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Theorem mplcoe1 21960

Description: Decompose a polynomial into a finite sum of monomials. (Contributed by Mario Carneiro, 9-Jan-2015.)

**Hypotheses**
Ref	Expression
mplcoe1.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplcoe1.d	⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
mplcoe1.z	⊢ 0 = (0_g‘𝑅)
mplcoe1.o	⊢ 1 = (1_r‘𝑅)
mplcoe1.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplcoe1.b	⊢ 𝐵 = (Base‘𝑃)
mplcoe1.n	⊢ · = ( ·_𝑠 ‘𝑃)
mplcoe1.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mplcoe1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
mplcoe1	⊢ (𝜑 → 𝑋 = (𝑃 Σ_g (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))

Distinct variable groups: 𝑦,𝑘, 1 𝐵,𝑘 𝑓,𝑘,𝑦,𝐼 𝜑,𝑘,𝑦 𝑅,𝑓,𝑦 𝐷,𝑘,𝑦 𝑃,𝑘 0 ,𝑓,𝑘,𝑦 𝑓,𝑋,𝑘,𝑦 𝑘,𝑊,𝑦 · ,𝑘 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑦,𝑓) 𝐷(𝑓) 𝑃(𝑦,𝑓) 𝑅(𝑘) · (𝑦,𝑓) 1 (𝑓) 𝑊(𝑓)

Proof of Theorem mplcoe1 Dummy variables 𝑤 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplcoe1.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
2		eqid 2729	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		mplcoe1.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
4		mplcoe1.d	. . . . . 6 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
5		mplcoe1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6	1, 2, 3, 4, 5	mplelf 21923	. . . . 5 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
7	6	feqmptd 6895	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑦 ∈ 𝐷 ↦ (𝑋‘𝑦)))
8		iftrue 4484	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 supp 0 ) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
9	8	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
10		eldif 3915	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 )) ↔ (𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )))
11		ssidd 3961	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ (𝑋 supp 0 ))
12		ovex 7386	. . . . . . . . . . . . 13 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
13	4, 12	rabex2 5283	. . . . . . . . . . . 12 ⊢ 𝐷 ∈ V
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ V)
15		mplcoe1.z	. . . . . . . . . . . . 13 ⊢ 0 = (0_g‘𝑅)
16	15	fvexi 6840	. . . . . . . . . . . 12 ⊢ 0 ∈ V
17	16	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ V)
18	6, 11, 14, 17	suppssr 8135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋‘𝑦) = 0 )
19	18	ifeq2d 4499	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), (𝑋‘𝑦)) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ))
20		ifid 4519	. . . . . . . . 9 ⊢ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), (𝑋‘𝑦)) = (𝑋‘𝑦)
21	19, 20	eqtr3di 2779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
22	10, 21	sylan2br 595	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
23	22	anassrs 467	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
24	9, 23	pm2.61dan 812	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
25	24	mpteq2dva 5188	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 )) = (𝑦 ∈ 𝐷 ↦ (𝑋‘𝑦)))
26	7, 25	eqtr4d 2767	. . 3 ⊢ (𝜑 → 𝑋 = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 )))
27		suppssdm 8117	. . . . 5 ⊢ (𝑋 supp 0 ) ⊆ dom 𝑋
28	27, 6	fssdm 6675	. . . 4 ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ 𝐷)
29		eqid 2729	. . . . . . . . 9 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
30		eqid 2729	. . . . . . . . 9 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
31	1, 29, 30, 15, 3	mplelbas 21916	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 finSupp 0 ))
32	31	simprbi 496	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑋 finSupp 0 )
33	5, 32	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 finSupp 0 )
34	33	fsuppimpd 9278	. . . . 5 ⊢ (𝜑 → (𝑋 supp 0 ) ∈ Fin)
35		sseq1 3963	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐷 ↔ ∅ ⊆ 𝐷))
36		mpteq1 5184	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ ∅ ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
37		mpt0 6628	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ∅ ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅
38	36, 37	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅)
39	38	oveq2d 7369	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σ_g ∅))
40		eqid 2729	. . . . . . . . . . 11 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
41	40	gsum0 18576	. . . . . . . . . 10 ⊢ (𝑃 Σ_g ∅) = (0_g‘𝑃)
42	39, 41	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (0_g‘𝑃))
43		noel 4291	. . . . . . . . . . . 12 ⊢ ¬ 𝑦 ∈ ∅
44		eleq2 2817	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ ∅))
45	43, 44	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → ¬ 𝑦 ∈ 𝑤)
46	45	iffalsed 4489	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ) = 0 )
47	46	mpteq2dv 5189	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )) = (𝑦 ∈ 𝐷 ↦ 0 ))
48	42, 47	eqeq12d 2745	. . . . . . . 8 ⊢ (𝑤 = ∅ → ((𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )) ↔ (0_g‘𝑃) = (𝑦 ∈ 𝐷 ↦ 0 )))
49	35, 48	imbi12d 344	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ))) ↔ (∅ ⊆ 𝐷 → (0_g‘𝑃) = (𝑦 ∈ 𝐷 ↦ 0 ))))
50	49	imbi2d 340	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )))) ↔ (𝜑 → (∅ ⊆ 𝐷 → (0_g‘𝑃) = (𝑦 ∈ 𝐷 ↦ 0 )))))
51		sseq1 3963	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐷 ↔ 𝑥 ⊆ 𝐷))
52		mpteq1 5184	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
53	52	oveq2d 7369	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
54		eleq2 2817	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑥))
55	54	ifbid 4502	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ) = if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))
56	55	mpteq2dv 5189	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )))
57	53, 56	eqeq12d 2745	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )) ↔ (𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))))
58	51, 57	imbi12d 344	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((𝑤 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ))) ↔ (𝑥 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )))))
59	58	imbi2d 340	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝜑 → (𝑤 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )))) ↔ (𝜑 → (𝑥 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))))))
60		sseq1 3963	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑤 ⊆ 𝐷 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐷))
61		mpteq1 5184	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
62	61	oveq2d 7369	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
63		eleq2 2817	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ (𝑥 ∪ {𝑧})))
64	63	ifbid 4502	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ))
65	64	mpteq2dv 5189	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 )))
66	62, 65	eqeq12d 2745	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → ((𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )) ↔ (𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ))))
67	60, 66	imbi12d 344	. . . . . . 7 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → ((𝑤 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ))) ↔ ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 )))))
68	67	imbi2d 340	. . . . . 6 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → ((𝜑 → (𝑤 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )))) ↔ (𝜑 → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ))))))
69		sseq1 3963	. . . . . . . 8 ⊢ (𝑤 = (𝑋 supp 0 ) → (𝑤 ⊆ 𝐷 ↔ (𝑋 supp 0 ) ⊆ 𝐷))
70		mpteq1 5184	. . . . . . . . . 10 ⊢ (𝑤 = (𝑋 supp 0 ) → (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
71	70	oveq2d 7369	. . . . . . . . 9 ⊢ (𝑤 = (𝑋 supp 0 ) → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σ_g (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
72		eleq2 2817	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑋 supp 0 ) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ (𝑋 supp 0 )))
73	72	ifbid 4502	. . . . . . . . . 10 ⊢ (𝑤 = (𝑋 supp 0 ) → if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ))
74	73	mpteq2dv 5189	. . . . . . . . 9 ⊢ (𝑤 = (𝑋 supp 0 ) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 )))
75	71, 74	eqeq12d 2745	. . . . . . . 8 ⊢ (𝑤 = (𝑋 supp 0 ) → ((𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )) ↔ (𝑃 Σ_g (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ))))
76	69, 75	imbi12d 344	. . . . . . 7 ⊢ (𝑤 = (𝑋 supp 0 ) → ((𝑤 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ))) ↔ ((𝑋 supp 0 ) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 )))))
77	76	imbi2d 340	. . . . . 6 ⊢ (𝑤 = (𝑋 supp 0 ) → ((𝜑 → (𝑤 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )))) ↔ (𝜑 → ((𝑋 supp 0 ) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ))))))
78		mplcoe1.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
79		mplcoe1.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
80		ringgrp 20141	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
81	79, 80	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Grp)
82	1, 4, 15, 40, 78, 81	mpl0 21931	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝑃) = (𝐷 × { 0 }))
83		fconstmpt 5685	. . . . . . . 8 ⊢ (𝐷 × { 0 }) = (𝑦 ∈ 𝐷 ↦ 0 )
84	82, 83	eqtrdi 2780	. . . . . . 7 ⊢ (𝜑 → (0_g‘𝑃) = (𝑦 ∈ 𝐷 ↦ 0 ))
85	84	a1d 25	. . . . . 6 ⊢ (𝜑 → (∅ ⊆ 𝐷 → (0_g‘𝑃) = (𝑦 ∈ 𝐷 ↦ 0 )))
86		ssun1 4131	. . . . . . . . . . 11 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑧})
87		sstr2 3944	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ (𝑥 ∪ {𝑧}) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → 𝑥 ⊆ 𝐷))
88	86, 87	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → 𝑥 ⊆ 𝐷)
89	88	imim1i 63	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))))
90		oveq1 7360	. . . . . . . . . . . 12 ⊢ ((𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) → ((𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))(+_g‘𝑃)((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))(+_g‘𝑃)((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))))
91		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
92	1, 78, 79	mplringd 21948	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ Ring)
93		ringcmn 20185	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
94	92, 93	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ CMnd)
95	94	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ CMnd)
96		simprll 778	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥 ∈ Fin)
97		simprr 772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑥 ∪ {𝑧}) ⊆ 𝐷)
98	97	unssad 4146	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥 ⊆ 𝐷)
99	98	sselda 3937	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝐷)
100	78	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐼 ∈ 𝑊)
101	79	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ Ring)
102	1, 100, 101	mpllmodd 21949	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑃 ∈ LMod)
103	6	ffvelcdmda 7022	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋‘𝑘) ∈ (Base‘𝑅))
104	1, 78, 79	mplsca 21938	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
105	104	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 = (Scalar‘𝑃))
106	105	fveq2d 6830	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
107	103, 106	eleqtrd 2830	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋‘𝑘) ∈ (Base‘(Scalar‘𝑃)))
108		mplcoe1.o	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (1_r‘𝑅)
109		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑘 ∈ 𝐷)
110	1, 3, 15, 108, 4, 100, 101, 109	mplmon 21958	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵)
111		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
112		mplcoe1.n	. . . . . . . . . . . . . . . . . 18 ⊢ · = ( ·_𝑠 ‘𝑃)
113		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
114	3, 111, 112, 113	lmodvscl 20799	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ LMod ∧ (𝑋‘𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵) → ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
115	102, 107, 110, 114	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
116	115	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘 ∈ 𝐷) → ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
117	99, 116	syldan 591	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘 ∈ 𝑥) → ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
118		vex 3442	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
119	118	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧 ∈ V)
120		simprlr 779	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ¬ 𝑧 ∈ 𝑥)
121	1, 78, 79	mpllmodd 21949	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ LMod)
122	121	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ LMod)
123	6	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑋:𝐷⟶(Base‘𝑅))
124	97	unssbd 4147	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → {𝑧} ⊆ 𝐷)
125	118	snss 4739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐷 ↔ {𝑧} ⊆ 𝐷)
126	124, 125	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧 ∈ 𝐷)
127	123, 126	ffvelcdmd 7023	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋‘𝑧) ∈ (Base‘𝑅))
128	104	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 = (Scalar‘𝑃))
129	128	fveq2d 6830	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
130	127, 129	eleqtrd 2830	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋‘𝑧) ∈ (Base‘(Scalar‘𝑃)))
131	78	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐼 ∈ 𝑊)
132	79	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 ∈ Ring)
133	1, 3, 15, 108, 4, 131, 132, 126	mplmon 21958	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) ∈ 𝐵)
134	3, 111, 112, 113	lmodvscl 20799	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ LMod ∧ (𝑋‘𝑧) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) ∈ 𝐵) → ((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) ∈ 𝐵)
135	122, 130, 133, 134	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) ∈ 𝐵)
136		fveq2 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝑋‘𝑘) = (𝑋‘𝑧))
137		equequ2 2026	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑧 → (𝑦 = 𝑘 ↔ 𝑦 = 𝑧))
138	137	ifbid 4502	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → if(𝑦 = 𝑘, 1 , 0 ) = if(𝑦 = 𝑧, 1 , 0 ))
139	138	mpteq2dv 5189	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))
140	136, 139	oveq12d 7371	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑧 → ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))))
141	3, 91, 95, 96, 117, 119, 120, 135, 140	gsumunsn 19857	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = ((𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))(+_g‘𝑃)((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))))
142		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
143	123	ffvelcdmda 7022	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → (𝑋‘𝑦) ∈ (Base‘𝑅))
144	2, 15	ring0cl 20170	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
145	79, 144	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ (Base‘𝑅))
146	145	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → 0 ∈ (Base‘𝑅))
147	143, 146	ifcld 4525	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ) ∈ (Base‘𝑅))
148	147	fmpttd 7053	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )):𝐷⟶(Base‘𝑅))
149		fvex 6839	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) ∈ V
150	149, 13	elmap 8805	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ ((Base‘𝑅) ↑_m 𝐷) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )):𝐷⟶(Base‘𝑅))
151	148, 150	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ ((Base‘𝑅) ↑_m 𝐷))
152	29, 2, 4, 30, 131	psrbas 21858	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑_m 𝐷))
153	151, 152	eleqtrrd 2831	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)))
154	13	mptex 7163	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ V
155		funmpt 6524	. . . . . . . . . . . . . . . . . . 19 ⊢ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))
156	154, 155, 16	3pm3.2i 1340	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∧ 0 ∈ V)
157	156	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∧ 0 ∈ V))
158		eldifn 4085	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝐷 ∖ 𝑥) → ¬ 𝑦 ∈ 𝑥)
159	158	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷 ∖ 𝑥)) → ¬ 𝑦 ∈ 𝑥)
160	159	iffalsed 4489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷 ∖ 𝑥)) → if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ) = 0 )
161	13	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐷 ∈ V)
162	160, 161	suppss2 8140	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) supp 0 ) ⊆ 𝑥)
163		suppssfifsupp 9289	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ V ∧ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∧ 0 ∈ V) ∧ (𝑥 ∈ Fin ∧ ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) supp 0 ) ⊆ 𝑥)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) finSupp 0 )
164	157, 96, 162, 163	syl12anc 836	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) finSupp 0 )
165	1, 29, 30, 15, 3	mplelbas 21916	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ 𝐵 ↔ ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) finSupp 0 ))
166	153, 164, 165	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ 𝐵)
167	1, 3, 142, 91, 166, 135	mpladd 21934	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))(+_g‘𝑃)((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∘_f (+_g‘𝑅)((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))))
168		ovexd 7388	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → ((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 )) ∈ V)
169		eqidd 2730	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )))
170		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑅) = (._r‘𝑅)
171	1, 112, 2, 3, 170, 4, 127, 133	mplvsca 21940	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) = ((𝐷 × {(𝑋‘𝑧)}) ∘_f (._r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))))
172	127	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → (𝑋‘𝑧) ∈ (Base‘𝑅))
173	2, 108	ringidcl 20168	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
174	173, 144	ifcld 4525	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ Ring → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
175	79, 174	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
176	175	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
177		fconstmpt 5685	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 × {(𝑋‘𝑧)}) = (𝑦 ∈ 𝐷 ↦ (𝑋‘𝑧))
178	177	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝐷 × {(𝑋‘𝑧)}) = (𝑦 ∈ 𝐷 ↦ (𝑋‘𝑧)))
179		eqidd 2730	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))
180	161, 172, 176, 178, 179	offval2 7637	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝐷 × {(𝑋‘𝑧)}) ∘_f (._r‘𝑅)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ ((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 ))))
181	171, 180	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ ((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 ))))
182	161, 147, 168, 169, 181	offval2 7637	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∘_f (+_g‘𝑅)((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))) = (𝑦 ∈ 𝐷 ↦ (if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )(+_g‘𝑅)((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 )))))
183	132, 80	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 ∈ Grp)
184	2, 142, 15	grplid 18864	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑧) ∈ (Base‘𝑅)) → ( 0 (+_g‘𝑅)(𝑋‘𝑧)) = (𝑋‘𝑧))
185	183, 127, 184	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ( 0 (+_g‘𝑅)(𝑋‘𝑧)) = (𝑋‘𝑧))
186	185	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+_g‘𝑅)(𝑋‘𝑧)) = (𝑋‘𝑧))
187		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ {𝑧})
188		velsn 4595	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
189	187, 188	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 = 𝑧)
190	189	fveq2d 6830	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → (𝑋‘𝑦) = (𝑋‘𝑧))
191	186, 190	eqtr4d 2767	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+_g‘𝑅)(𝑋‘𝑧)) = (𝑋‘𝑦))
192	120	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑧 ∈ 𝑥)
193	189, 192	eqneltrd 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑦 ∈ 𝑥)
194	193	iffalsed 4489	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ) = 0 )
195	189	iftrued 4486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 1 )
196	195	oveq2d 7369	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋‘𝑧)(._r‘𝑅) 1 ))
197	2, 170, 108	ringridm 20173	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑧) ∈ (Base‘𝑅)) → ((𝑋‘𝑧)(._r‘𝑅) 1 ) = (𝑋‘𝑧))
198	132, 127, 197	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋‘𝑧)(._r‘𝑅) 1 ) = (𝑋‘𝑧))
199	198	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋‘𝑧)(._r‘𝑅) 1 ) = (𝑋‘𝑧))
200	196, 199	eqtrd 2764	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 )) = (𝑋‘𝑧))
201	194, 200	oveq12d 7371	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → (if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )(+_g‘𝑅)((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 ))) = ( 0 (+_g‘𝑅)(𝑋‘𝑧)))
202		elun2 4136	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ {𝑧} → 𝑦 ∈ (𝑥 ∪ {𝑧}))
203	202	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ (𝑥 ∪ {𝑧}))
204	203	iftrued 4486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
205	191, 201, 204	3eqtr4d 2774	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → (if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )(+_g‘𝑅)((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 ))) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ))
206	81	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → 𝑅 ∈ Grp)
207	2, 142, 15	grprid 18865	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Grp ∧ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ) ∈ (Base‘𝑅)) → (if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )(+_g‘𝑅) 0 ) = if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))
208	206, 147, 207	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → (if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )(+_g‘𝑅) 0 ) = if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))
209	208	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )(+_g‘𝑅) 0 ) = if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))
210		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 ∈ {𝑧})
211	210, 188	sylnib 328	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 = 𝑧)
212	211	iffalsed 4489	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 0 )
213	212	oveq2d 7369	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 )) = ((𝑋‘𝑧)(._r‘𝑅) 0 ))
214	2, 170, 15	ringrz 20197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑧) ∈ (Base‘𝑅)) → ((𝑋‘𝑧)(._r‘𝑅) 0 ) = 0 )
215	132, 127, 214	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋‘𝑧)(._r‘𝑅) 0 ) = 0 )
216	215	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋‘𝑧)(._r‘𝑅) 0 ) = 0 )
217	213, 216	eqtrd 2764	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 )) = 0 )
218	217	oveq2d 7369	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )(+_g‘𝑅)((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 ))) = (if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )(+_g‘𝑅) 0 ))
219		elun 4106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 ∈ {𝑧}))
220		orcom 870	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝑥 ∨ 𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦 ∈ 𝑥))
221	219, 220	bitri 275	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦 ∈ 𝑥))
222		biorf 936	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑦 ∈ {𝑧} → (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ {𝑧} ∨ 𝑦 ∈ 𝑥)))
223	221, 222	bitr4id 290	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑦 ∈ {𝑧} → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦 ∈ 𝑥))
224	223	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦 ∈ 𝑥))
225	224	ifbid 4502	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ) = if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))
226	209, 218, 225	3eqtr4d 2774	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )(+_g‘𝑅)((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 ))) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ))
227	205, 226	pm2.61dan 812	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → (if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )(+_g‘𝑅)((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 ))) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ))
228	227	mpteq2dva 5188	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ (if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )(+_g‘𝑅)((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 )))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 )))
229	167, 182, 228	3eqtrrd 2769	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 )) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))(+_g‘𝑃)((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))))
230	141, 229	eqeq12d 2745	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 )) ↔ ((𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))(+_g‘𝑃)((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))(+_g‘𝑃)((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))))))
231	90, 230	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) → (𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ))))
232	231	expr 456	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥)) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → ((𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) → (𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 )))))
233	232	a2d 29	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥)) → (((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 )))))
234	89, 233	syl5 34	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥)) → ((𝑥 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 )))))
235	234	expcom 413	. . . . . . 7 ⊢ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) → (𝜑 → ((𝑥 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))) → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ))))))
236	235	a2d 29	. . . . . 6 ⊢ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) → ((𝜑 → (𝑥 ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ 𝑥 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )))) → (𝜑 → ((𝑥 ∪ {𝑧}) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ (𝑥 ∪ {𝑧}) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ))))))
237	50, 59, 68, 77, 85, 236	findcard2s 9089	. . . . 5 ⊢ ((𝑋 supp 0 ) ∈ Fin → (𝜑 → ((𝑋 supp 0 ) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 )))))
238	34, 237	mpcom 38	. . . 4 ⊢ (𝜑 → ((𝑋 supp 0 ) ⊆ 𝐷 → (𝑃 Σ_g (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ))))
239	28, 238	mpd 15	. . 3 ⊢ (𝜑 → (𝑃 Σ_g (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 )))
240	26, 239	eqtr4d 2767	. 2 ⊢ (𝜑 → 𝑋 = (𝑃 Σ_g (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
241	28	resmptd 5995	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ↾ (𝑋 supp 0 )) = (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))))
242	241	oveq2d 7369	. . 3 ⊢ (𝜑 → (𝑃 Σ_g ((𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ↾ (𝑋 supp 0 ))) = (𝑃 Σ_g (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
243	115	fmpttd 7053	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))):𝐷⟶𝐵)
244	6, 11, 14, 17	suppssr 8135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋‘𝑘) = 0 )
245	244	oveq1d 7368	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ( 0 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
246		eldifi 4084	. . . . . . 7 ⊢ (𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 )) → 𝑘 ∈ 𝐷)
247	105	fveq2d 6830	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (0_g‘𝑅) = (0_g‘(Scalar‘𝑃)))
248	15, 247	eqtrid 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 0 = (0_g‘(Scalar‘𝑃)))
249	248	oveq1d 7368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ( 0 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ((0_g‘(Scalar‘𝑃)) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
250		eqid 2729	. . . . . . . . . 10 ⊢ (0_g‘(Scalar‘𝑃)) = (0_g‘(Scalar‘𝑃))
251	3, 111, 112, 250, 40	lmod0vs 20816	. . . . . . . . 9 ⊢ ((𝑃 ∈ LMod ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵) → ((0_g‘(Scalar‘𝑃)) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0_g‘𝑃))
252	102, 110, 251	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((0_g‘(Scalar‘𝑃)) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0_g‘𝑃))
253	249, 252	eqtrd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ( 0 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0_g‘𝑃))
254	246, 253	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ( 0 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0_g‘𝑃))
255	245, 254	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0_g‘𝑃))
256	255, 14	suppss2 8140	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) supp (0_g‘𝑃)) ⊆ (𝑋 supp 0 ))
257	13	mptex 7163	. . . . . . 7 ⊢ (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V
258		funmpt 6524	. . . . . . 7 ⊢ Fun (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
259		fvex 6839	. . . . . . 7 ⊢ (0_g‘𝑃) ∈ V
260	257, 258, 259	3pm3.2i 1340	. . . . . 6 ⊢ ((𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V ∧ Fun (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∧ (0_g‘𝑃) ∈ V)
261	260	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V ∧ Fun (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∧ (0_g‘𝑃) ∈ V))
262		suppssfifsupp 9289	. . . . 5 ⊢ ((((𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V ∧ Fun (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∧ (0_g‘𝑃) ∈ V) ∧ ((𝑋 supp 0 ) ∈ Fin ∧ ((𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) supp (0_g‘𝑃)) ⊆ (𝑋 supp 0 ))) → (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) finSupp (0_g‘𝑃))
263	261, 34, 256, 262	syl12anc 836	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) finSupp (0_g‘𝑃))
264	3, 40, 94, 14, 243, 256, 263	gsumres 19810	. . 3 ⊢ (𝜑 → (𝑃 Σ_g ((𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ↾ (𝑋 supp 0 ))) = (𝑃 Σ_g (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
265	242, 264	eqtr3d 2766	. 2 ⊢ (𝜑 → (𝑃 Σ_g (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σ_g (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
266	240, 265	eqtrd 2764	1 ⊢ (𝜑 → 𝑋 = (𝑃 Σ_g (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {crab 3396 Vcvv 3438 ∖ cdif 3902 ∪ cun 3903 ⊆ wss 3905 ∅c0 4286 ifcif 4478 {csn 4579 class class class wbr 5095 ↦ cmpt 5176 × cxp 5621 ^◡ccnv 5622 ↾ cres 5625 “ cima 5626 Fun wfun 6480 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ∘_f cof 7615 supp csupp 8100 ↑_m cmap 8760 Fincfn 8879 finSupp cfsupp 9270 ℕcn 12146 ℕ₀cn0 12402 Basecbs 17138 +_gcplusg 17179 ._rcmulr 17180 Scalarcsca 17182 ·_𝑠 cvsca 17183 0_gc0g 17361 Σ_g cgsu 17362 Grpcgrp 18830 CMndccmn 19677 1_rcur 20084 Ringcrg 20136 LModclmod 20781 mPwSer cmps 21829 mPoly cmpl 21831

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-ofr 7618 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-sup 9351 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-fzo 13576 df-seq 13927 df-hash 14256 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17363 df-gsum 17364 df-prds 17369 df-pws 17371 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-mulg 18965 df-subg 19020 df-ghm 19110 df-cntz 19214 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-subrng 20449 df-subrg 20473 df-lmod 20783 df-lss 20853 df-psr 21834 df-mpl 21836

This theorem is referenced by: mplbas2 21965 mplcoe4 21994 ply1coe 22201

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