Theorem mplmonmul 21943

Description: The product of two monomials adds the exponent vectors together. For example, the product of (𝑥↑2)(𝑦↑2) with (𝑦↑1)(𝑧↑3) is (𝑥↑2)(𝑦↑3)(𝑧↑3), where the exponent vectors ⟨2, 2, 0⟩ and ⟨0, 1, 3⟩ are added to give ⟨2, 3, 3⟩. (Contributed by Mario Carneiro, 9-Jan-2015.)

Hypotheses

Ref	Expression
mplmon.s	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplmon.b	⊢ 𝐵 = (Base‘𝑃)
mplmon.z	⊢ 0 = (0g‘𝑅)
mplmon.o	⊢ 1 = (1r‘𝑅)
mplmon.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
mplmon.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplmon.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mplmon.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
mplmonmul.t	⊢ · = (.r‘𝑃)
mplmonmul.x	⊢ (𝜑 → 𝑌 ∈ 𝐷)

Assertion

Ref	Expression
mplmonmul	⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 )))

Distinct variable groups:   𝑦,𝐷   𝑓,𝐼   𝜑,𝑦   𝑦,𝑓,𝑋   𝑦, 0   𝑦, 1   𝑦,𝑅   𝑓,𝑌,𝑦

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑦,𝑓)   𝐷(𝑓)   𝑃(𝑦,𝑓)   𝑅(𝑓)   · (𝑦,𝑓)   1 (𝑓)   𝐼(𝑦)   𝑊(𝑦,𝑓)   0 (𝑓)

Proof of Theorem mplmonmul

Dummy variables 𝑗 𝑘 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplmon.s	. . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
2		mplmon.b	. . 3 ⊢ 𝐵 = (Base‘𝑃)
3		eqid 2729	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		mplmonmul.t	. . 3 ⊢ · = (.r‘𝑃)
5		mplmon.d	. . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
6		mplmon.z	. . . 4 ⊢ 0 = (0g‘𝑅)
7		mplmon.o	. . . 4 ⊢ 1 = (1r‘𝑅)
8		mplmon.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
9		mplmon.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
10		mplmon.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
11	1, 2, 6, 7, 5, 8, 9, 10	mplmon 21942	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)
12		mplmonmul.x	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
13	1, 2, 6, 7, 5, 8, 9, 12	mplmon 21942	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵)
14	1, 2, 3, 4, 5, 11, 13	mplmul 21920	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))))))
15		eqeq1 2733	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝑦 = (𝑋 ∘f + 𝑌) ↔ 𝑘 = (𝑋 ∘f + 𝑌)))
16	15	ifbid 4512	. . . 4 ⊢ (𝑦 = 𝑘 → if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ))
17	16	cbvmptv 5211	. . 3 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ))
18		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
19	18	snssd 4773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → {𝑋} ⊆ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
20	19	resmptd 6011	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ↾ {𝑋}) = (𝑗 ∈ {𝑋} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))))
21	20	oveq2d 7403	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ↾ {𝑋})) = (𝑅 Σg (𝑗 ∈ {𝑋} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))))))
22	9	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑅 ∈ Ring)
23		ringmnd 20152	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
24	22, 23	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑅 ∈ Mnd)
25	10	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑋 ∈ 𝐷)
26		iftrue 4494	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 )
27		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
28	7	fvexi 6872	. . . . . . . . . . . . 13 ⊢ 1 ∈ V
29	26, 27, 28	fvmpt 6968	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐷 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋) = 1 )
30	25, 29	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋) = 1 )
31		ssrab2 4043	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ⊆ 𝐷
32		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘 ∈ 𝐷)
33		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}
34	5, 33	psrbagconcl 21836	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝐷 ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑋) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
35	32, 18, 34	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑋) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
36	31, 35	sselid 3944	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑋) ∈ 𝐷)
37		eqeq1 2733	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑘 ∘f − 𝑋) → (𝑦 = 𝑌 ↔ (𝑘 ∘f − 𝑋) = 𝑌))
38	37	ifbid 4512	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑘 ∘f − 𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ))
39		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))
40	6	fvexi 6872	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
41	28, 40	ifex 4539	. . . . . . . . . . . . 13 ⊢ if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ) ∈ V
42	38, 39, 41	fvmpt 6968	. . . . . . . . . . . 12 ⊢ ((𝑘 ∘f − 𝑋) ∈ 𝐷 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑋)) = if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ))
43	36, 42	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑋)) = if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ))
44	30, 43	oveq12d 7405	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑋))) = ( 1 (.r‘𝑅)if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 )))
45		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
46	45, 7	ringidcl 20174	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
47	45, 6	ring0cl 20176	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
48	46, 47	ifcld 4535	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
49	22, 48	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
50	45, 3, 7	ringlidm 20178	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅)) → ( 1 (.r‘𝑅)if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 )) = if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ))
51	22, 49, 50	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ( 1 (.r‘𝑅)if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 )) = if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ))
52	5	psrbagf 21827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝐷 → 𝑘:𝐼⟶ℕ0)
53	32, 52	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0)
54	53	ffvelcdmda 7056	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
55	10	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ 𝐷)
56	5	psrbagf 21827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ 𝐷 → 𝑋:𝐼⟶ℕ0)
57	55, 56	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0)
58	57	ffvelcdmda 7056	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℕ0)
59	58	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℕ0)
60	5	psrbagf 21827	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑌 ∈ 𝐷 → 𝑌:𝐼⟶ℕ0)
61	12, 60	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑌:𝐼⟶ℕ0)
62	61	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0)
63	62	ffvelcdmda 7056	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈ ℕ0)
64	63	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈ ℕ0)
65		nn0cn 12452	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ)
66		nn0cn 12452	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋‘𝑧) ∈ ℕ0 → (𝑋‘𝑧) ∈ ℂ)
67		nn0cn 12452	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌‘𝑧) ∈ ℕ0 → (𝑌‘𝑧) ∈ ℂ)
68		subadd 11424	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘‘𝑧) ∈ ℂ ∧ (𝑋‘𝑧) ∈ ℂ ∧ (𝑌‘𝑧) ∈ ℂ) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧)))
69	65, 66, 67, 68	syl3an 1160	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘‘𝑧) ∈ ℕ0 ∧ (𝑋‘𝑧) ∈ ℕ0 ∧ (𝑌‘𝑧) ∈ ℕ0) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧)))
70	54, 59, 64, 69	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧)))
71		eqcom 2736	. . . . . . . . . . . . . . 15 ⊢ (((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧) ↔ (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))
72	70, 71	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧))))
73	72	ralbidva 3154	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (∀𝑧 ∈ 𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧))))
74		mpteqb 6987	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ 𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) ∈ V → ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)) ↔ ∀𝑧 ∈ 𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧)))
75		ovexd 7422	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐼 → ((𝑘‘𝑧) − (𝑋‘𝑧)) ∈ V)
76	74, 75	mprg 3050	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)) ↔ ∀𝑧 ∈ 𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧))
77		mpteqb 6987	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ 𝐼 (𝑘‘𝑧) ∈ V → ((𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧))))
78		fvexd 6873	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐼 → (𝑘‘𝑧) ∈ V)
79	77, 78	mprg 3050	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))
80	73, 76, 79	3bitr4g 314	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)) ↔ (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧)))))
81	8	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝐼 ∈ 𝑊)
82	53	feqmptd 6929	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)))
83	57	feqmptd 6929	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧)))
84	83	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧)))
85	81, 54, 59, 82, 84	offval2 7673	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑋) = (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))))
86	62	feqmptd 6929	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))
87	86	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))
88	85, 87	eqeq12d 2745	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑘 ∘f − 𝑋) = 𝑌 ↔ (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧))))
89	8	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐼 ∈ 𝑊)
90	89, 58, 63, 83, 86	offval2 7673	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘f + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))))
91	90	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑋 ∘f + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))))
92	82, 91	eqeq12d 2745	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 = (𝑋 ∘f + 𝑌) ↔ (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧)))))
93	80, 88, 92	3bitr4d 311	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑘 ∘f − 𝑋) = 𝑌 ↔ 𝑘 = (𝑋 ∘f + 𝑌)))
94	93	ifbid 4512	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ))
95	44, 51, 94	3eqtrd 2768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑋))) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ))
96	94, 49	eqeltrrd 2829	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ) ∈ (Base‘𝑅))
97	95, 96	eqeltrd 2828	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑋))) ∈ (Base‘𝑅))
98		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋))
99		oveq2 7395	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑋 → (𝑘 ∘f − 𝑗) = (𝑘 ∘f − 𝑋))
100	99	fveq2d 6862	. . . . . . . . . 10 ⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑋)))
101	98, 100	oveq12d 7405	. . . . . . . . 9 ⊢ (𝑗 = 𝑋 → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))) = (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑋))))
102	45, 101	gsumsn 19884	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ 𝑋 ∈ 𝐷 ∧ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑋))) ∈ (Base‘𝑅)) → (𝑅 Σg (𝑗 ∈ {𝑋} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))))) = (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑋))))
103	24, 25, 97, 102	syl3anc 1373	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑅 Σg (𝑗 ∈ {𝑋} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))))) = (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑋))))
104	21, 103, 95	3eqtrd 2768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ↾ {𝑋})) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ))
105	6	gsum0 18611	. . . . . . 7 ⊢ (𝑅 Σg ∅) = 0
106		disjsn 4675	. . . . . . . . 9 ⊢ (({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
107	9	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑅 ∈ Ring)
108	1, 45, 2, 5, 11	mplelf 21907	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
109	108	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
110		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
111	31, 110	sselid 3944	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑗 ∈ 𝐷)
112	109, 111	ffvelcdmd 7057	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅))
113	1, 45, 2, 5, 13	mplelf 21907	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
114	113	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
115		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘 ∈ 𝐷)
116	5, 33	psrbagconcl 21836	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ 𝐷 ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
117	115, 110, 116	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
118	31, 117	sselid 3944	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ 𝐷)
119	114, 118	ffvelcdmd 7057	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅))
120	45, 3	ringcl 20159	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅) ∧ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅)) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))) ∈ (Base‘𝑅))
121	107, 112, 119, 120	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))) ∈ (Base‘𝑅))
122	121	fmpttd 7087	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))):{𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}⟶(Base‘𝑅))
123		ffn 6688	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))):{𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}⟶(Base‘𝑅) → (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) Fn {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
124		fnresdisj 6638	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) Fn {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} → (({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∩ {𝑋}) = ∅ ↔ ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ↾ {𝑋}) = ∅))
125	122, 123, 124	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∩ {𝑋}) = ∅ ↔ ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ↾ {𝑋}) = ∅))
126	125	biimpa 476	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∩ {𝑋}) = ∅) → ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ↾ {𝑋}) = ∅)
127	106, 126	sylan2br 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ↾ {𝑋}) = ∅)
128	127	oveq2d 7403	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ↾ {𝑋})) = (𝑅 Σg ∅))
129		breq1 5110	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 ∘r ≤ (𝑋 ∘f + 𝑌) ↔ 𝑋 ∘r ≤ (𝑋 ∘f + 𝑌)))
130	58	nn0red 12504	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℝ)
131		nn0addge1 12488	. . . . . . . . . . . . . 14 ⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℕ0) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))
132	130, 63, 131	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))
133	132	ralrimiva 3125	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))
134		ovexd 7422	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → ((𝑋‘𝑧) + (𝑌‘𝑧)) ∈ V)
135	89, 58, 134, 83, 90	ofrfval2 7674	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘r ≤ (𝑋 ∘f + 𝑌) ↔ ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))))
136	133, 135	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∘r ≤ (𝑋 ∘f + 𝑌))
137	129, 55, 136	elrabd 3661	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌)})
138		breq2 5111	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑋 ∘f + 𝑌) → (𝑥 ∘r ≤ 𝑘 ↔ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌)))
139	138	rabbidv 3413	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑋 ∘f + 𝑌) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌)})
140	139	eleq2d 2814	. . . . . . . . . 10 ⊢ (𝑘 = (𝑋 ∘f + 𝑌) → (𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↔ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌)}))
141	137, 140	syl5ibrcom 247	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑘 = (𝑋 ∘f + 𝑌) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}))
142	141	con3dimp 408	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ¬ 𝑘 = (𝑋 ∘f + 𝑌))
143	142	iffalsed 4499	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ) = 0 )
144	105, 128, 143	3eqtr4a 2790	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ↾ {𝑋})) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ))
145	104, 144	pm2.61dan 812	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ↾ {𝑋})) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ))
146	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ Ring)
147		ringcmn 20191	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
148	146, 147	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ CMnd)
149	5	psrbaglefi 21835	. . . . . . 7 ⊢ (𝑘 ∈ 𝐷 → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ Fin)
150	149	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ Fin)
151		ssdif 4107	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ⊆ 𝐷 → ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}))
152	31, 151	ax-mp 5	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})
153	152	sseli 3942	. . . . . . . . . 10 ⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋}))
154	108	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
155		eldifsni 4754	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋)
156	155	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋)
157	156	neneqd 2930	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
158	157	iffalsed 4499	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
159		ovex 7420	. . . . . . . . . . . . . 14 ⊢ (ℕ0 ↑m 𝐼) ∈ V
160	5, 159	rabex2 5296	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ V
161	160	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐷 ∈ V)
162	158, 161	suppss2 8179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
163	40	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 0 ∈ V)
164	154, 162, 161, 163	suppssr 8174	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
165	153, 164	sylan2 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
166	165	oveq1d 7402	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋})) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))) = ( 0 (.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))))
167		eldifi 4094	. . . . . . . . 9 ⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
168	45, 3, 6	ringlz 20202	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅)) → ( 0 (.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))) = 0 )
169	107, 119, 168	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ( 0 (.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))) = 0 )
170	167, 169	sylan2 593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋})) → ( 0 (.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))) = 0 )
171	166, 170	eqtrd 2764	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋})) → (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))) = 0 )
172	160	rabex 5294	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ V
173	172	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ V)
174	171, 173	suppss2 8179	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) supp 0 ) ⊆ {𝑋})
175	160	mptrabex 7199	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ∈ V
176		funmpt 6554	. . . . . . . . 9 ⊢ Fun (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))))
177	175, 176, 40	3pm3.2i 1340	. . . . . . . 8 ⊢ ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ∈ V ∧ Fun (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ∧ 0 ∈ V)
178	177	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ∈ V ∧ Fun (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ∧ 0 ∈ V))
179		snfi 9014	. . . . . . . 8 ⊢ {𝑋} ∈ Fin
180	179	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑋} ∈ Fin)
181		suppssfifsupp 9331	. . . . . . 7 ⊢ ((((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ∈ V ∧ Fun (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ∧ 0 ∈ V) ∧ ({𝑋} ∈ Fin ∧ ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) supp 0 ) ⊆ {𝑋})) → (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) finSupp 0 )
182	178, 180, 174, 181	syl12anc 836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) finSupp 0 )
183	45, 6, 148, 150, 122, 174, 182	gsumres 19843	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑅 Σg ((𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))) ↾ {𝑋})) = (𝑅 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))))))
184	145, 183	eqtr3d 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ) = (𝑅 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))))))
185	184	mpteq2dva 5200	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 )) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))))))
186	17, 185	eqtrid 2776	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 )) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↦ (((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗)(.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)))))))
187	14, 186	eqtr4d 2767	1 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3405  Vcvv 3447   ∖ cdif 3911   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  ifcif 4488  {csn 4589   class class class wbr 5107   ↦ cmpt 5188  ^◡ccnv 5637   ↾ cres 5640   “ cima 5641  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651   ∘_r cofr 7652   supp csupp 8139   ↑_m cmap 8799  Fincfn 8918   finSupp cfsupp 9312  ℂcc 11066  ℝcr 11067   + caddc 11071   ≤ cle 11209   − cmin 11405  ℕcn 12186  ℕ₀cn0 12442  Basecbs 17179  ._rcmulr 17221  0_gc0g 17402   Σ_g cgsu 17403  Mndcmnd 18661  CMndccmn 19710  1_rcur 20090  Ringcrg 20142   mPoly cmpl 21815

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-ofr 7654  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-tset 17239  df-0g 17404  df-gsum 17405  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-minusg 18869  df-mulg 19000  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-psr 21818  df-mpl 21820

This theorem is referenced by:  mplcoe3  21945  mplcoe5  21947  mplmon2mul  21976