Theorem evlslem2 22295

Description: A linear function on the polynomial ring which is multiplicative on scaled monomials is generally multiplicative. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 11-Apr-2024.)

Hypotheses

Ref	Expression
evlslem2.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
evlslem2.b	⊢ 𝐵 = (Base‘𝑃)
evlslem2.m	⊢ · = (.r‘𝑆)
evlslem2.z	⊢ 0 = (0g‘𝑅)
evlslem2.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
evlslem2.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
evlslem2.r	⊢ (𝜑 → 𝑅 ∈ CRing)
evlslem2.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlslem2.e1	⊢ (𝜑 → 𝐸 ∈ (𝑃 GrpHom 𝑆))
evlslem2.e2	⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷))) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑗 ∘f + 𝑖), ((𝑥‘𝑗)(.r‘𝑅)(𝑦‘𝑖)), 0 ))) = ((𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) · (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))

Assertion

Ref	Expression
evlslem2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘(𝑥(.r‘𝑃)𝑦)) = ((𝐸‘𝑥) · (𝐸‘𝑦)))

Distinct variable groups:   𝜑,𝑖,𝑗,𝑘,𝑦   𝐵,𝑖,𝑗,𝑘,𝑥,𝑦   𝐷,𝑖,𝑗,𝑘,𝑥,𝑦   𝑖,𝐸,𝑗   ℎ,𝐼,𝑖,𝑗,𝑘   · ,𝑖,𝑗   𝑃,𝑖,𝑗,𝑘,𝑥,𝑦   𝑅,ℎ,𝑖,𝑗,𝑘   𝑆,𝑖,𝑗   𝑖,𝑊,𝑗,𝑘   0 ,ℎ,𝑖,𝑗,𝑘,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, ℎ)   𝐵(ℎ)   𝐷(ℎ)   𝑃(ℎ)   𝑅(𝑥, 𝑦)   𝑆(𝑥, 𝑦, ℎ, 𝑘)   · (𝑥, 𝑦, ℎ, 𝑘)   𝐸(𝑥, 𝑦, ℎ, 𝑘)   𝐼(𝑥, 𝑦)   𝑊(𝑥, 𝑦, ℎ)

Proof of Theorem evlslem2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlslem2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
2		eqid 2760	. . . . 5 ⊢ (.r‘𝑃) = (.r‘𝑃)
3		eqid 2760	. . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃)
4		evlslem2.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
5		ovex 7446	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
6	4, 5	rabex2 5305	. . . . . 6 ⊢ 𝐷 ∈ V
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ V)
8		evlslem2.p	. . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
9		evlslem2.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
10		evlslem2.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ CRing)
11		crngring 20384	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	10, 11	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
13	8, 9, 12	mplringd 22237	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Ring)
14	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ Ring)
15		evlslem2.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
16		eqid 2760	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
17	9	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝐼 ∈ 𝑊)
18	12	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝑅 ∈ Ring)
19		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
20	8, 16, 1, 4, 19	mplelf 22212	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥:𝐷⟶(Base‘𝑅))
21	20	ffvelcdmda 7077	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝑥‘𝑗) ∈ (Base‘𝑅))
22		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝑗 ∈ 𝐷)
23	8, 4, 15, 16, 17, 18, 1, 21, 22	mplmon2cl 22284	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ 𝐵)
24	9	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝐼 ∈ 𝑊)
25	12	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝑅 ∈ Ring)
26		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
27	8, 16, 1, 4, 26	mplelf 22212	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦:𝐷⟶(Base‘𝑅))
28	27	ffvelcdmda 7077	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝑦‘𝑖) ∈ (Base‘𝑅))
29		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝑖 ∈ 𝐷)
30	8, 4, 15, 16, 24, 25, 1, 28, 29	mplmon2cl 22284	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ 𝐵)
31	6	mptex 7222	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∈ V
32		funmpt 6571	. . . . . . . . . . . 12 ⊢ Fun (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
33		fvex 6891	. . . . . . . . . . . 12 ⊢ (0g‘𝑃) ∈ V
34	31, 32, 33	3pm3.2i 1358	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∈ V ∧ Fun (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∧ (0g‘𝑃) ∈ V)
35	34	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∈ V ∧ Fun (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∧ (0g‘𝑃) ∈ V))
36		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
37	8, 1, 15, 36	mplelsfi 22209	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 finSupp 0 )
38	37	fsuppimpd 9339	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ∈ Fin)
39	8, 16, 1, 4, 36	mplelf 22212	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐷⟶(Base‘𝑅))
40		ssidd 3954	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 ))
41	6	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V)
42	15	fvexi 6892	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
43	42	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 0 ∈ V)
44	39, 40, 41, 43	suppssr 8193	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑗) = 0 )
45	44	ifeq1d 4502	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = if(𝑘 = 𝑗, 0 , 0 ))
46		ifid 4523	. . . . . . . . . . . . . 14 ⊢ if(𝑘 = 𝑗, 0 , 0 ) = 0
47	45, 46	eqtrdi 2811	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = 0 )
48	47	mpteq2dv 5199	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (𝑘 ∈ 𝐷 ↦ 0 ))
49		ringgrp 20377	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
50	12, 49	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ Grp)
51	8, 4, 15, 3, 9, 50	mpl0 22220	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝑃) = (𝐷 × { 0 }))
52		fconstmpt 5717	. . . . . . . . . . . . . 14 ⊢ (𝐷 × { 0 }) = (𝑘 ∈ 𝐷 ↦ 0 )
53	51, 52	eqtrdi 2811	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝑃) = (𝑘 ∈ 𝐷 ↦ 0 ))
54	53	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (0g‘𝑃) = (𝑘 ∈ 𝐷 ↦ 0 ))
55	48, 54	eqtr4d 2798	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (0g‘𝑃))
56	55, 41	suppss2 8198	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) supp (0g‘𝑃)) ⊆ (𝑦 supp 0 ))
57		suppssfifsupp 9350	. . . . . . . . . 10 ⊢ ((((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∈ V ∧ Fun (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∧ (0g‘𝑃) ∈ V) ∧ ((𝑦 supp 0 ) ∈ Fin ∧ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) supp (0g‘𝑃)) ⊆ (𝑦 supp 0 ))) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃))
58	35, 38, 56, 57	syl12anc 850	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃))
59	58	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃))
60		fveq1 6877	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦‘𝑗) = (𝑥‘𝑗))
61	60	ifeq1d 4502	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))
62	61	mpteq2dv 5199	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))
63	62	mpteq2dv 5199	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) = (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))
64	63	breq1d 5113	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃) ↔ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0g‘𝑃)))
65	64	cbvralvw 3240	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃) ↔ ∀𝑥 ∈ 𝐵 (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0g‘𝑃))
66	59, 65	sylib 221	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0g‘𝑃))
67	66	r19.21bi 3254	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0g‘𝑃))
68	67	adantrr 730	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0g‘𝑃))
69		equequ2 2059	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑘 = 𝑖 ↔ 𝑘 = 𝑗))
70		fveq2 6878	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑦‘𝑖) = (𝑦‘𝑗))
71	69, 70	ifbieq1d 4507	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ) = if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))
72	71	mpteq2dv 5199	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
73	72	cbvmptv 5209	. . . . . 6 ⊢ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) = (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
74	58	adantrl 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃))
75	73, 74	eqbrtrid 5140	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) finSupp (0g‘𝑃))
76	1, 2, 3, 7, 7, 14, 23, 30, 68, 75	gsumdixp 20459	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑃 Σg (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))(.r‘𝑃)(𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))) = (𝑃 Σg (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
77	76	fveq2d 6882	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘((𝑃 Σg (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))(.r‘𝑃)(𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = (𝐸‘(𝑃 Σg (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
78		ringcmn 20423	. . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
79	13, 78	syl 18	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ CMnd)
80	79	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ CMnd)
81		evlslem2.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ CRing)
82		crngring 20384	. . . . . . 7 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
83	81, 82	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring)
84	83	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ Ring)
85		ringmnd 20382	. . . . 5 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Mnd)
86	84, 85	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ Mnd)
87	6, 6	xpex 7752	. . . . 5 ⊢ (𝐷 × 𝐷) ∈ V
88	87	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐷 × 𝐷) ∈ V)
89		evlslem2.e1	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑃 GrpHom 𝑆))
90		ghmmhm 19353	. . . . . 6 ⊢ (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸 ∈ (𝑃 MndHom 𝑆))
91	89, 90	syl 18	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝑃 MndHom 𝑆))
92	91	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐸 ∈ (𝑃 MndHom 𝑆))
93	13	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑃 ∈ Ring)
94	23	adantrr 730	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ 𝐵)
95	30	adantrl 729	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ 𝐵)
96	1, 2	ringcl 20389	. . . . . . 7 ⊢ ((𝑃 ∈ Ring ∧ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ 𝐵 ∧ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ 𝐵) → ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) ∈ 𝐵)
97	93, 94, 95, 96	syl3anc 1398	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) ∈ 𝐵)
98	97	ralrimivva 3205	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑗 ∈ 𝐷 ∀𝑖 ∈ 𝐷 ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) ∈ 𝐵)
99		eqid 2760	. . . . . 6 ⊢ (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) = (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))
100	99	fmpo 8065	. . . . 5 ⊢ (∀𝑗 ∈ 𝐷 ∀𝑖 ∈ 𝐷 ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) ∈ 𝐵 ↔ (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))):(𝐷 × 𝐷)⟶𝐵)
101	98, 100	sylib 221	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))):(𝐷 × 𝐷)⟶𝐵)
102	6, 6	mpoex 8078	. . . . . . 7 ⊢ (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V
103	99	mpofun 7537	. . . . . . 7 ⊢ Fun (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))
104	102, 103, 33	3pm3.2i 1358	. . . . . 6 ⊢ ((𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V ∧ Fun (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∧ (0g‘𝑃) ∈ V)
105	104	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V ∧ Fun (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∧ (0g‘𝑃) ∈ V))
106	68	fsuppimpd 9339	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)) ∈ Fin)
107	75	fsuppimpd 9339	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃)) ∈ Fin)
108		xpfi 9289	. . . . . 6 ⊢ ((((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)) ∈ Fin ∧ ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃)) ∈ Fin) → (((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)) × ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃))) ∈ Fin)
109	106, 107, 108	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)) × ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃))) ∈ Fin)
110	1, 3, 2, 14, 23, 30, 7, 7	evlslem4 22292	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) supp (0g‘𝑃)) ⊆ (((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)) × ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃))))
111		suppssfifsupp 9350	. . . . 5 ⊢ ((((𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V ∧ Fun (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∧ (0g‘𝑃) ∈ V) ∧ ((((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)) × ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃))) ∈ Fin ∧ ((𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) supp (0g‘𝑃)) ⊆ (((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)) × ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃))))) → (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) finSupp (0g‘𝑃))
112	105, 109, 110, 111	syl12anc 850	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) finSupp (0g‘𝑃))
113	1, 3, 80, 86, 88, 92, 101, 112	gsummhm 20065	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = (𝐸‘(𝑃 Σg (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
114	9	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝐼 ∈ 𝑊)
115	10	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑅 ∈ CRing)
116		eqid 2760	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
117		simprl 783	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑗 ∈ 𝐷)
118		simprr 785	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑖 ∈ 𝐷)
119	21	adantrr 730	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑥‘𝑗) ∈ (Base‘𝑅))
120	28	adantrl 729	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑦‘𝑖) ∈ (Base‘𝑅))
121	8, 4, 15, 16, 114, 115, 2, 116, 117, 118, 119, 120	mplmon2mul 22285	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑗 ∘f + 𝑖), ((𝑥‘𝑗)(.r‘𝑅)(𝑦‘𝑖)), 0 )))
122	121	fveq2d 6882	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝐸‘((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) = (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑗 ∘f + 𝑖), ((𝑥‘𝑗)(.r‘𝑅)(𝑦‘𝑖)), 0 ))))
123		evlslem2.e2	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷))) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑗 ∘f + 𝑖), ((𝑥‘𝑗)(.r‘𝑅)(𝑦‘𝑖)), 0 ))) = ((𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) · (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))
124	123	anassrs 473	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = (𝑗 ∘f + 𝑖), ((𝑥‘𝑗)(.r‘𝑅)(𝑦‘𝑖)), 0 ))) = ((𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) · (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))
125	122, 124	eqtrd 2795	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝐸‘((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) = ((𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) · (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))
126	125	3impb 1132	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷) → (𝐸‘((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) = ((𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) · (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))
127	126	mpoeq3dva 7490	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ (𝐸‘((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))) = (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) · (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
128	127	oveq2d 7429	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σg (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ (𝐸‘((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = (𝑆 Σg (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) · (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
129		eqidd 2761	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) = (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))
130		eqid 2760	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
131	1, 130	ghmf 19347	. . . . . . . . 9 ⊢ (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸:𝐵⟶(Base‘𝑆))
132	89, 131	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐸:𝐵⟶(Base‘𝑆))
133	132	feqmptd 6946	. . . . . . 7 ⊢ (𝜑 → 𝐸 = (𝑧 ∈ 𝐵 ↦ (𝐸‘𝑧)))
134	133	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐸 = (𝑧 ∈ 𝐵 ↦ (𝐸‘𝑧)))
135		fveq2 6878	. . . . . 6 ⊢ (𝑧 = ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) → (𝐸‘𝑧) = (𝐸‘((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))
136	97, 129, 134, 135	fmpoco 8092	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸 ∘ (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))) = (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ (𝐸‘((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
137	136	oveq2d 7429	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = (𝑆 Σg (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ (𝐸‘((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
138		eqidd 2761	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) = (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))
139		fveq2 6878	. . . . . . . 8 ⊢ (𝑧 = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) → (𝐸‘𝑧) = (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))
140	23, 138, 134, 139	fmptco 7123	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) = (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))))
141	140	oveq2d 7429	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) = (𝑆 Σg (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
142		eqidd 2761	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) = (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))
143		fveq2 6878	. . . . . . . 8 ⊢ (𝑧 = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) → (𝐸‘𝑧) = (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))
144	30, 142, 134, 143	fmptco 7123	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) = (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))
145	144	oveq2d 7429	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))) = (𝑆 Σg (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
146	141, 145	oveq12d 7431	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σg (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = ((𝑆 Σg (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σg (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
147		evlslem2.m	. . . . . 6 ⊢ · = (.r‘𝑆)
148		eqid 2760	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
149	132	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
150	149, 23	ffvelcdmd 7078	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) ∈ (Base‘𝑆))
151	132	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
152	151, 30	ffvelcdmd 7078	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) ∈ (Base‘𝑆))
153	6	mptex 7222	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) ∈ V
154		funmpt 6571	. . . . . . . . 9 ⊢ Fun (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))
155		fvex 6891	. . . . . . . . 9 ⊢ (0g‘𝑆) ∈ V
156	153, 154, 155	3pm3.2i 1358	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) ∈ V ∧ Fun (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) ∧ (0g‘𝑆) ∈ V)
157	156	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) ∈ V ∧ Fun (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) ∧ (0g‘𝑆) ∈ V))
158		ssidd 3954	. . . . . . . . 9 ⊢ (𝜑 → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)) ⊆ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)))
159	3, 148	ghmid 19349	. . . . . . . . . 10 ⊢ (𝐸 ∈ (𝑃 GrpHom 𝑆) → (𝐸‘(0g‘𝑃)) = (0g‘𝑆))
160	89, 159	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐸‘(0g‘𝑃)) = (0g‘𝑆))
161	6	mptex 7222	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ V
162	161	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ V)
163	33	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝑃) ∈ V)
164	158, 160, 162, 163	suppssfv 8200	. . . . . . . 8 ⊢ (𝜑 → ((𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) supp (0g‘𝑆)) ⊆ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)))
165	164	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) supp (0g‘𝑆)) ⊆ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)))
166		suppssfifsupp 9350	. . . . . . 7 ⊢ ((((𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) ∈ V ∧ Fun (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) ∧ (0g‘𝑆) ∈ V) ∧ (((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)) ∈ Fin ∧ ((𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) supp (0g‘𝑆)) ⊆ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)))) → (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) finSupp (0g‘𝑆))
167	157, 106, 165, 166	syl12anc 850	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) finSupp (0g‘𝑆))
168	6	mptex 7222	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V
169		funmpt 6571	. . . . . . . . 9 ⊢ Fun (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))
170	168, 169, 155	3pm3.2i 1358	. . . . . . . 8 ⊢ ((𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V ∧ Fun (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∧ (0g‘𝑆) ∈ V)
171	170	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V ∧ Fun (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∧ (0g‘𝑆) ∈ V))
172		ssidd 3954	. . . . . . . . 9 ⊢ (𝜑 → ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃)) ⊆ ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃)))
173	6	mptex 7222	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ V
174	173	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ V)
175	172, 160, 174, 163	suppssfv 8200	. . . . . . . 8 ⊢ (𝜑 → ((𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) supp (0g‘𝑆)) ⊆ ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃)))
176	175	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) supp (0g‘𝑆)) ⊆ ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃)))
177		suppssfifsupp 9350	. . . . . . 7 ⊢ ((((𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V ∧ Fun (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∧ (0g‘𝑆) ∈ V) ∧ (((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃)) ∈ Fin ∧ ((𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) supp (0g‘𝑆)) ⊆ ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃)))) → (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) finSupp (0g‘𝑆))
178	171, 107, 176, 177	syl12anc 850	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) finSupp (0g‘𝑆))
179	130, 147, 148, 7, 7, 84, 150, 152, 167, 178	gsumdixp 20459	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑆 Σg (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σg (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = (𝑆 Σg (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) · (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
180	146, 179	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σg (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = (𝑆 Σg (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) · (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
181	128, 137, 180	3eqtr4d 2805	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = ((𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σg (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
182	77, 113, 181	3eqtr2d 2801	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘((𝑃 Σg (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))(.r‘𝑃)(𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = ((𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σg (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
183	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐼 ∈ 𝑊)
184	12	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ Ring)
185	8, 4, 15, 1, 183, 184, 19	mplcoe4 22287	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = (𝑃 Σg (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))))
186	8, 4, 15, 1, 183, 184, 26	mplcoe4 22287	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 = (𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))
187	185, 186	oveq12d 7431	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝑃)𝑦) = ((𝑃 Σg (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))(.r‘𝑃)(𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
188	187	fveq2d 6882	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘(𝑥(.r‘𝑃)𝑦)) = (𝐸‘((𝑃 Σg (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))(.r‘𝑃)(𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
189	185	fveq2d 6882	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑥) = (𝐸‘(𝑃 Σg (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
190	23	fmpttd 7108	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))):𝐷⟶𝐵)
191	1, 3, 80, 86, 7, 92, 190, 68	gsummhm 20065	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) = (𝐸‘(𝑃 Σg (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
192	189, 191	eqtr4d 2798	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑥) = (𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
193	186	fveq2d 6882	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑦) = (𝐸‘(𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
194	30	fmpttd 7108	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))):𝐷⟶𝐵)
195	1, 3, 80, 86, 7, 92, 194, 75	gsummhm 20065	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))) = (𝐸‘(𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
196	193, 195	eqtr4d 2798	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑦) = (𝑆 Σg (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
197	192, 196	oveq12d 7431	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐸‘𝑥) · (𝐸‘𝑦)) = ((𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σg (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
198	182, 188, 197	3eqtr4d 2805	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘(𝑥(.r‘𝑃)𝑦)) = ((𝐸‘𝑥) · (𝐸‘𝑦)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450   ∖ cdif 3896   ⊆ wss 3899  ifcif 4482  {csn 4584   class class class wbr 5103   ↦ cmpt 5186   × cxp 5653  ^◡ccnv 5654   “ cima 5658   ∘ ccom 5659  Fun wfun 6527  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415   ∘_f cof 7676   supp csupp 8158   ↑_m cmap 8826  Fincfn 8952   finSupp cfsupp 9331   + caddc 11127  ℕcn 12257  ℕ₀cn0 12528  Basecbs 17301  ._rcmulr 17343  0_gc0g 17524   Σ_g cgsu 17525  Mndcmnd 18836   MndHom cmhm 18889  Grpcgrp 19057   GrpHom cghm 19340  CMndccmn 19907  Ringcrg 20372  CRingccrg 20373   mPoly cmpl 22121

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-ofr 7679  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-pm 8829  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-sup 9412  df-oi 9482  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-uz 12888  df-fz 13562  df-fzo 13710  df-seq 14066  df-hash 14395  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-mulr 17356  df-sca 17358  df-vsca 17359  df-ip 17360  df-tset 17361  df-ple 17362  df-ds 17364  df-hom 17366  df-cco 17367  df-0g 17526  df-gsum 17527  df-prds 17532  df-pws 17534  df-mre 17670  df-mrc 17671  df-acs 17673  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-mhm 18891  df-submnd 18892  df-grp 19060  df-minusg 19061  df-sbg 19062  df-mulg 19191  df-subg 19246  df-ghm 19341  df-cntz 19444  df-cmn 19909  df-abl 19910  df-mgp 20274  df-rng 20288  df-ur 20321  df-ring 20374  df-cring 20375  df-subrng 20708  df-subrg 20732  df-lmod 21046  df-lss 21116  df-assa 22068  df-psr 22124  df-mpl 22126

This theorem is used by:  evlslem1  22298