Theorem mplmonmul 19861

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 2778	. . 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 19860	. . 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 19860	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵)
14	1, 2, 3, 4, 5, 11, 13	mplmul 19840	. 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 2782	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝑦 = (𝑋 ∘𝑓 + 𝑌) ↔ 𝑘 = (𝑋 ∘𝑓 + 𝑌)))
16	15	ifbid 4329	. . . 4 ⊢ (𝑦 = 𝑘 → if(𝑦 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ))
17	16	cbvmptv 4985	. . 3 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ))
18		simpr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘})
19	18	snssd 4571	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → {𝑋} ⊆ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘})
20	19	resmptd 5702	. . . . . . . 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 6938	. . . . . . 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 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑅 ∈ Ring)
23		ringmnd 18943	. . . . . . . . 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 716	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑋 ∈ 𝐷)
26		iftrue 4313	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 )
27		eqid 2778	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
28	7	fvexi 6460	. . . . . . . . . . . . 13 ⊢ 1 ∈ V
29	26, 27, 28	fvmpt 6542	. . . . . . . . . . . 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 3908	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ⊆ 𝐷
32	8	ad2antrr 716	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝐼 ∈ 𝑊)
33		simplr 759	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘 ∈ 𝐷)
34		eqid 2778	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}
35	5, 34	psrbagconcl 19770	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑋) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘})
36	32, 33, 18, 35	syl3anc 1439	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑋) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘})
37	31, 36	sseldi 3819	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑋) ∈ 𝐷)
38		eqeq1 2782	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑘 ∘𝑓 − 𝑋) → (𝑦 = 𝑌 ↔ (𝑘 ∘𝑓 − 𝑋) = 𝑌))
39	38	ifbid 4329	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑘 ∘𝑓 − 𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘 ∘𝑓 − 𝑋) = 𝑌, 1 , 0 ))
40		eqid 2778	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))
41	6	fvexi 6460	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
42	28, 41	ifex 4355	. . . . . . . . . . . . 13 ⊢ if((𝑘 ∘𝑓 − 𝑋) = 𝑌, 1 , 0 ) ∈ V
43	39, 40, 42	fvmpt 6542	. . . . . . . . . . . 12 ⊢ ((𝑘 ∘𝑓 − 𝑋) ∈ 𝐷 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓 − 𝑋)) = if((𝑘 ∘𝑓 − 𝑋) = 𝑌, 1 , 0 ))
44	37, 43	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 ))
45	30, 44	oveq12d 6940	. . . . . . . . . 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 )))
46		eqid 2778	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
47	46, 7	ringidcl 18955	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
48	46, 6	ring0cl 18956	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
49	47, 48	ifcld 4352	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → if((𝑘 ∘𝑓 − 𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
50	22, 49	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → if((𝑘 ∘𝑓 − 𝑋) = 𝑌, 1 , 0 ) ∈ (Base‘𝑅))
51	46, 3, 7	ringlidm 18958	. . . . . . . . . . 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 ))
52	22, 50, 51	syl2anc 579	. . . . . . . . . 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 ))
53	5	psrbagf 19762	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷) → 𝑘:𝐼⟶ℕ0)
54	32, 33, 53	syl2anc 579	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0)
55	54	ffvelrnda 6623	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
56	8	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐼 ∈ 𝑊)
57	10	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ 𝐷)
58	5	psrbagf 19762	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0)
59	56, 57, 58	syl2anc 579	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0)
60	59	ffvelrnda 6623	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℕ0)
61	60	adantlr 705	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℕ0)
62	5	psrbagf 19762	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑌 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0)
63	8, 12, 62	syl2anc 579	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑌:𝐼⟶ℕ0)
64	63	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0)
65	64	ffvelrnda 6623	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈ ℕ0)
66	65	adantlr 705	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈ ℕ0)
67		nn0cn 11653	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ)
68		nn0cn 11653	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋‘𝑧) ∈ ℕ0 → (𝑋‘𝑧) ∈ ℂ)
69		nn0cn 11653	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌‘𝑧) ∈ ℕ0 → (𝑌‘𝑧) ∈ ℂ)
70		subadd 10625	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘‘𝑧) ∈ ℂ ∧ (𝑋‘𝑧) ∈ ℂ ∧ (𝑌‘𝑧) ∈ ℂ) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧)))
71	67, 68, 69, 70	syl3an 1160	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘‘𝑧) ∈ ℕ0 ∧ (𝑋‘𝑧) ∈ ℕ0 ∧ (𝑌‘𝑧) ∈ ℕ0) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧)))
72	55, 61, 66, 71	syl3anc 1439	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ ((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧)))
73		eqcom 2785	. . . . . . . . . . . . . . 15 ⊢ (((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧) ↔ (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))
74	72, 73	syl6bb 279	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧) ↔ (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧))))
75	74	ralbidva 3167	. . . . . . . . . . . . 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))))
76		mpteqb 6560	. . . . . . . . . . . . . 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)))
77		ovexd 6956	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐼 → ((𝑘‘𝑧) − (𝑋‘𝑧)) ∈ V)
78	76, 77	mprg 3108	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)) ↔ ∀𝑧 ∈ 𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧))
79		mpteqb 6560	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ 𝐼 (𝑘‘𝑧) ∈ V → ((𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧))))
80		fvex 6459	. . . . . . . . . . . . . . 15 ⊢ (𝑘‘𝑧) ∈ V
81	80	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐼 → (𝑘‘𝑧) ∈ V)
82	79, 81	mprg 3108	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))
83	75, 78, 82	3bitr4g 306	. . . . . . . . . . . 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)))))
84	54	feqmptd 6509	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)))
85	59	feqmptd 6509	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧)))
86	85	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧)))
87	32, 55, 61, 84, 86	offval2 7191	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑋) = (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))))
88	64	feqmptd 6509	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))
89	88	adantr 474	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))
90	87, 89	eqeq12d 2793	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑘 ∘𝑓 − 𝑋) = 𝑌 ↔ (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧))))
91	56, 60, 65, 85, 88	offval2 7191	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘𝑓 + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))))
92	91	adantr 474	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑋 ∘𝑓 + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))))
93	84, 92	eqeq12d 2793	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 = (𝑋 ∘𝑓 + 𝑌) ↔ (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧)))))
94	83, 90, 93	3bitr4d 303	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑘 ∘𝑓 − 𝑋) = 𝑌 ↔ 𝑘 = (𝑋 ∘𝑓 + 𝑌)))
95	94	ifbid 4329	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → if((𝑘 ∘𝑓 − 𝑋) = 𝑌, 1 , 0 ) = if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ))
96	45, 52, 95	3eqtrd 2818	. . . . . . . . 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 ))
97	95, 50	eqeltrrd 2860	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ) ∈ (Base‘𝑅))
98	96, 97	eqeltrd 2859	. . . . . . . 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))
99		fveq2 6446	. . . . . . . . . 10 ⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋))
100		oveq2 6930	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑋 → (𝑘 ∘𝑓 − 𝑗) = (𝑘 ∘𝑓 − 𝑋))
101	100	fveq2d 6450	. . . . . . . . . 10 ⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓 − 𝑗)) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓 − 𝑋)))
102	99, 101	oveq12d 6940	. . . . . . . . 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))))
103	46, 102	gsumsn 18740	. . . . . . . 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))))
104	24, 25, 98, 103	syl3anc 1439	. . . . . . 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))))
105	21, 104, 96	3eqtrd 2818	. . . . . 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 ))
106	6	gsum0 17664	. . . . . . 7 ⊢ (𝑅 Σg ∅) = 0
107		disjsn 4478	. . . . . . . . 9 ⊢ (({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘})
108	9	ad2antrr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑅 ∈ Ring)
109	1, 46, 2, 5, 11	mplelf 19830	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
110	109	ad2antrr 716	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
111		simpr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘})
112	31, 111	sseldi 3819	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑗 ∈ 𝐷)
113	110, 112	ffvelrnd 6624	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅))
114	1, 46, 2, 5, 13	mplelf 19830	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
115	114	ad2antrr 716	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
116	8	ad2antrr 716	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝐼 ∈ 𝑊)
117		simplr 759	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → 𝑘 ∈ 𝐷)
118	5, 34	psrbagconcl 19770	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘})
119	116, 117, 111, 118	syl3anc 1439	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘})
120	31, 119	sseldi 3819	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → (𝑘 ∘𝑓 − 𝑗) ∈ 𝐷)
121	115, 120	ffvelrnd 6624	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓 − 𝑗)) ∈ (Base‘𝑅))
122	46, 3	ringcl 18948	. . . . . . . . . . . . 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))
123	108, 113, 121, 122	syl3anc 1439	. . . . . . . . . . . 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))
124	123	fmpttd 6649	. . . . . . . . . . 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))
125		ffn 6291	. . . . . . . . . . 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})
126		fnresdisj 6247	. . . . . . . . . . 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}) = ∅))
127	124, 125, 126	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}) = ∅))
128	127	biimpa 470	. . . . . . . . 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}) = ∅)
129	107, 128	sylan2br 588	. . . . . . . 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}) = ∅)
130	129	oveq2d 6938	. . . . . . 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 ∅))
131	60	nn0red 11703	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℝ)
132		nn0addge1 11690	. . . . . . . . . . . . . 14 ⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℕ0) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))
133	131, 65, 132	syl2anc 579	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))
134	133	ralrimiva 3148	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))
135		ovexd 6956	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → ((𝑋‘𝑧) + (𝑌‘𝑧)) ∈ V)
136	56, 60, 135, 85, 91	ofrfval2 7192	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 + 𝑌) ↔ ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))))
137	134, 136	mpbird 249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 + 𝑌))
138		breq1 4889	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 + 𝑌) ↔ 𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 + 𝑌)))
139	138	elrab 3572	. . . . . . . . . . 11 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 + 𝑌)} ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ∘𝑟 ≤ (𝑋 ∘𝑓 + 𝑌)))
140	57, 137, 139	sylanbrc 578	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 + 𝑌)})
141		breq2 4890	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑋 ∘𝑓 + 𝑌) → (𝑥 ∘𝑟 ≤ 𝑘 ↔ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 + 𝑌)))
142	141	rabbidv 3386	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑋 ∘𝑓 + 𝑌) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 + 𝑌)})
143	142	eleq2d 2845	. . . . . . . . . 10 ⊢ (𝑘 = (𝑋 ∘𝑓 + 𝑌) → (𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ↔ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ (𝑋 ∘𝑓 + 𝑌)}))
144	140, 143	syl5ibrcom 239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑘 = (𝑋 ∘𝑓 + 𝑌) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}))
145	144	con3dimp 399	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ¬ 𝑘 = (𝑋 ∘𝑓 + 𝑌))
146	145	iffalsed 4318	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → if(𝑘 = (𝑋 ∘𝑓 + 𝑌), 1 , 0 ) = 0 )
147	106, 130, 146	3eqtr4a 2840	. . . . . 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 ))
148	105, 147	pm2.61dan 803	. . . . 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 ))
149	9	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ Ring)
150		ringcmn 18968	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
151	149, 150	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ CMnd)
152	5	psrbaglefi 19769	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ Fin)
153	8, 152	sylan 575	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ Fin)
154		ssdif 3968	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ⊆ 𝐷 → ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}))
155	31, 154	ax-mp 5	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})
156	155	sseli 3817	. . . . . . . . . 10 ⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋}))
157	109	adantr 474	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
158		eldifsni 4553	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋)
159	158	adantl 475	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋)
160	159	neneqd 2974	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
161	160	iffalsed 4318	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
162		ovex 6954	. . . . . . . . . . . . . 14 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V
163	5, 162	rabex2 5051	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ V
164	163	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐷 ∈ V)
165	161, 164	suppss2 7611	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
166	41	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 0 ∈ V)
167	157, 165, 164, 166	suppssr 7608	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
168	156, 167	sylan2 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
169	168	oveq1d 6937	. . . . . . . 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))))
170		eldifi 3955	. . . . . . . . 9 ⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘})
171	46, 3, 6	ringlz 18974	. . . . . . . . . 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 )
172	108, 121, 171	syl2anc 579	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘}) → ( 0 (.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘𝑓 − 𝑗))) = 0 )
173	170, 172	sylan2 586	. . . . . . . 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 )
174	169, 173	eqtrd 2814	. . . . . . 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 )
175	163	rabex 5049	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ V
176	175	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘𝑟 ≤ 𝑘} ∈ V)
177	174, 176	suppss2 7611	. . . . . 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})
178	163	mptrabex 6760	. . . . . . . . 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
179		funmpt 6173	. . . . . . . . 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))))
180	178, 179, 41	3pm3.2i 1395	. . . . . . . 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)
181	180	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))
182		snfi 8326	. . . . . . . 8 ⊢ {𝑋} ∈ Fin
183	182	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑋} ∈ Fin)
184		suppssfifsupp 8578	. . . . . . 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 )
185	181, 183, 177, 184	syl12anc 827	. . . . . 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 )
186	46, 6, 151, 153, 124, 177, 185	gsumres 18700	. . . . 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))))))
187	148, 186	eqtr3d 2816	. . . 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))))))
188	187	mpteq2dva 4979	. . 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)))))))
189	17, 188	syl5eq 2826	. 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)))))))
190	14, 189	eqtr4d 2817	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 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  {crab 3094  Vcvv 3398   ∖ cdif 3789   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  ifcif 4307  {csn 4398   class class class wbr 4886   ↦ cmpt 4965  ^◡ccnv 5354   ↾ cres 5357   “ cima 5358  Fun wfun 6129   Fn wfn 6130  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   ∘_𝑓 cof 7172   ∘_𝑟 cofr 7173   supp csupp 7576   ↑_𝑚 cmap 8140  Fincfn 8241   finSupp cfsupp 8563  ℂcc 10270  ℝcr 10271   + caddc 10275   ≤ cle 10412   − cmin 10606  ℕcn 11374  ℕ₀cn0 11642  Basecbs 16255  ._rcmulr 16339  0_gc0g 16486   Σ_g cgsu 16487  Mndcmnd 17680  CMndccmn 18579  1_rcur 18888  Ringcrg 18934   mPoly cmpl 19750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-ofr 7175  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-seq 13120  df-hash 13436  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-mulr 16352  df-sca 16354  df-vsca 16355  df-tset 16357  df-0g 16488  df-gsum 16489  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-minusg 17813  df-mulg 17928  df-cntz 18133  df-cmn 18581  df-abl 18582  df-mgp 18877  df-ur 18889  df-ring 18936  df-psr 19753  df-mpl 19755

This theorem is referenced by:  mplcoe3  19863  mplcoe5  19865  mplmon2mul  19897