Theorem mplmonmul 20704

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 2798	. . 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 20703	. . 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 20703	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) ∈ 𝐵)
14	1, 2, 3, 4, 5, 11, 13	mplmul 20682	. 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 2802	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝑦 = (𝑋 ∘f + 𝑌) ↔ 𝑘 = (𝑋 ∘f + 𝑌)))
16	15	ifbid 4447	. . . 4 ⊢ (𝑦 = 𝑘 → if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 ) = if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ))
17	16	cbvmptv 5133	. . 3 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ))
18		simpr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
19	18	snssd 4702	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → {𝑋} ⊆ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
20	19	resmptd 5875	. . . . . . . 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 7151	. . . . . . 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 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑅 ∈ Ring)
23		ringmnd 19300	. . . . . . . . 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 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑋 ∈ 𝐷)
26		iftrue 4431	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑋, 1 , 0 ) = 1 )
27		eqid 2798	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))
28	7	fvexi 6659	. . . . . . . . . . . . 13 ⊢ 1 ∈ V
29	26, 27, 28	fvmpt 6745	. . . . . . . . . . . 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 4007	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ⊆ 𝐷
32	8	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝐼 ∈ 𝑊)
33		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘 ∈ 𝐷)
34		eqid 2798	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}
35	5, 34	psrbagconcl 20611	. . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑋) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
37	31, 36	sseldi 3913	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑋) ∈ 𝐷)
38		eqeq1 2802	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑘 ∘f − 𝑋) → (𝑦 = 𝑌 ↔ (𝑘 ∘f − 𝑋) = 𝑌))
39	38	ifbid 4447	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑘 ∘f − 𝑋) → if(𝑦 = 𝑌, 1 , 0 ) = if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ))
40		eqid 2798	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))
41	6	fvexi 6659	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
42	28, 41	ifex 4473	. . . . . . . . . . . . 13 ⊢ if((𝑘 ∘f − 𝑋) = 𝑌, 1 , 0 ) ∈ V
43	39, 40, 42	fvmpt 6745	. . . . . . . . . . . 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 7153	. . . . . . . . . 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 2798	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
47	46, 7	ringidcl 19314	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
48	46, 6	ring0cl 19315	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
49	47, 48	ifcld 4470	. . . . . . . . . . . 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 19317	. . . . . . . . . . 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 587	. . . . . . . . . 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 20603	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷) → 𝑘:𝐼⟶ℕ0)
54	32, 33, 53	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0)
55	54	ffvelrnda 6828	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
56	8	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐼 ∈ 𝑊)
57	10	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ 𝐷)
58	5	psrbagf 20603	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0)
59	56, 57, 58	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0)
60	59	ffvelrnda 6828	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℕ0)
61	60	adantlr 714	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℕ0)
62	5	psrbagf 20603	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑌 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0)
63	8, 12, 62	syl2anc 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑌:𝐼⟶ℕ0)
64	63	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌:𝐼⟶ℕ0)
65	64	ffvelrnda 6828	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈ ℕ0)
66	65	adantlr 714	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑌‘𝑧) ∈ ℕ0)
67		nn0cn 11895	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ)
68		nn0cn 11895	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋‘𝑧) ∈ ℕ0 → (𝑋‘𝑧) ∈ ℂ)
69		nn0cn 11895	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌‘𝑧) ∈ ℕ0 → (𝑌‘𝑧) ∈ ℂ)
70		subadd 10878	. . . . . . . . . . . . . . . . 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 1157	. . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . 15 ⊢ (((𝑋‘𝑧) + (𝑌‘𝑧)) = (𝑘‘𝑧) ↔ (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))
74	72, 73	syl6bb 290	. . . . . . . . . . . . . 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 3161	. . . . . . . . . . . . 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 6764	. . . . . . . . . . . . . 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 7170	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐼 → ((𝑘‘𝑧) − (𝑋‘𝑧)) ∈ V)
78	76, 77	mprg 3120	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)) ↔ ∀𝑧 ∈ 𝐼 ((𝑘‘𝑧) − (𝑋‘𝑧)) = (𝑌‘𝑧))
79		mpteqb 6764	. . . . . . . . . . . . . 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		fvexd 6660	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐼 → (𝑘‘𝑧) ∈ V)
81	79, 80	mprg 3120	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))) ↔ ∀𝑧 ∈ 𝐼 (𝑘‘𝑧) = ((𝑋‘𝑧) + (𝑌‘𝑧)))
82	75, 78, 81	3bitr4g 317	. . . . . . . . . . . 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)))))
83	54	feqmptd 6708	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)))
84	59	feqmptd 6708	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧)))
85	84	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑋 = (𝑧 ∈ 𝐼 ↦ (𝑋‘𝑧)))
86	32, 55, 61, 83, 85	offval2 7406	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑋) = (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))))
87	64	feqmptd 6708	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))
88	87	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑌 = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧)))
89	86, 88	eqeq12d 2814	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑘 ∘f − 𝑋) = 𝑌 ↔ (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑋‘𝑧))) = (𝑧 ∈ 𝐼 ↦ (𝑌‘𝑧))))
90	56, 60, 65, 84, 87	offval2 7406	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘f + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))))
91	90	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑋 ∘f + 𝑌) = (𝑧 ∈ 𝐼 ↦ ((𝑋‘𝑧) + (𝑌‘𝑧))))
92	83, 91	eqeq12d 2814	. . . . . . . . . . . 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	82, 89, 92	3bitr4d 314	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑘 ∘f − 𝑋) = 𝑌 ↔ 𝑘 = (𝑋 ∘f + 𝑌)))
94	93	ifbid 4447	. . . . . . . . . 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	45, 52, 94	3eqtrd 2837	. . . . . . . . 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, 50	eqeltrrd 2891	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ) ∈ (Base‘𝑅))
97	95, 96	eqeltrd 2890	. . . . . . . 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 6645	. . . . . . . . . 10 ⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑋))
99		oveq2 7143	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑋 → (𝑘 ∘f − 𝑗) = (𝑘 ∘f − 𝑋))
100	99	fveq2d 6649	. . . . . . . . . 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 7153	. . . . . . . . 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	46, 101	gsumsn 19067	. . . . . . . 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 1368	. . . . . . 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 2837	. . . . . 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 17886	. . . . . . 7 ⊢ (𝑅 Σg ∅) = 0
106		disjsn 4607	. . . . . . . . 9 ⊢ (({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
107	9	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑅 ∈ Ring)
108	1, 46, 2, 5, 11	mplelf 20671	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
109	108	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
110		simpr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
111	31, 110	sseldi 3913	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑗 ∈ 𝐷)
112	109, 111	ffvelrnd 6829	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) ∈ (Base‘𝑅))
113	1, 46, 2, 5, 13	mplelf 20671	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
114	113	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 )):𝐷⟶(Base‘𝑅))
115	8	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝐼 ∈ 𝑊)
116		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → 𝑘 ∈ 𝐷)
117	5, 34	psrbagconcl 20611	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷 ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
118	115, 116, 110, 117	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
119	31, 118	sseldi 3913	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ 𝐷)
120	114, 119	ffvelrnd 6829	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅))
121	46, 3	ringcl 19307	. . . . . . . . . . . . 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))
122	107, 112, 120, 121	syl3anc 1368	. . . . . . . . . . . 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))
123	122	fmpttd 6856	. . . . . . . . . . 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))
124		ffn 6487	. . . . . . . . . . 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})
125		fnresdisj 6439	. . . . . . . . . . 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}) = ∅))
126	123, 124, 125	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}) = ∅))
127	126	biimpa 480	. . . . . . . . 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}) = ∅)
128	106, 127	sylan2br 597	. . . . . . . 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}) = ∅)
129	128	oveq2d 7151	. . . . . . 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 ∅))
130		breq1 5033	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 ∘r ≤ (𝑋 ∘f + 𝑌) ↔ 𝑋 ∘r ≤ (𝑋 ∘f + 𝑌)))
131	60	nn0red 11944	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ∈ ℝ)
132		nn0addge1 11931	. . . . . . . . . . . . . 14 ⊢ (((𝑋‘𝑧) ∈ ℝ ∧ (𝑌‘𝑧) ∈ ℕ0) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))
133	131, 65, 132	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))
134	133	ralrimiva 3149	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧)))
135		ovexd 7170	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑧 ∈ 𝐼) → ((𝑋‘𝑧) + (𝑌‘𝑧)) ∈ V)
136	56, 60, 135, 84, 90	ofrfval2 7407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑋 ∘r ≤ (𝑋 ∘f + 𝑌) ↔ ∀𝑧 ∈ 𝐼 (𝑋‘𝑧) ≤ ((𝑋‘𝑧) + (𝑌‘𝑧))))
137	134, 136	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∘r ≤ (𝑋 ∘f + 𝑌))
138	130, 57, 137	elrabd 3630	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌)})
139		breq2 5034	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑋 ∘f + 𝑌) → (𝑥 ∘r ≤ 𝑘 ↔ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌)))
140	139	rabbidv 3427	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑋 ∘f + 𝑌) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} = {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌)})
141	140	eleq2d 2875	. . . . . . . . . 10 ⊢ (𝑘 = (𝑋 ∘f + 𝑌) → (𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ↔ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝑋 ∘f + 𝑌)}))
142	138, 141	syl5ibrcom 250	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑘 = (𝑋 ∘f + 𝑌) → 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}))
143	142	con3dimp 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ¬ 𝑘 = (𝑋 ∘f + 𝑌))
144	143	iffalsed 4436	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ ¬ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → if(𝑘 = (𝑋 ∘f + 𝑌), 1 , 0 ) = 0 )
145	105, 129, 144	3eqtr4a 2859	. . . . . 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 ))
146	104, 145	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 ))
147	9	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ Ring)
148		ringcmn 19327	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
149	147, 148	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ CMnd)
150	5	psrbaglefi 20610	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ Fin)
151	8, 150	sylan 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ Fin)
152		ssdif 4067	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ⊆ 𝐷 → ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋}))
153	31, 152	ax-mp 5	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) ⊆ (𝐷 ∖ {𝑋})
154	153	sseli 3911	. . . . . . . . . 10 ⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ (𝐷 ∖ {𝑋}))
155	108	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )):𝐷⟶(Base‘𝑅))
156		eldifsni 4683	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐷 ∖ {𝑋}) → 𝑦 ≠ 𝑋)
157	156	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → 𝑦 ≠ 𝑋)
158	157	neneqd 2992	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → ¬ 𝑦 = 𝑋)
159	158	iffalsed 4436	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑦 ∈ (𝐷 ∖ {𝑋})) → if(𝑦 = 𝑋, 1 , 0 ) = 0 )
160		ovex 7168	. . . . . . . . . . . . . 14 ⊢ (ℕ0 ↑m 𝐼) ∈ V
161	5, 160	rabex2 5201	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ V
162	161	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐷 ∈ V)
163	159, 162	suppss2 7847	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) supp 0 ) ⊆ {𝑋})
164	41	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 0 ∈ V)
165	155, 163, 162, 164	suppssr 7844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ (𝐷 ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
166	154, 165	sylan2 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋})) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 ))‘𝑗) = 0 )
167	166	oveq1d 7150	. . . . . . . 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))))
168		eldifi 4054	. . . . . . . . 9 ⊢ (𝑗 ∈ ({𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∖ {𝑋}) → 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘})
169	46, 3, 6	ringlz 19333	. . . . . . . . . 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 )
170	107, 120, 169	syl2anc 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘}) → ( 0 (.r‘𝑅)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))‘(𝑘 ∘f − 𝑗))) = 0 )
171	168, 170	sylan2 595	. . . . . . . 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 )
172	167, 171	eqtrd 2833	. . . . . . 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 )
173	161	rabex 5199	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ V
174	173	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑘} ∈ V)
175	172, 174	suppss2 7847	. . . . . 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})
176	161	mptrabex 6965	. . . . . . . . 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
177		funmpt 6362	. . . . . . . . 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))))
178	176, 177, 41	3pm3.2i 1336	. . . . . . . 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)
179	178	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))
180		snfi 8577	. . . . . . . 8 ⊢ {𝑋} ∈ Fin
181	180	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑋} ∈ Fin)
182		suppssfifsupp 8832	. . . . . . 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 )
183	179, 181, 175, 182	syl12anc 835	. . . . . 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 )
184	46, 6, 149, 151, 123, 175, 183	gsumres 19026	. . . . 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))))))
185	146, 184	eqtr3d 2835	. . . 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))))))
186	185	mpteq2dva 5125	. . 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)))))))
187	17, 186	syl5eq 2845	. 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)))))))
188	14, 187	eqtr4d 2836	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  {crab 3110  Vcvv 3441   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  ifcif 4425  {csn 4525   class class class wbr 5030   ↦ cmpt 5110  ^◡ccnv 5518   ↾ cres 5521   “ cima 5522  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘_f cof 7387   ∘_r cofr 7388   supp csupp 7813   ↑_m cmap 8389  Fincfn 8492   finSupp cfsupp 8817  ℂcc 10524  ℝcr 10525   + caddc 10529   ≤ cle 10665   − cmin 10859  ℕcn 11625  ℕ₀cn0 11885  Basecbs 16475  ._rcmulr 16558  0_gc0g 16705   Σ_g cgsu 16706  Mndcmnd 17903  CMndccmn 18898  1_rcur 19244  Ringcrg 19290   mPoly cmpl 20591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-ofr 7390  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-tset 16576  df-0g 16707  df-gsum 16708  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-mulg 18217  df-cntz 18439  df-cmn 18900  df-abl 18901  df-mgp 19233  df-ur 19245  df-ring 19292  df-psr 20594  df-mpl 20596

This theorem is referenced by:  mplcoe3  20706  mplcoe5  20708  mplmon2mul  20740