Theorem evlslem2 22230

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 2763	. . . . 5 ⊢ (.r‘𝑃) = (.r‘𝑃)
3		eqid 2763	. . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃)
4		evlslem2.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
5		ovex 7443	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
6	4, 5	rabex2 5311	. . . . . 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 20322	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	10, 11	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
13	8, 9, 12	mplringd 22172	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Ring)
14	13	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ Ring)
15		evlslem2.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
16		eqid 2763	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
17	9	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝐼 ∈ 𝑊)
18	12	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝑅 ∈ Ring)
19		simprl 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
20	8, 16, 1, 4, 19	mplelf 22147	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥:𝐷⟶(Base‘𝑅))
21	20	ffvelcdmda 7079	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝑥‘𝑗) ∈ (Base‘𝑅))
22		simpr 489	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝑗 ∈ 𝐷)
23	8, 4, 15, 16, 17, 18, 1, 21, 22	mplmon2cl 22219	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ 𝐵)
24	9	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝐼 ∈ 𝑊)
25	12	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝑅 ∈ Ring)
26		simprr 784	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
27	8, 16, 1, 4, 26	mplelf 22147	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦:𝐷⟶(Base‘𝑅))
28	27	ffvelcdmda 7079	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝑦‘𝑖) ∈ (Base‘𝑅))
29		simpr 489	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝑖 ∈ 𝐷)
30	8, 4, 15, 16, 24, 25, 1, 28, 29	mplmon2cl 22219	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ 𝐵)
31	6	mptex 7221	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∈ V
32		funmpt 6574	. . . . . . . . . . . 12 ⊢ Fun (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
33		fvex 6894	. . . . . . . . . . . 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 489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
37	8, 1, 15, 36	mplelsfi 22144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 finSupp 0 )
38	37	fsuppimpd 9325	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ∈ Fin)
39	8, 16, 1, 4, 36	mplelf 22147	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐷⟶(Base‘𝑅))
40		ssidd 3960	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 ))
41	6	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V)
42	15	fvexi 6895	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
43	42	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 0 ∈ V)
44	39, 40, 41, 43	suppssr 8187	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑗) = 0 )
45	44	ifeq1d 4507	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = if(𝑘 = 𝑗, 0 , 0 ))
46		ifid 4528	. . . . . . . . . . . . . 14 ⊢ if(𝑘 = 𝑗, 0 , 0 ) = 0
47	45, 46	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = 0 )
48	47	mpteq2dv 5205	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (𝑘 ∈ 𝐷 ↦ 0 ))
49		ringgrp 20315	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
50	12, 49	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ Grp)
51	8, 4, 15, 3, 9, 50	mpl0 22155	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝑃) = (𝐷 × { 0 }))
52		fconstmpt 5723	. . . . . . . . . . . . . 14 ⊢ (𝐷 × { 0 }) = (𝑘 ∈ 𝐷 ↦ 0 )
53	51, 52	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝑃) = (𝑘 ∈ 𝐷 ↦ 0 ))
54	53	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (0g‘𝑃) = (𝑘 ∈ 𝐷 ↦ 0 ))
55	48, 54	eqtr4d 2801	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (0g‘𝑃))
56	55, 41	suppss2 8192	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) supp (0g‘𝑃)) ⊆ (𝑦 supp 0 ))
57		suppssfifsupp 9336	. . . . . . . . . 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 849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃))
59	58	ralrimiva 3157	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃))
60		fveq1 6880	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦‘𝑗) = (𝑥‘𝑗))
61	60	ifeq1d 4507	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))
62	61	mpteq2dv 5205	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))
63	62	mpteq2dv 5205	. . . . . . . . . 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 5119	. . . . . . . . 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 3243	. . . . . . . 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 3257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0g‘𝑃))
68	67	adantrr 729	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0g‘𝑃))
69		equequ2 2056	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑘 = 𝑖 ↔ 𝑘 = 𝑗))
70		fveq2 6881	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑦‘𝑖) = (𝑦‘𝑗))
71	69, 70	ifbieq1d 4512	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ) = if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))
72	71	mpteq2dv 5205	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
73	72	cbvmptv 5215	. . . . . 6 ⊢ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) = (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
74	58	adantrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃))
75	73, 74	eqbrtrid 5146	. . . . 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 20396	. . . 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 6885	. . 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 20361	. . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
79	13, 78	syl 18	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ CMnd)
80	79	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ CMnd)
81		evlslem2.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ CRing)
82		crngring 20322	. . . . . . 7 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
83	81, 82	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring)
84	83	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ Ring)
85		ringmnd 20320	. . . . 5 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Mnd)
86	84, 85	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ Mnd)
87	6, 6	xpex 7748	. . . . 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 19291	. . . . . 6 ⊢ (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸 ∈ (𝑃 MndHom 𝑆))
91	89, 90	syl 18	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝑃 MndHom 𝑆))
92	91	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐸 ∈ (𝑃 MndHom 𝑆))
93	13	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑃 ∈ Ring)
94	23	adantrr 729	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ 𝐵)
95	30	adantrl 728	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ 𝐵)
96	1, 2	ringcl 20327	. . . . . . 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 3208	. . . . 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 2763	. . . . . 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 8061	. . . . 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 8072	. . . . . . 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 7534	. . . . . . 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 9325	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)) ∈ Fin)
107	75	fsuppimpd 9325	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃)) ∈ Fin)
108		xpfi 9275	. . . . . 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 595	. . . . 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 22227	. . . . 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 9336	. . . . 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 849	. . . 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 20003	. . 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 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝐼 ∈ 𝑊)
115	10	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑅 ∈ CRing)
116		eqid 2763	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
117		simprl 782	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑗 ∈ 𝐷)
118		simprr 784	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑖 ∈ 𝐷)
119	21	adantrr 729	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑥‘𝑗) ∈ (Base‘𝑅))
120	28	adantrl 728	. . . . . . . . . 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 22220	. . . . . . . . 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 6885	. . . . . . . 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 472	. . . . . . . 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 2798	. . . . . . 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 7487	. . . . 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 7426	. . . 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 2764	. . . . . 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 2763	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
131	1, 130	ghmf 19285	. . . . . . . . 9 ⊢ (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸:𝐵⟶(Base‘𝑆))
132	89, 131	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐸:𝐵⟶(Base‘𝑆))
133	132	feqmptd 6949	. . . . . . 7 ⊢ (𝜑 → 𝐸 = (𝑧 ∈ 𝐵 ↦ (𝐸‘𝑧)))
134	133	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐸 = (𝑧 ∈ 𝐵 ↦ (𝐸‘𝑧)))
135		fveq2 6881	. . . . . 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 8086	. . . . 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 7426	. . . 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 2764	. . . . . . . 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 6881	. . . . . . . 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 7125	. . . . . . 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 7426	. . . . . 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 2764	. . . . . . . 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 6881	. . . . . . . 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 7125	. . . . . . 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 7426	. . . . . 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 7428	. . . . 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 2763	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
149	132	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
150	149, 23	ffvelcdmd 7080	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) ∈ (Base‘𝑆))
151	132	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
152	151, 30	ffvelcdmd 7080	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) ∈ (Base‘𝑆))
153	6	mptex 7221	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) ∈ V
154		funmpt 6574	. . . . . . . . 9 ⊢ Fun (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))
155		fvex 6894	. . . . . . . . 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 3960	. . . . . . . . 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 19287	. . . . . . . . . 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 7221	. . . . . . . . . 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 8194	. . . . . . . 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 485	. . . . . . 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 9336	. . . . . . 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 849	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) finSupp (0g‘𝑆))
168	6	mptex 7221	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V
169		funmpt 6574	. . . . . . . . 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 3960	. . . . . . . . 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 7221	. . . . . . . . . 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 8194	. . . . . . . 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 485	. . . . . . 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 9336	. . . . . . 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 849	. . . . . 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 20396	. . . . 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 2798	. . . 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 2808	. . 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 2804	. 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 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐼 ∈ 𝑊)
184	12	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ Ring)
185	8, 4, 15, 1, 183, 184, 19	mplcoe4 22222	. . . 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 22222	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 = (𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))
187	185, 186	oveq12d 7428	. . 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 6885	. 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 6885	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑥) = (𝐸‘(𝑃 Σg (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
190	23	fmpttd 7110	. . . . 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 20003	. . . 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 2801	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑥) = (𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
193	186	fveq2d 6885	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑦) = (𝐸‘(𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
194	30	fmpttd 7110	. . . . 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 20003	. . . 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 2801	. . 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 7428	. 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 2808	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘(𝑥(.r‘𝑃)𝑦)) = ((𝐸‘𝑥) · (𝐸‘𝑦)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  ifcif 4487  {csn 4589   class class class wbr 5109   ↦ cmpt 5192   × cxp 5659  ^◡ccnv 5660   “ cima 5664   ∘ ccom 5665  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ∘_f cof 7672   supp csupp 8152   ↑_m cmap 8820  Fincfn 8939   finSupp cfsupp 9317   + caddc 11098  ℕcn 12228  ℕ₀cn0 12499  Basecbs 17264  ._rcmulr 17306  0_gc0g 17487   Σ_g cgsu 17488  Mndcmnd 18787   MndHom cmhm 18834  Grpcgrp 18995   GrpHom cghm 19278  CMndccmn 19845  Ringcrg 20310  CRingccrg 20311   mPoly cmpl 22056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-ofr 7675  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-sup 9398  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-fz 13531  df-fzo 13679  df-seq 14034  df-hash 14363  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-hom 17329  df-cco 17330  df-0g 17489  df-gsum 17490  df-prds 17495  df-pws 17497  df-mre 17633  df-mrc 17634  df-acs 17636  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836  df-submnd 18837  df-grp 18998  df-minusg 18999  df-sbg 19000  df-mulg 19129  df-subg 19184  df-ghm 19279  df-cntz 19382  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-ring 20312  df-cring 20313  df-subrng 20645  df-subrg 20669  df-lmod 20983  df-lss 21053  df-assa 22003  df-psr 22059  df-mpl 22061

This theorem is referenced by:  evlslem1  22233