127	sstrid 3988	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → {𝑧} ⊆ 𝐼)
129		vex 3465	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑧 ∈ V
130	129	snss 4791	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐼 ↔ {𝑧} ⊆ 𝐼)
131	128, 130	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑧 ∈ 𝐼)
132	107, 131	ffvelcdmd 7094	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑌‘𝑧) ∈ ℕ0)
133	3	snifpsrbag 21872	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑊 ∧ (𝑌‘𝑧) ∈ ℕ0) → (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0)) ∈ 𝐷)
134	105, 132, 133	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0)) ∈ 𝐷)
135	92, 104, 93, 94, 3, 105, 106, 124, 125, 134	mplmonmul 21996	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 ))(.r‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0)), 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) ∘f + (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0))), 1 , 0 )))
136		mplcoe2.m	. . . . . . . . . . . . . . . 16 ⊢ ↑ = (.g‘𝐺)
137		mplcoe2.v	. . . . . . . . . . . . . . . 16 ⊢ 𝑉 = (𝐼 mVar 𝑅)
138	92, 3, 93, 94, 105, 48, 136, 137, 106, 131, 132	mplcoe3 21998	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0)), 1 , 0 )) = ((𝑌‘𝑧) ↑ (𝑉‘𝑧)))
139	138	oveq2d 7435	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 ))(.r‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0)), 1 , 0 ))) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 ))(.r‘𝑃)((𝑌‘𝑧) ↑ (𝑉‘𝑧))))
140	132	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) → (𝑌‘𝑧) ∈ ℕ0)
141		ifcl 4575	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑌‘𝑧) ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑖 = 𝑧, (𝑌‘𝑧), 0) ∈ ℕ0)
142	140, 109, 141	sylancl 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) → if(𝑖 = 𝑧, (𝑌‘𝑧), 0) ∈ ℕ0)
143		eqidd 2726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)))
144		eqidd 2726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0)) = (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0)))
145	105, 111, 142, 143, 144	offval2 7705	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) ∘f + (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0))) = (𝑖 ∈ 𝐼 ↦ (if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) + if(𝑖 = 𝑧, (𝑌‘𝑧), 0))))
146	108	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌‘𝑖) ∈ ℕ0)
147	146	nn0cnd 12567	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌‘𝑖) ∈ ℂ)
148	147	addlidd 11447	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → (0 + (𝑌‘𝑖)) = (𝑌‘𝑖))
149		elsni 4647	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ {𝑧} → 𝑖 = 𝑧)
150	149	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 = 𝑧)
151		simprlr 778	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ¬ 𝑧 ∈ 𝑥)
152	151	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → ¬ 𝑧 ∈ 𝑥)
153	150, 152	eqneltrd 2845	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → ¬ 𝑖 ∈ 𝑥)
154	153	iffalsed 4541	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) = 0)
155	150	iftrued 4538	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌‘𝑧), 0) = (𝑌‘𝑧))
156	150	fveq2d 6900	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → (𝑌‘𝑖) = (𝑌‘𝑧))
157	155, 156	eqtr4d 2768	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌‘𝑧), 0) = (𝑌‘𝑖))
158	154, 157	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → (if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) + if(𝑖 = 𝑧, (𝑌‘𝑧), 0)) = (0 + (𝑌‘𝑖)))
159		simpr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 ∈ {𝑧})
160	126, 159	sselid 3974	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → 𝑖 ∈ (𝑥 ∪ {𝑧}))
161	160	iftrued 4538	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0) = (𝑌‘𝑖))
162	148, 158, 161	3eqtr4d 2775	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑖 ∈ {𝑧}) → (if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) + if(𝑖 = 𝑧, (𝑌‘𝑧), 0)) = if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0))
163	111	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) ∈ ℕ0)
164	163	nn0cnd 12567	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) ∈ ℂ)
165	164	addridd 11446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) + 0) = if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0))
166		simpr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → ¬ 𝑖 ∈ {𝑧})
167		velsn 4646	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ {𝑧} ↔ 𝑖 = 𝑧)
168	166, 167	sylnib 327	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → ¬ 𝑖 = 𝑧)
169	168	iffalsed 4541	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖 = 𝑧, (𝑌‘𝑧), 0) = 0)
170	169	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) + if(𝑖 = 𝑧, (𝑌‘𝑧), 0)) = (if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) + 0))
171		elun 4145	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑖 ∈ 𝑥 ∨ 𝑖 ∈ {𝑧}))
172		orcom 868	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ 𝑥 ∨ 𝑖 ∈ {𝑧}) ↔ (𝑖 ∈ {𝑧} ∨ 𝑖 ∈ 𝑥))
173	171, 172	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑖 ∈ {𝑧} ∨ 𝑖 ∈ 𝑥))
174		biorf 934	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑖 ∈ {𝑧} → (𝑖 ∈ 𝑥 ↔ (𝑖 ∈ {𝑧} ∨ 𝑖 ∈ 𝑥)))
175	173, 174	bitr4id 289	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑖 ∈ {𝑧} → (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑖 ∈ 𝑥))
176	175	adantl 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (𝑖 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑖 ∈ 𝑥))
177	176	ifbid 4553	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0) = if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0))
178	165, 170, 177	3eqtr4d 2775	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) ∧ ¬ 𝑖 ∈ {𝑧}) → (if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) + if(𝑖 = 𝑧, (𝑌‘𝑧), 0)) = if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0))
179	162, 178	pm2.61dan 811	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑖 ∈ 𝐼) → (if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) + if(𝑖 = 𝑧, (𝑌‘𝑧), 0)) = if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0))
180	179	mpteq2dva 5249	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑖 ∈ 𝐼 ↦ (if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0) + if(𝑖 = 𝑧, (𝑌‘𝑧), 0))) = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)))
181	145, 180	eqtrd 2765	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) ∘f + (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0))) = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)))
182	181	eqeq2d 2736	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑦 = ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) ∘f + (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0))) ↔ 𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0))))
183	182	ifbid 4553	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → if(𝑦 = ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) ∘f + (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0))), 1 , 0 ) = if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 ))
184	183	mpteq2dv 5251	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)) ∘f + (𝑖 ∈ 𝐼 ↦ if(𝑖 = 𝑧, (𝑌‘𝑧), 0))), 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )))
185	135, 139, 184	3eqtr3rd 2774	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 ))(.r‘𝑃)((𝑌‘𝑧) ↑ (𝑉‘𝑧))))
186	48, 104	mgpbas 20092	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑃) = (Base‘𝐺)
187	48, 125	mgpplusg 20090	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑃) = (+g‘𝐺)
188		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
189		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))
190	92, 2, 95	mplringd 21985	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ Ring)
191	48	ringmgp 20191	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
192	190, 191	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ Mnd)
193	192	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝐺 ∈ Mnd)
194	1	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑌 ∈ 𝐷)
195		mplcoe5.c	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)))
196		fveq2 6896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → (𝑉‘𝑥) = (𝑉‘𝑎))
197	196	oveq2d 7435	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑎)))
198	196	oveq1d 7434	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)) = ((𝑉‘𝑎)(+g‘𝐺)(𝑉‘𝑦)))
199	197, 198	eqeq12d 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)) ↔ ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑎)) = ((𝑉‘𝑎)(+g‘𝐺)(𝑉‘𝑦))))
200		fveq2 6896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑏 → (𝑉‘𝑦) = (𝑉‘𝑏))
201	200	oveq1d 7434	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑎)) = ((𝑉‘𝑏)(+g‘𝐺)(𝑉‘𝑎)))
202	200	oveq2d 7435	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → ((𝑉‘𝑎)(+g‘𝐺)(𝑉‘𝑦)) = ((𝑉‘𝑎)(+g‘𝐺)(𝑉‘𝑏)))
203	201, 202	eqeq12d 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑎)) = ((𝑉‘𝑎)(+g‘𝐺)(𝑉‘𝑦)) ↔ ((𝑉‘𝑏)(+g‘𝐺)(𝑉‘𝑎)) = ((𝑉‘𝑎)(+g‘𝐺)(𝑉‘𝑏))))
204	199, 203	cbvral2vw 3228	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 ((𝑉‘𝑦)(+g‘𝐺)(𝑉‘𝑥)) = ((𝑉‘𝑥)(+g‘𝐺)(𝑉‘𝑦)) ↔ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐼 ((𝑉‘𝑏)(+g‘𝐺)(𝑉‘𝑎)) = ((𝑉‘𝑎)(+g‘𝐺)(𝑉‘𝑏)))
205	195, 204	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐼 ((𝑉‘𝑏)(+g‘𝐺)(𝑉‘𝑎)) = ((𝑉‘𝑎)(+g‘𝐺)(𝑉‘𝑏)))
206	205	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐼 ((𝑉‘𝑏)(+g‘𝐺)(𝑉‘𝑎)) = ((𝑉‘𝑎)(+g‘𝐺)(𝑉‘𝑏)))
207	92, 3, 93, 94, 105, 48, 136, 137, 106, 194, 206, 127	mplcoe5lem 21999	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ran (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
208	99, 127	sstrid 3988	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → 𝑥 ⊆ 𝐼)
209	208	sselda 3976	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝐼)
210	192	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐺 ∈ Mnd)
211	7	ffvelcdmda 7093	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑌‘𝑘) ∈ ℕ0)
212	2	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐼 ∈ 𝑊)
213	95	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑅 ∈ Ring)
214		simpr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
215	92, 137, 104, 212, 213, 214	mvrcl 21954	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑉‘𝑘) ∈ (Base‘𝑃))
216	186, 136, 210, 211, 215	mulgnn0cld 19058	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∈ (Base‘𝑃))
217	216	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝐼) → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∈ (Base‘𝑃))
218	209, 217	syldan 589	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 ∈ 𝑥) → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) ∈ (Base‘𝑃))
219	92, 137, 104, 105, 106, 131	mvrcl 21954	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝑉‘𝑧) ∈ (Base‘𝑃))
220	186, 136, 193, 132, 219	mulgnn0cld 19058	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑌‘𝑧) ↑ (𝑉‘𝑧)) ∈ (Base‘𝑃))
221		fveq2 6896	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (𝑌‘𝑘) = (𝑌‘𝑧))
222		fveq2 6896	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (𝑉‘𝑘) = (𝑉‘𝑧))
223	221, 222	oveq12d 7437	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) = ((𝑌‘𝑧) ↑ (𝑉‘𝑧)))
224	223	adantl 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) ∧ 𝑘 = 𝑧) → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) = ((𝑌‘𝑧) ↑ (𝑉‘𝑧)))
225	186, 187, 188, 189, 193, 115, 207, 218, 131, 151, 220, 224	gsumzunsnd 19923	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) = ((𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))(.r‘𝑃)((𝑌‘𝑧) ↑ (𝑉‘𝑧))))
226	185, 225	eqeq12d 2741	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) ↔ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 ))(.r‘𝑃)((𝑌‘𝑧) ↑ (𝑉‘𝑧))) = ((𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))(.r‘𝑃)((𝑌‘𝑧) ↑ (𝑉‘𝑧)))))
227	103, 226	imbitrrid 245	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐼)) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))))
228	227	expr 455	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥)) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))))
229	228	a2d 29	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥)) → (((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))))
230	102, 229	syl5 34	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥)) → ((𝑥 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))))
231	230	expcom 412	. . . . . . 7 ⊢ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) → (𝜑 → ((𝑥 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))))))
232	231	a2d 29	. . . . . 6 ⊢ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) → ((𝜑 → (𝑥 ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑥, (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝑥 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))) → (𝜑 → ((𝑥 ∪ {𝑧}) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (𝑥 ∪ {𝑧}), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))))))
233	55, 67, 79, 91, 98, 232	findcard2s 9190	. . . . 5 ⊢ ((◡𝑌 “ ℕ) ∈ Fin → (𝜑 → ((◡𝑌 “ ℕ) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (◡𝑌 “ ℕ) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))))
234	32, 233	mpcom 38	. . . 4 ⊢ (𝜑 → ((◡𝑌 “ ℕ) ⊆ 𝐼 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (◡𝑌 “ ℕ) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))))
235	31, 234	mpd 15	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ (◡𝑌 “ ℕ) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
236	31	resmptd 6045	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ↾ (◡𝑌 “ ℕ)) = (𝑘 ∈ (◡𝑌 “ ℕ) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))
237	236	oveq2d 7435	. . 3 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ↾ (◡𝑌 “ ℕ))) = (𝐺 Σg (𝑘 ∈ (◡𝑌 “ ℕ) ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
238	216	fmpttd 7124	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))):𝐼⟶(Base‘𝑃))
239		ssidd 4000	. . . . 5 ⊢ (𝜑 → 𝐼 ⊆ 𝐼)
240	92, 3, 93, 94, 2, 48, 136, 137, 95, 1, 195, 239	mplcoe5lem 21999	. . . 4 ⊢ (𝜑 → ran (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
241	7, 15, 2, 17	suppssr 8201	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ (◡𝑌 “ ℕ))) → (𝑌‘𝑘) = 0)
242	241	oveq1d 7434	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ (◡𝑌 “ ℕ))) → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) = (0 ↑ (𝑉‘𝑘)))
243		eldifi 4123	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐼 ∖ (◡𝑌 “ ℕ)) → 𝑘 ∈ 𝐼)
244	243, 215	sylan2 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ (◡𝑌 “ ℕ))) → (𝑉‘𝑘) ∈ (Base‘𝑃))
245	186, 50, 136	mulg0 19038	. . . . . . 7 ⊢ ((𝑉‘𝑘) ∈ (Base‘𝑃) → (0 ↑ (𝑉‘𝑘)) = (1r‘𝑃))
246	244, 245	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ (◡𝑌 “ ℕ))) → (0 ↑ (𝑉‘𝑘)) = (1r‘𝑃))
247	242, 246	eqtrd 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐼 ∖ (◡𝑌 “ ℕ))) → ((𝑌‘𝑘) ↑ (𝑉‘𝑘)) = (1r‘𝑃))
248	247, 2	suppss2 8206	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) supp (1r‘𝑃)) ⊆ (◡𝑌 “ ℕ))
249	2	mptexd 7236	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ∈ V)
250		funmpt 6592	. . . . . 6 ⊢ Fun (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))
251	250	a1i 11	. . . . 5 ⊢ (𝜑 → Fun (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))))
252		fvexd 6911	. . . . 5 ⊢ (𝜑 → (1r‘𝑃) ∈ V)
253		suppssfifsupp 9405	. . . . 5 ⊢ ((((𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ∈ V ∧ Fun (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ∧ (1r‘𝑃) ∈ V) ∧ ((◡𝑌 “ ℕ) ∈ Fin ∧ ((𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) supp (1r‘𝑃)) ⊆ (◡𝑌 “ ℕ))) → (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) finSupp (1r‘𝑃))
254	249, 251, 252, 32, 248, 253	syl32anc 1375	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) finSupp (1r‘𝑃))
255	186, 50, 188, 192, 2, 238, 240, 248, 254	gsumzres 19876	. . 3 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘))) ↾ (◡𝑌 “ ℕ))) = (𝐺 Σg (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
256	235, 237, 255	3eqtr2d 2771	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ (◡𝑌 “ ℕ), (𝑌‘𝑖), 0)), 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))
257	29, 256	eqtrd 2765	1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝐺 Σg (𝑘 ∈ 𝐼 ↦ ((𝑌‘𝑘) ↑ (𝑉‘𝑘)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  ∀wral 3050  {crab 3418  Vcvv 3461   ∖ cdif 3941   ∪ cun 3942   ⊆ wss 3944  ∅c0 4322  ifcif 4530  {csn 4630   class class class wbr 5149   ↦ cmpt 5232   × cxp 5676  ^◡ccnv 5677   ↾ cres 5680   “ cima 5681  Fun wfun 6543  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419   ∘_f cof 7683   supp csupp 8165   ↑_m cmap 8845  Fincfn 8964   finSupp cfsupp 9387  0cc0 11140   + caddc 11143  ℕcn 12245  ℕ₀cn0 12505  Basecbs 17183  +_gcplusg 17236  ._rcmulr 17237  0_gc0g 17424   Σ_g cgsu 17425  Mndcmnd 18697  ._gcmg 19031  Cntzccntz 19278  mulGrpcmgp 20086  1_rcur 20133  Ringcrg 20185   mVar cmvr 21855   mPoly cmpl 21856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-ofr 7686  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9388  df-sup 9467  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-fz 13520  df-fzo 13663  df-seq 14003  df-hash 14326  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-hom 17260  df-cco 17261  df-0g 17426  df-gsum 17427  df-prds 17432  df-pws 17434  df-mre 17569  df-mrc 17570  df-acs 17572  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18743  df-submnd 18744  df-grp 18901  df-minusg 18902  df-mulg 19032  df-subg 19086  df-ghm 19176  df-cntz 19280  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-srg 20139  df-ring 20187  df-subrng 20495  df-subrg 20520  df-psr 21859  df-mvr 21860  df-mpl 21861

This theorem is referenced by:  mplcoe2  22001  ply1coe  22242