Theorem evlslem2 20896

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 2739	. . . . 5 ⊢ (.r‘𝑃) = (.r‘𝑃)
3		eqid 2739	. . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃)
4		evlslem2.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
5		ovex 7206	. . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V
6	4, 5	rabex2 5203	. . . . . 6 ⊢ 𝐷 ∈ V
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ V)
8		evlslem2.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
9		evlslem2.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ CRing)
10		crngring 19431	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
12		evlslem2.p	. . . . . . . 8 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
13	12	mplring 20837	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
14	8, 11, 13	syl2anc 587	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Ring)
15	14	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ Ring)
16		evlslem2.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
17		eqid 2739	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
18	8	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝐼 ∈ 𝑊)
19	11	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝑅 ∈ Ring)
20		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
21	12, 17, 1, 4, 20	mplelf 20817	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥:𝐷⟶(Base‘𝑅))
22	21	ffvelrnda 6864	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝑥‘𝑗) ∈ (Base‘𝑅))
23		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝑗 ∈ 𝐷)
24	12, 4, 16, 17, 18, 19, 1, 22, 23	mplmon2cl 20883	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ 𝐵)
25	8	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝐼 ∈ 𝑊)
26	11	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝑅 ∈ Ring)
27		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
28	12, 17, 1, 4, 27	mplelf 20817	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦:𝐷⟶(Base‘𝑅))
29	28	ffvelrnda 6864	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝑦‘𝑖) ∈ (Base‘𝑅))
30		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝑖 ∈ 𝐷)
31	12, 4, 16, 17, 25, 26, 1, 29, 30	mplmon2cl 20883	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ 𝐵)
32	6	mptex 6999	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∈ V
33		funmpt 6378	. . . . . . . . . . . 12 ⊢ Fun (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
34		fvex 6690	. . . . . . . . . . . 12 ⊢ (0g‘𝑃) ∈ V
35	32, 33, 34	3pm3.2i 1340	. . . . . . . . . . 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)
36	35	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))
37		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
38	9	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ CRing)
39	12, 1, 16, 37, 38	mplelsfi 20814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 finSupp 0 )
40	39	fsuppimpd 8916	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ∈ Fin)
41	12, 17, 1, 4, 37	mplelf 20817	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐷⟶(Base‘𝑅))
42		ssidd 3901	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 ))
43	6	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V)
44	16	fvexi 6691	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
45	44	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 0 ∈ V)
46	41, 42, 43, 45	suppssr 7894	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑗) = 0 )
47	46	ifeq1d 4434	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = if(𝑘 = 𝑗, 0 , 0 ))
48		ifid 4455	. . . . . . . . . . . . . 14 ⊢ if(𝑘 = 𝑗, 0 , 0 ) = 0
49	47, 48	eqtrdi 2790	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = 0 )
50	49	mpteq2dv 5127	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (𝑘 ∈ 𝐷 ↦ 0 ))
51		ringgrp 19424	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
52	11, 51	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ Grp)
53	12, 4, 16, 3, 8, 52	mpl0 20825	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝑃) = (𝐷 × { 0 }))
54		fconstmpt 5586	. . . . . . . . . . . . . 14 ⊢ (𝐷 × { 0 }) = (𝑘 ∈ 𝐷 ↦ 0 )
55	53, 54	eqtrdi 2790	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝑃) = (𝑘 ∈ 𝐷 ↦ 0 ))
56	55	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (0g‘𝑃) = (𝑘 ∈ 𝐷 ↦ 0 ))
57	50, 56	eqtr4d 2777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (0g‘𝑃))
58	57, 43	suppss2 7898	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) supp (0g‘𝑃)) ⊆ (𝑦 supp 0 ))
59		suppssfifsupp 8924	. . . . . . . . . 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))
60	36, 40, 58, 59	syl12anc 836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃))
61	60	ralrimiva 3097	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃))
62		fveq1 6676	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦‘𝑗) = (𝑥‘𝑗))
63	62	ifeq1d 4434	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))
64	63	mpteq2dv 5127	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))
65	64	mpteq2dv 5127	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) = (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))
66	65	breq1d 5041	. . . . . . . . 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)))
67	66	cbvralvw 3350	. . . . . . . 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))
68	61, 67	sylib 221	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0g‘𝑃))
69	68	r19.21bi 3122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0g‘𝑃))
70	69	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0g‘𝑃))
71		equequ2 2038	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑘 = 𝑖 ↔ 𝑘 = 𝑗))
72		fveq2 6677	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑦‘𝑖) = (𝑦‘𝑗))
73	71, 72	ifbieq1d 4439	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ) = if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))
74	73	mpteq2dv 5127	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
75	74	cbvmptv 5134	. . . . . 6 ⊢ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) = (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
76	60	adantrl 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0g‘𝑃))
77	75, 76	eqbrtrid 5066	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) finSupp (0g‘𝑃))
78	1, 2, 3, 7, 7, 15, 24, 31, 70, 77	gsumdixp 19484	. . . 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 ))))))
79	78	fveq2d 6681	. . 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 )))))))
80		ringcmn 19456	. . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
81	14, 80	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ CMnd)
82	81	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ CMnd)
83		evlslem2.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ CRing)
84		crngring 19431	. . . . . . 7 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
85	83, 84	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring)
86	85	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ Ring)
87		ringmnd 19429	. . . . 5 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Mnd)
88	86, 87	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ Mnd)
89	6, 6	xpex 7497	. . . . 5 ⊢ (𝐷 × 𝐷) ∈ V
90	89	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐷 × 𝐷) ∈ V)
91		evlslem2.e1	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑃 GrpHom 𝑆))
92		ghmmhm 18489	. . . . . 6 ⊢ (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸 ∈ (𝑃 MndHom 𝑆))
93	91, 92	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝑃 MndHom 𝑆))
94	93	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐸 ∈ (𝑃 MndHom 𝑆))
95	14	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑃 ∈ Ring)
96	24	adantrr 717	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ 𝐵)
97	31	adantrl 716	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ 𝐵)
98	1, 2	ringcl 19436	. . . . . . 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)
99	95, 96, 97, 98	syl3anc 1372	. . . . . 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)
100	99	ralrimivva 3104	. . . . 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)
101		eqid 2739	. . . . . 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 ))))
102	101	fmpo 7794	. . . . 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)
103	100, 102	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)
104	6, 6	mpoex 7806	. . . . . . 7 ⊢ (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V
105	101	mpofun 7293	. . . . . . 7 ⊢ Fun (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(.r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))
106	104, 105, 34	3pm3.2i 1340	. . . . . 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)
107	106	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))
108	70	fsuppimpd 8916	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0g‘𝑃)) ∈ Fin)
109	77	fsuppimpd 8916	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0g‘𝑃)) ∈ Fin)
110		xpfi 8866	. . . . . 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)
111	108, 109, 110	syl2anc 587	. . . . 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)
112	1, 3, 2, 15, 24, 31, 7, 7	evlslem4 20891	. . . . 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))))
113		suppssfifsupp 8924	. . . . 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))
114	107, 111, 112, 113	syl12anc 836	. . . 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))
115	1, 3, 82, 88, 90, 94, 103, 114	gsummhm 19180	. . 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 )))))))
116	8	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝐼 ∈ 𝑊)
117	9	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑅 ∈ CRing)
118		eqid 2739	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
119		simprl 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑗 ∈ 𝐷)
120		simprr 773	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑖 ∈ 𝐷)
121	22	adantrr 717	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑥‘𝑗) ∈ (Base‘𝑅))
122	29	adantrl 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑦‘𝑖) ∈ (Base‘𝑅))
123	12, 4, 16, 17, 116, 117, 2, 118, 119, 120, 121, 122	mplmon2mul 20884	. . . . . . . . 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 )))
124	123	fveq2d 6681	. . . . . . . 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 ))))
125		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 )))))
126	125	anassrs 471	. . . . . . . 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 )))))
127	124, 126	eqtrd 2774	. . . . . . 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 )))))
128	127	3impb 1116	. . . . . 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 )))))
129	128	mpoeq3dva 7248	. . . . 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 ))))))
130	129	oveq2d 7189	. . . 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 )))))))
131		eqidd 2740	. . . . . 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 )))))
132		eqid 2739	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
133	1, 132	ghmf 18483	. . . . . . . . 9 ⊢ (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸:𝐵⟶(Base‘𝑆))
134	91, 133	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐸:𝐵⟶(Base‘𝑆))
135	134	feqmptd 6740	. . . . . . 7 ⊢ (𝜑 → 𝐸 = (𝑧 ∈ 𝐵 ↦ (𝐸‘𝑧)))
136	135	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐸 = (𝑧 ∈ 𝐵 ↦ (𝐸‘𝑧)))
137		fveq2 6677	. . . . . 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 )))))
138	99, 131, 136, 137	fmpoco 7819	. . . . 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 ))))))
139	138	oveq2d 7189	. . . 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 )))))))
140		eqidd 2740	. . . . . . . 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 ))))
141		fveq2 6677	. . . . . . . 8 ⊢ (𝑧 = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) → (𝐸‘𝑧) = (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))
142	24, 140, 136, 141	fmptco 6904	. . . . . . 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 )))))
143	142	oveq2d 7189	. . . . . 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 ))))))
144		eqidd 2740	. . . . . . . 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 ))))
145		fveq2 6677	. . . . . . . 8 ⊢ (𝑧 = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) → (𝐸‘𝑧) = (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))
146	31, 144, 136, 145	fmptco 6904	. . . . . . 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 )))))
147	146	oveq2d 7189	. . . . . 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 ))))))
148	143, 147	oveq12d 7191	. . . . 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 )))))))
149		evlslem2.m	. . . . . 6 ⊢ · = (.r‘𝑆)
150		eqid 2739	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
151	134	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
152	151, 24	ffvelrnd 6865	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) ∈ (Base‘𝑆))
153	134	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
154	153, 31	ffvelrnd 6865	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) ∈ (Base‘𝑆))
155	6	mptex 6999	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) ∈ V
156		funmpt 6378	. . . . . . . . 9 ⊢ Fun (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))
157		fvex 6690	. . . . . . . . 9 ⊢ (0g‘𝑆) ∈ V
158	155, 156, 157	3pm3.2i 1340	. . . . . . . 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)
159	158	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))
160		ssidd 3901	. . . . . . . . 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)))
161	3, 150	ghmid 18485	. . . . . . . . . 10 ⊢ (𝐸 ∈ (𝑃 GrpHom 𝑆) → (𝐸‘(0g‘𝑃)) = (0g‘𝑆))
162	91, 161	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐸‘(0g‘𝑃)) = (0g‘𝑆))
163	6	mptex 6999	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ V
164	163	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ V)
165	34	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝑃) ∈ V)
166	160, 162, 164, 165	suppssfv 7900	. . . . . . . 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)))
167	166	adantr 484	. . . . . . 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)))
168		suppssfifsupp 8924	. . . . . . 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))
169	159, 108, 167, 168	syl12anc 836	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) finSupp (0g‘𝑆))
170	6	mptex 6999	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V
171		funmpt 6378	. . . . . . . . 9 ⊢ Fun (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))
172	170, 171, 157	3pm3.2i 1340	. . . . . . . 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)
173	172	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))
174		ssidd 3901	. . . . . . . . 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)))
175	6	mptex 6999	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ V
176	175	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ V)
177	174, 162, 176, 165	suppssfv 7900	. . . . . . . 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)))
178	177	adantr 484	. . . . . . 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)))
179		suppssfifsupp 8924	. . . . . . 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))
180	173, 109, 178, 179	syl12anc 836	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) finSupp (0g‘𝑆))
181	132, 149, 150, 7, 7, 86, 152, 154, 169, 180	gsumdixp 19484	. . . . 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 )))))))
182	148, 181	eqtrd 2774	. . . 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 )))))))
183	130, 139, 182	3eqtr4d 2784	. . 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 )))))))
184	79, 115, 183	3eqtr2d 2780	. 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 )))))))
185	8	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐼 ∈ 𝑊)
186	11	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ Ring)
187	12, 4, 16, 1, 185, 186, 20	mplcoe4 20886	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = (𝑃 Σg (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))))
188	12, 4, 16, 1, 185, 186, 27	mplcoe4 20886	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 = (𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))
189	187, 188	oveq12d 7191	. . 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 ))))))
190	189	fveq2d 6681	. 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 )))))))
191	187	fveq2d 6681	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑥) = (𝐸‘(𝑃 Σg (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
192	24	fmpttd 6892	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))):𝐷⟶𝐵)
193	1, 3, 82, 88, 7, 94, 192, 70	gsummhm 19180	. . . 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 ))))))
194	191, 193	eqtr4d 2777	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑥) = (𝑆 Σg (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
195	188	fveq2d 6681	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑦) = (𝐸‘(𝑃 Σg (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
196	31	fmpttd 6892	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))):𝐷⟶𝐵)
197	1, 3, 82, 88, 7, 94, 196, 77	gsummhm 19180	. . . 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 ))))))
198	195, 197	eqtr4d 2777	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑦) = (𝑆 Σg (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
199	194, 198	oveq12d 7191	. 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 )))))))
200	184, 190, 199	3eqtr4d 2784	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘(𝑥(.r‘𝑃)𝑦)) = ((𝐸‘𝑥) · (𝐸‘𝑦)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3054  {crab 3058  Vcvv 3399   ∖ cdif 3841   ⊆ wss 3844  ifcif 4415  {csn 4517   class class class wbr 5031   ↦ cmpt 5111   × cxp 5524  ^◡ccnv 5525   “ cima 5529   ∘ ccom 5530  Fun wfun 6334  ⟶wf 6336  ‘cfv 6340  (class class class)co 7173   ∈ cmpo 7175   ∘_f cof 7426   supp csupp 7859   ↑_m cmap 8440  Fincfn 8558   finSupp cfsupp 8909   + caddc 10621  ℕcn 11719  ℕ₀cn0 11979  Basecbs 16589  ._rcmulr 16672  0_gc0g 16819   Σ_g cgsu 16820  Mndcmnd 18030   MndHom cmhm 18073  Grpcgrp 18222   GrpHom cghm 18476  CMndccmn 19027  Ringcrg 19419  CRingccrg 19420   mPoly cmpl 20722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482  ax-cnex 10674  ax-resscn 10675  ax-1cn 10676  ax-icn 10677  ax-addcl 10678  ax-addrcl 10679  ax-mulcl 10680  ax-mulrcl 10681  ax-mulcom 10682  ax-addass 10683  ax-mulass 10684  ax-distr 10685  ax-i2m1 10686  ax-1ne0 10687  ax-1rid 10688  ax-rnegex 10689  ax-rrecex 10690  ax-cnre 10691  ax-pre-lttri 10692  ax-pre-lttrn 10693  ax-pre-ltadd 10694  ax-pre-mulgt0 10695

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-int 4838  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-se 5485  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-isom 6349  df-riota 7130  df-ov 7176  df-oprab 7177  df-mpo 7178  df-of 7428  df-ofr 7429  df-om 7603  df-1st 7717  df-2nd 7718  df-supp 7860  df-wrecs 7979  df-recs 8040  df-rdg 8078  df-1o 8134  df-er 8323  df-map 8442  df-pm 8443  df-ixp 8511  df-en 8559  df-dom 8560  df-sdom 8561  df-fin 8562  df-fsupp 8910  df-oi 9050  df-card 9444  df-pnf 10758  df-mnf 10759  df-xr 10760  df-ltxr 10761  df-le 10762  df-sub 10953  df-neg 10954  df-nn 11720  df-2 11782  df-3 11783  df-4 11784  df-5 11785  df-6 11786  df-7 11787  df-8 11788  df-9 11789  df-n0 11980  df-z 12066  df-uz 12328  df-fz 12985  df-fzo 13128  df-seq 13464  df-hash 13786  df-struct 16591  df-ndx 16592  df-slot 16593  df-base 16595  df-sets 16596  df-ress 16597  df-plusg 16684  df-mulr 16685  df-sca 16687  df-vsca 16688  df-tset 16690  df-0g 16821  df-gsum 16822  df-mre 16963  df-mrc 16964  df-acs 16966  df-mgm 17971  df-sgrp 18020  df-mnd 18031  df-mhm 18075  df-submnd 18076  df-grp 18225  df-minusg 18226  df-sbg 18227  df-mulg 18346  df-subg 18397  df-ghm 18477  df-cntz 18568  df-cmn 19029  df-abl 19030  df-mgp 19362  df-ur 19374  df-ring 19421  df-cring 19422  df-subrg 19655  df-lmod 19758  df-lss 19826  df-assa 20672  df-psr 20725  df-mpl 20727

This theorem is referenced by:  evlslem1  20899