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Theorem mplcoe1 22020

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 2740	. . . . . 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 21979	. . . . 5 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
7	6	feqmptd 6902	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑦 ∈ 𝐷 ↦ (𝑋‘𝑦)))
8		iftrue 4467	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 supp 0 ) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
9	8	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
10		eldif 3900	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 )) ↔ (𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )))
11		ssidd 3945	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ (𝑋 supp 0 ))
12		ovex 7396	. . . . . . . . . . . . 13 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
13	4, 12	rabex2 5276	. . . . . . . . . . . 12 ⊢ 𝐷 ∈ V
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ V)
15		mplcoe1.z	. . . . . . . . . . . . 13 ⊢ 0 = (0_g‘𝑅)
16	15	fvexi 6848	. . . . . . . . . . . 12 ⊢ 0 ∈ V
17	16	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ V)
18	6, 11, 14, 17	suppssr 8142	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋‘𝑦) = 0 )
19	18	ifeq2d 4482	. . . . . . . . 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 4502	. . . . . . . . 9 ⊢ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), (𝑋‘𝑦)) = (𝑋‘𝑦)
21	19, 20	eqtr3di 2790	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
22	10, 21	sylan2br 601	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ (𝑋 supp 0 ))) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
23	22	anassrs 468	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ (𝑋 supp 0 )) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
24	9, 23	pm2.61dan 818	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
25	24	mpteq2dva 5172	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 )) = (𝑦 ∈ 𝐷 ↦ (𝑋‘𝑦)))
26	7, 25	eqtr4d 2778	. . 3 ⊢ (𝜑 → 𝑋 = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 )))
27		suppssdm 8124	. . . . 5 ⊢ (𝑋 supp 0 ) ⊆ dom 𝑋
28	27, 6	fssdm 6681	. . . 4 ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ 𝐷)
29		eqid 2740	. . . . . . . . 9 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
30		eqid 2740	. . . . . . . . 9 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
31	1, 29, 30, 15, 3	mplelbas 21972	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑋 finSupp 0 ))
32	31	simprbi 498	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑋 finSupp 0 )
33	5, 32	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 finSupp 0 )
34	33	fsuppimpd 9279	. . . . 5 ⊢ (𝜑 → (𝑋 supp 0 ) ∈ Fin)
35		sseq1 3947	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐷 ↔ ∅ ⊆ 𝐷))
36		mpteq1 5168	. . . . . . . . . . . 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 6634	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ∅ ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅
38	36, 37	eqtrdi 2791	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) = ∅)
39	38	oveq2d 7379	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (𝑃 Σ_g ∅))
40		eqid 2740	. . . . . . . . . . 11 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
41	40	gsum0 18650	. . . . . . . . . 10 ⊢ (𝑃 Σ_g ∅) = (0_g‘𝑃)
42	39, 41	eqtrdi 2791	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑃 Σ_g (𝑘 ∈ 𝑤 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))) = (0_g‘𝑃))
43		noel 4273	. . . . . . . . . . . 12 ⊢ ¬ 𝑦 ∈ ∅
44		eleq2 2829	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ ∅))
45	43, 44	mtbiri 328	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → ¬ 𝑦 ∈ 𝑤)
46	45	iffalsed 4472	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ) = 0 )
47	46	mpteq2dv 5173	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )) = (𝑦 ∈ 𝐷 ↦ 0 ))
48	42, 47	eqeq12d 2756	. . . . . . . 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 345	. . . . . . 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 341	. . . . . 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 3947	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐷 ↔ 𝑥 ⊆ 𝐷))
52		mpteq1 5168	. . . . . . . . . 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 7379	. . . . . . . . 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 2829	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑥))
55	54	ifbid 4485	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ) = if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))
56	55	mpteq2dv 5173	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )))
57	53, 56	eqeq12d 2756	. . . . . . . 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 345	. . . . . . 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 341	. . . . . 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 3947	. . . . . . . 8 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑤 ⊆ 𝐷 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐷))
61		mpteq1 5168	. . . . . . . . . 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 7379	. . . . . . . . 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 2829	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ (𝑥 ∪ {𝑧})))
64	63	ifbid 4485	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 ∪ {𝑧}) → if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ) = if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ))
65	64	mpteq2dv 5173	. . . . . . . . 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 2756	. . . . . . . 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 345	. . . . . . 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 341	. . . . . 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 3947	. . . . . . . 8 ⊢ (𝑤 = (𝑋 supp 0 ) → (𝑤 ⊆ 𝐷 ↔ (𝑋 supp 0 ) ⊆ 𝐷))
70		mpteq1 5168	. . . . . . . . . 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 7379	. . . . . . . . 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 2829	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑋 supp 0 ) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ (𝑋 supp 0 )))
73	72	ifbid 4485	. . . . . . . . . 10 ⊢ (𝑤 = (𝑋 supp 0 ) → if(𝑦 ∈ 𝑤, (𝑋‘𝑦), 0 ) = if(𝑦 ∈ (𝑋 supp 0 ), (𝑋‘𝑦), 0 ))
74	73	mpteq2dv 5173	. . . . . . . . 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 2756	. . . . . . . 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 345	. . . . . . 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 341	. . . . . 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 20217	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
81	79, 80	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Grp)
82	1, 4, 15, 40, 78, 81	mpl0 21987	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝑃) = (𝐷 × { 0 }))
83		fconstmpt 5687	. . . . . . . 8 ⊢ (𝐷 × { 0 }) = (𝑦 ∈ 𝐷 ↦ 0 )
84	82, 83	eqtrdi 2791	. . . . . . 7 ⊢ (𝜑 → (0_g‘𝑃) = (𝑦 ∈ 𝐷 ↦ 0 ))
85	84	a1d 25	. . . . . 6 ⊢ (𝜑 → (∅ ⊆ 𝐷 → (0_g‘𝑃) = (𝑦 ∈ 𝐷 ↦ 0 )))
86		ssun1 4114	. . . . . . . . . . 11 ⊢ 𝑥 ⊆ (𝑥 ∪ {𝑧})
87		sstr2 3929	. . . . . . . . . . 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 7370	. . . . . . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
92	1, 78, 79	mplringd 22004	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ Ring)
93		ringcmn 20261	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
94	92, 93	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ CMnd)
95	94	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ CMnd)
96		simprll 784	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥 ∈ Fin)
97		simprr 778	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑥 ∪ {𝑧}) ⊆ 𝐷)
98	97	unssad 4129	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑥 ⊆ 𝐷)
99	98	sselda 3922	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝐷)
100	78	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐼 ∈ 𝑊)
101	79	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ Ring)
102	1, 100, 101	mpllmodd 22006	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑃 ∈ LMod)
103	6	ffvelcdmda 7032	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋‘𝑘) ∈ (Base‘𝑅))
104	1, 78, 79	mplsca 21994	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
105	104	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 = (Scalar‘𝑃))
106	105	fveq2d 6838	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
107	103, 106	eleqtrd 2842	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋‘𝑘) ∈ (Base‘(Scalar‘𝑃)))
108		mplcoe1.o	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (1_r‘𝑅)
109		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑘 ∈ 𝐷)
110	1, 3, 15, 108, 4, 100, 101, 109	mplmon 22018	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) ∈ 𝐵)
111		eqid 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
112		mplcoe1.n	. . . . . . . . . . . . . . . . . 18 ⊢ · = ( ·_𝑠 ‘𝑃)
113		eqid 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
114	3, 111, 112, 113	lmodvscl 20875	. . . . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
116	115	adantlr 721	. . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑘 ∈ 𝑥) → ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) ∈ 𝐵)
118		vex 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
119	118	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧 ∈ V)
120		simprlr 785	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ¬ 𝑧 ∈ 𝑥)
121	1, 78, 79	mpllmodd 22006	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ LMod)
122	121	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑃 ∈ LMod)
123	6	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑋:𝐷⟶(Base‘𝑅))
124	97	unssbd 4130	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → {𝑧} ⊆ 𝐷)
125	118	snss 4723	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐷 ↔ {𝑧} ⊆ 𝐷)
126	124, 125	sylibr 235	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑧 ∈ 𝐷)
127	123, 126	ffvelcdmd 7033	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋‘𝑧) ∈ (Base‘𝑅))
128	104	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 = (Scalar‘𝑃))
129	128	fveq2d 6838	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
130	127, 129	eleqtrd 2842	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑋‘𝑧) ∈ (Base‘(Scalar‘𝑃)))
131	78	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝐼 ∈ 𝑊)
132	79	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → 𝑅 ∈ Ring)
133	1, 3, 15, 108, 4, 131, 132, 126	mplmon 22018	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )) ∈ 𝐵)
134	3, 111, 112, 113	lmodvscl 20875	. . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋‘𝑧) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 ))) ∈ 𝐵)
136		fveq2 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝑋‘𝑘) = (𝑋‘𝑧))
137		equequ2 2033	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑧 → (𝑦 = 𝑘 ↔ 𝑦 = 𝑧))
138	137	ifbid 4485	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → if(𝑦 = 𝑘, 1 , 0 ) = if(𝑦 = 𝑧, 1 , 0 ))
139	138	mpteq2dv 5173	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑧, 1 , 0 )))
140	136, 139	oveq12d 7381	. . . . . . . . . . . . . 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 19933	. . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
143	123	ffvelcdmda 7032	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → (𝑋‘𝑦) ∈ (Base‘𝑅))
144	2, 15	ring0cl 20246	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
145	79, 144	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ (Base‘𝑅))
146	145	ad2antrr 732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → 0 ∈ (Base‘𝑅))
147	143, 146	ifcld 4508	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ) ∈ (Base‘𝑅))
148	147	fmpttd 7063	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )):𝐷⟶(Base‘𝑅))
149		fvex 6847	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) ∈ V
150	149, 13	elmap 8816	. . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . 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 21916	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑_m 𝐷))
153	151, 152	eleqtrrd 2843	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ (Base‘(𝐼 mPwSer 𝑅)))
154	13	mptex 7174	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) ∈ V
155		funmpt 6530	. . . . . . . . . . . . . . . . . . 19 ⊢ Fun (𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ))
156	154, 155, 16	3pm3.2i 1346	. . . . . . . . . . . . . . . . . 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 4069	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝐷 ∖ 𝑥) → ¬ 𝑦 ∈ 𝑥)
159	158	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ (𝐷 ∖ 𝑥)) → ¬ 𝑦 ∈ 𝑥)
160	159	iffalsed 4472	. . . . . . . . . . . . . . . . . 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 8147	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 )) supp 0 ) ⊆ 𝑥)
163		suppssfifsupp 9290	. . . . . . . . . . . . . . . . 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 842	. . . . . . . . . . . . . . . 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 21972	. . . . . . . . . . . . . . . 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 589	. . . . . . . . . . . . . . 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 21990	. . . . . . . . . . . . . 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 7398	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → ((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 )) ∈ V)
169		eqidd 2741	. . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑅) = (._r‘𝑅)
171	1, 112, 2, 3, 170, 4, 127, 133	mplvsca 21996	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → (𝑋‘𝑧) ∈ (Base‘𝑅))
173	2, 108	ringidcl 20244	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
174	173, 144	ifcld 4508	. . . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → if(𝑦 = 𝑧, 1 , 0 ) ∈ (Base‘𝑅))
177		fconstmpt 5687	. . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . 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 7647	. . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . 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 7647	. . . . . . . . . . . . . 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 18941	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Grp ∧ (𝑋‘𝑧) ∈ (Base‘𝑅)) → ( 0 (+_g‘𝑅)(𝑋‘𝑧)) = (𝑋‘𝑧))
185	183, 127, 184	syl2anc 590	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ( 0 (+_g‘𝑅)(𝑋‘𝑧)) = (𝑋‘𝑧))
186	185	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+_g‘𝑅)(𝑋‘𝑧)) = (𝑋‘𝑧))
187		velsn 4578	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
188	187	bilani 505	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 = 𝑧)
189	188	fveq2d 6838	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → (𝑋‘𝑦) = (𝑋‘𝑧))
190	186, 189	eqtr4d 2778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ( 0 (+_g‘𝑅)(𝑋‘𝑧)) = (𝑋‘𝑦))
191	120	ad2antrr 732	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑧 ∈ 𝑥)
192	188, 191	eqneltrd 2860	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ¬ 𝑦 ∈ 𝑥)
193	192	iffalsed 4472	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ 𝑥, (𝑋‘𝑦), 0 ) = 0 )
194	188	iftrued 4469	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 1 )
195	194	oveq2d 7379	. . . . . . . . . . . . . . . . . . 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 ))
196	2, 170, 108	ringridm 20249	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑧) ∈ (Base‘𝑅)) → ((𝑋‘𝑧)(._r‘𝑅) 1 ) = (𝑋‘𝑧))
197	132, 127, 196	syl2anc 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋‘𝑧)(._r‘𝑅) 1 ) = (𝑋‘𝑧))
198	197	ad2antrr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋‘𝑧)(._r‘𝑅) 1 ) = (𝑋‘𝑧))
199	195, 198	eqtrd 2775	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → ((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 )) = (𝑋‘𝑧))
200	193, 199	oveq12d 7381	. . . . . . . . . . . . . . . . 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)))
201		elun2 4119	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ {𝑧} → 𝑦 ∈ (𝑥 ∪ {𝑧}))
202	201	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → 𝑦 ∈ (𝑥 ∪ {𝑧}))
203	202	iftrued 4469	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ 𝑦 ∈ {𝑧}) → if(𝑦 ∈ (𝑥 ∪ {𝑧}), (𝑋‘𝑦), 0 ) = (𝑋‘𝑦))
204	190, 200, 203	3eqtr4d 2785	. . . . . . . . . . . . . . . 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 ))
205	81	ad2antrr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) → 𝑅 ∈ Grp)
206	2, 142, 15	grprid 18942	. . . . . . . . . . . . . . . . . . 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 ))
207	205, 147, 206	syl2anc 590	. . . . . . . . . . . . . . . . . 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 ))
208	207	adantr 481	. . . . . . . . . . . . . . . . 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 ))
209		simpr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 ∈ {𝑧})
210	209, 187	sylnib 329	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ¬ 𝑦 = 𝑧)
211	210	iffalsed 4472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → if(𝑦 = 𝑧, 1 , 0 ) = 0 )
212	211	oveq2d 7379	. . . . . . . . . . . . . . . . . . 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 ))
213	2, 170, 15	ringrz 20273	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑧) ∈ (Base‘𝑅)) → ((𝑋‘𝑧)(._r‘𝑅) 0 ) = 0 )
214	132, 127, 213	syl2anc 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) → ((𝑋‘𝑧)(._r‘𝑅) 0 ) = 0 )
215	214	ad2antrr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋‘𝑧)(._r‘𝑅) 0 ) = 0 )
216	212, 215	eqtrd 2775	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → ((𝑋‘𝑧)(._r‘𝑅)if(𝑦 = 𝑧, 1 , 0 )) = 0 )
217	216	oveq2d 7379	. . . . . . . . . . . . . . . . 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 ))
218		elun 4090	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 ∈ {𝑧}))
219		orcom 876	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝑥 ∨ 𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦 ∈ 𝑥))
220	218, 219	bitri 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ (𝑦 ∈ {𝑧} ∨ 𝑦 ∈ 𝑥))
221		biorf 942	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑦 ∈ {𝑧} → (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ {𝑧} ∨ 𝑦 ∈ 𝑥)))
222	220, 221	bitr4id 291	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑦 ∈ {𝑧} → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦 ∈ 𝑥))
223	222	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ((𝑥 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑥) ∧ (𝑥 ∪ {𝑧}) ⊆ 𝐷)) ∧ 𝑦 ∈ 𝐷) ∧ ¬ 𝑦 ∈ {𝑧}) → (𝑦 ∈ (𝑥 ∪ {𝑧}) ↔ 𝑦 ∈ 𝑥))
224	223	ifbid 4485	. . . . . . . . . . . . . . . . 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 ))
225	208, 217, 224	3eqtr4d 2785	. . . . . . . . . . . . . . . 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 ))
226	204, 225	pm2.61dan 818	. . . . . . . . . . . . . . 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 ))
227	226	mpteq2dva 5172	. . . . . . . . . . . . . 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 )))
228	167, 182, 227	3eqtrrd 2780	. . . . . . . . . . . . 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 )))))
229	141, 228	eqeq12d 2756	. . . . . . . . . . . 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 ))))))
230	90, 229	imbitrrid 247	. . . . . . . . . . 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 ))))
231	230	expr 457	. . . . . . . . . 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 )))))
232	231	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 )))))
233	89, 232	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 )))))
234	233	expcom 414	. . . . . . 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 ))))))
235	234	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 ))))))
236	50, 59, 68, 77, 85, 235	findcard2s 9097	. . . . 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 )))))
237	34, 236	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 ))))
238	28, 237	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 )))
239	26, 238	eqtr4d 2778	. 2 ⊢ (𝜑 → 𝑋 = (𝑃 Σ_g (𝑘 ∈ (𝑋 supp 0 ) ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
240	28	resmptd 5999	. . . 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 )))))
241	240	oveq2d 7379	. . 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 ))))))
242	115	fmpttd 7063	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))):𝐷⟶𝐵)
243	6, 11, 14, 17	suppssr 8142	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → (𝑋‘𝑘) = 0 )
244	243	oveq1d 7378	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ( 0 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
245		eldifi 4068	. . . . . . 7 ⊢ (𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 )) → 𝑘 ∈ 𝐷)
246	105	fveq2d 6838	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (0_g‘𝑅) = (0_g‘(Scalar‘𝑃)))
247	15, 246	eqtrid 2787	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 0 = (0_g‘(Scalar‘𝑃)))
248	247	oveq1d 7378	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ( 0 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = ((0_g‘(Scalar‘𝑃)) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
249		eqid 2740	. . . . . . . . . 10 ⊢ (0_g‘(Scalar‘𝑃)) = (0_g‘(Scalar‘𝑃))
250	3, 111, 112, 249, 40	lmod0vs 20892	. . . . . . . . 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))
251	102, 110, 250	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((0_g‘(Scalar‘𝑃)) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0_g‘𝑃))
252	248, 251	eqtrd 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ( 0 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0_g‘𝑃))
253	245, 252	sylan2 599	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ( 0 · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0_g‘𝑃))
254	244, 253	eqtrd 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 ∖ (𝑋 supp 0 ))) → ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))) = (0_g‘𝑃))
255	254, 14	suppss2 8147	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) supp (0_g‘𝑃)) ⊆ (𝑋 supp 0 ))
256	13	mptex 7174	. . . . . . 7 ⊢ (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) ∈ V
257		funmpt 6530	. . . . . . 7 ⊢ Fun (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))
258		fvex 6847	. . . . . . 7 ⊢ (0_g‘𝑃) ∈ V
259	256, 257, 258	3pm3.2i 1346	. . . . . 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)
260	259	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))
261		suppssfifsupp 9290	. . . . 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))
262	260, 34, 255, 261	syl12anc 842	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ ((𝑋‘𝑘) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 )))) finSupp (0_g‘𝑃))
263	3, 40, 94, 14, 242, 255, 262	gsumres 19886	. . 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 ))))))
264	241, 263	eqtr3d 2777	. 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 ))))))
265	239, 264	eqtrd 2775	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 207 ∧ wa 396 ∨ wo 853 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 {crab 3392 Vcvv 3432 ∖ cdif 3887 ∪ cun 3888 ⊆ wss 3890 ∅c0 4268 ifcif 4461 {csn 4562 class class class wbr 5079 ↦ cmpt 5160 × cxp 5623 ^◡ccnv 5624 ↾ cres 5627 “ cima 5628 Fun wfun 6486 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ∘_f cof 7625 supp csupp 8107 ↑_m cmap 8770 Fincfn 8890 finSupp cfsupp 9271 ℕcn 12172 ℕ₀cn0 12435 Basecbs 17177 +_gcplusg 17218 ._rcmulr 17219 Scalarcsca 17221 ·_𝑠 cvsca 17222 0_gc0g 17400 Σ_g cgsu 17401 Grpcgrp 18907 CMndccmn 19753 1_rcur 20160 Ringcrg 20212 LModclmod 20857 mPwSer cmps 21886 mPoly cmpl 21888

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-ofr 7628 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-sup 9352 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-fz 13460 df-fzo 13607 df-seq 13962 df-hash 14291 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17402 df-gsum 17403 df-prds 17408 df-pws 17410 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-mhm 18749 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-mulg 19042 df-subg 19097 df-ghm 19186 df-cntz 19290 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-subrng 20525 df-subrg 20549 df-lmod 20859 df-lss 20929 df-psr 21891 df-mpl 21893

This theorem is referenced by: mplbas2 22025 mplcoe4 22054 ply1coe 22291

W3C validator