evlslem2 - Metamath Proof Explorer

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Theorem evlslem2 21861

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 = (0_g‘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 2730	. . . . 5 ⊢ (._r‘𝑃) = (._r‘𝑃)
3		eqid 2730	. . . . 5 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
4		evlslem2.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
5		ovex 7444	. . . . . . 7 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
6	4, 5	rabex2 5333	. . . . . 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 20139	. . . . . . . 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 21797	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
14	8, 11, 13	syl2anc 582	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Ring)
15	14	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ Ring)
16		evlslem2.z	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
17		eqid 2730	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
18	8	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝐼 ∈ 𝑊)
19	11	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝑅 ∈ Ring)
20		simprl 767	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
21	12, 17, 1, 4, 20	mplelf 21776	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥:𝐷⟶(Base‘𝑅))
22	21	ffvelcdmda 7085	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝑥‘𝑗) ∈ (Base‘𝑅))
23		simpr 483	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝑗 ∈ 𝐷)
24	12, 4, 16, 17, 18, 19, 1, 22, 23	mplmon2cl 21848	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ 𝐵)
25	8	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝐼 ∈ 𝑊)
26	11	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝑅 ∈ Ring)
27		simprr 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
28	12, 17, 1, 4, 27	mplelf 21776	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦:𝐷⟶(Base‘𝑅))
29	28	ffvelcdmda 7085	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝑦‘𝑖) ∈ (Base‘𝑅))
30		simpr 483	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝑖 ∈ 𝐷)
31	12, 4, 16, 17, 25, 26, 1, 29, 30	mplmon2cl 21848	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ 𝐵)
32	6	mptex 7226	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∈ V
33		funmpt 6585	. . . . . . . . . . . 12 ⊢ Fun (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
34		fvex 6903	. . . . . . . . . . . 12 ⊢ (0_g‘𝑃) ∈ V
35	32, 33, 34	3pm3.2i 1337	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∈ V ∧ Fun (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∧ (0_g‘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 ))) ∧ (0_g‘𝑃) ∈ V))
37		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
38	9	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑅 ∈ CRing)
39	12, 1, 16, 37, 38	mplelsfi 21773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 finSupp 0 )
40	39	fsuppimpd 9371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ∈ Fin)
41	12, 17, 1, 4, 37	mplelf 21776	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐷⟶(Base‘𝑅))
42		ssidd 4004	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 ))
43	6	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V)
44	16	fvexi 6904	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
45	44	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 0 ∈ V)
46	41, 42, 43, 45	suppssr 8183	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑗) = 0 )
47	46	ifeq1d 4546	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = if(𝑘 = 𝑗, 0 , 0 ))
48		ifid 4567	. . . . . . . . . . . . . 14 ⊢ if(𝑘 = 𝑗, 0 , 0 ) = 0
49	47, 48	eqtrdi 2786	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = 0 )
50	49	mpteq2dv 5249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (𝑘 ∈ 𝐷 ↦ 0 ))
51		ringgrp 20132	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
52	11, 51	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ Grp)
53	12, 4, 16, 3, 8, 52	mpl0 21784	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝑃) = (𝐷 × { 0 }))
54		fconstmpt 5737	. . . . . . . . . . . . . 14 ⊢ (𝐷 × { 0 }) = (𝑘 ∈ 𝐷 ↦ 0 )
55	53, 54	eqtrdi 2786	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0_g‘𝑃) = (𝑘 ∈ 𝐷 ↦ 0 ))
56	55	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (0_g‘𝑃) = (𝑘 ∈ 𝐷 ↦ 0 ))
57	50, 56	eqtr4d 2773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (0_g‘𝑃))
58	57, 43	suppss2 8187	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) supp (0_g‘𝑃)) ⊆ (𝑦 supp 0 ))
59		suppssfifsupp 9380	. . . . . . . . . 10 ⊢ ((((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∈ V ∧ Fun (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) ∧ (0_g‘𝑃) ∈ V) ∧ ((𝑦 supp 0 ) ∈ Fin ∧ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) supp (0_g‘𝑃)) ⊆ (𝑦 supp 0 ))) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0_g‘𝑃))
60	36, 40, 58, 59	syl12anc 833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0_g‘𝑃))
61	60	ralrimiva 3144	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0_g‘𝑃))
62		fveq1 6889	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦‘𝑗) = (𝑥‘𝑗))
63	62	ifeq1d 4546	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ) = if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))
64	63	mpteq2dv 5249	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))
65	64	mpteq2dv 5249	. . . . . . . . . 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 5157	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0_g‘𝑃) ↔ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0_g‘𝑃)))
67	66	cbvralvw 3232	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0_g‘𝑃) ↔ ∀𝑥 ∈ 𝐵 (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0_g‘𝑃))
68	61, 67	sylib 217	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0_g‘𝑃))
69	68	r19.21bi 3246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0_g‘𝑃))
70	69	adantrr 713	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) finSupp (0_g‘𝑃))
71		equequ2 2027	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑘 = 𝑖 ↔ 𝑘 = 𝑗))
72		fveq2 6890	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑦‘𝑖) = (𝑦‘𝑗))
73	71, 72	ifbieq1d 4551	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ) = if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))
74	73	mpteq2dv 5249	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) = (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
75	74	cbvmptv 5260	. . . . . 6 ⊢ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) = (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 )))
76	60	adantrl 712	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑦‘𝑗), 0 ))) finSupp (0_g‘𝑃))
77	75, 76	eqbrtrid 5182	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) finSupp (0_g‘𝑃))
78	1, 2, 3, 7, 7, 15, 24, 31, 70, 77	gsumdixp 20207	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑃 Σ_g (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))(._r‘𝑃)(𝑃 Σ_g (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))) = (𝑃 Σ_g (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 6894	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘((𝑃 Σ_g (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))(._r‘𝑃)(𝑃 Σ_g (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = (𝐸‘(𝑃 Σ_g (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(._r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
80		ringcmn 20170	. . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
81	14, 80	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ CMnd)
82	81	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ CMnd)
83		evlslem2.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ CRing)
84		crngring 20139	. . . . . . 7 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
85	83, 84	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring)
86	85	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ Ring)
87		ringmnd 20137	. . . . 5 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Mnd)
88	86, 87	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑆 ∈ Mnd)
89	6, 6	xpex 7742	. . . . 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 19140	. . . . . 6 ⊢ (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸 ∈ (𝑃 MndHom 𝑆))
93	91, 92	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝑃 MndHom 𝑆))
94	93	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐸 ∈ (𝑃 MndHom 𝑆))
95	14	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑃 ∈ Ring)
96	24	adantrr 713	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )) ∈ 𝐵)
97	31	adantrl 712	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )) ∈ 𝐵)
98	1, 2	ringcl 20144	. . . . . . 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 1369	. . . . . 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 3198	. . . . 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 2730	. . . . . 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 8056	. . . . 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 217	. . . 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 8068	. . . . . . 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 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 ))))
106	104, 105, 34	3pm3.2i 1337	. . . . . 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 )))) ∧ (0_g‘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 )))) ∧ (0_g‘𝑃) ∈ V))
108	70	fsuppimpd 9371	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)) ∈ Fin)
109	77	fsuppimpd 9371	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃)) ∈ Fin)
110		xpfi 9319	. . . . . 6 ⊢ ((((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)) ∈ Fin ∧ ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃)) ∈ Fin) → (((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)) × ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃))) ∈ Fin)
111	108, 109, 110	syl2anc 582	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)) × ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃))) ∈ Fin)
112	1, 3, 2, 15, 24, 31, 7, 7	evlslem4 21856	. . . . 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 (0_g‘𝑃)) ⊆ (((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)) × ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃))))
113		suppssfifsupp 9380	. . . . 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 )))) ∧ (0_g‘𝑃) ∈ V) ∧ ((((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)) × ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘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 (0_g‘𝑃)) ⊆ (((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)) × ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃))))) → (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(._r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) finSupp (0_g‘𝑃))
114	107, 111, 112, 113	syl12anc 833	. . . 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 (0_g‘𝑃))
115	1, 3, 82, 88, 90, 94, 103, 114	gsummhm 19847	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σ_g (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 Σ_g (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 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝐼 ∈ 𝑊)
117	9	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑅 ∈ CRing)
118		eqid 2730	. . . . . . . . . 10 ⊢ (._r‘𝑅) = (._r‘𝑅)
119		simprl 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑗 ∈ 𝐷)
120		simprr 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → 𝑖 ∈ 𝐷)
121	22	adantrr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑗 ∈ 𝐷 ∧ 𝑖 ∈ 𝐷)) → (𝑥‘𝑗) ∈ (Base‘𝑅))
122	29	adantrl 712	. . . . . . . . . 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 21849	. . . . . . . . 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 6894	. . . . . . . 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 466	. . . . . . . 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 2770	. . . . . . 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 1113	. . . . . 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 7488	. . . . 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 7427	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σ_g (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ (𝐸‘((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(._r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = (𝑆 Σ_g (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) · (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
131		eqidd 2731	. . . . . 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 2730	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
133	1, 132	ghmf 19134	. . . . . . . . 9 ⊢ (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸:𝐵⟶(Base‘𝑆))
134	91, 133	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐸:𝐵⟶(Base‘𝑆))
135	134	feqmptd 6959	. . . . . . 7 ⊢ (𝜑 → 𝐸 = (𝑧 ∈ 𝐵 ↦ (𝐸‘𝑧)))
136	135	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐸 = (𝑧 ∈ 𝐵 ↦ (𝐸‘𝑧)))
137		fveq2 6890	. . . . . 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 8083	. . . . 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 7427	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σ_g (𝐸 ∘ (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(._r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = (𝑆 Σ_g (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 2731	. . . . . . . 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 6890	. . . . . . . 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 7128	. . . . . . 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 7427	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σ_g (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) = (𝑆 Σ_g (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
144		eqidd 2731	. . . . . . . 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 6890	. . . . . . . 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 7128	. . . . . . 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 7427	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σ_g (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))) = (𝑆 Σ_g (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
148	143, 147	oveq12d 7429	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑆 Σ_g (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σ_g (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = ((𝑆 Σ_g (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σ_g (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
149		evlslem2.m	. . . . . 6 ⊢ · = (._r‘𝑆)
150		eqid 2730	. . . . . 6 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
151	134	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
152	151, 24	ffvelcdmd 7086	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑗 ∈ 𝐷) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) ∈ (Base‘𝑆))
153	134	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
154	153, 31	ffvelcdmd 7086	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑖 ∈ 𝐷) → (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) ∈ (Base‘𝑆))
155	6	mptex 7226	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) ∈ V
156		funmpt 6585	. . . . . . . . 9 ⊢ Fun (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))
157		fvex 6903	. . . . . . . . 9 ⊢ (0_g‘𝑆) ∈ V
158	155, 156, 157	3pm3.2i 1337	. . . . . . . 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 )))) ∧ (0_g‘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 )))) ∧ (0_g‘𝑆) ∈ V))
160		ssidd 4004	. . . . . . . . 9 ⊢ (𝜑 → ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)) ⊆ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)))
161	3, 150	ghmid 19136	. . . . . . . . . 10 ⊢ (𝐸 ∈ (𝑃 GrpHom 𝑆) → (𝐸‘(0_g‘𝑃)) = (0_g‘𝑆))
162	91, 161	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐸‘(0_g‘𝑃)) = (0_g‘𝑆))
163	6	mptex 7226	. . . . . . . . . 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 → (0_g‘𝑃) ∈ V)
166	160, 162, 164, 165	suppssfv 8189	. . . . . . . 8 ⊢ (𝜑 → ((𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) supp (0_g‘𝑆)) ⊆ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)))
167	166	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) supp (0_g‘𝑆)) ⊆ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)))
168		suppssfifsupp 9380	. . . . . . 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 )))) ∧ (0_g‘𝑆) ∈ V) ∧ (((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)) ∈ Fin ∧ ((𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) supp (0_g‘𝑆)) ⊆ ((𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))) supp (0_g‘𝑃)))) → (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) finSupp (0_g‘𝑆))
169	159, 108, 167, 168	syl12anc 833	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))) finSupp (0_g‘𝑆))
170	6	mptex 7226	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) ∈ V
171		funmpt 6585	. . . . . . . . 9 ⊢ Fun (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))
172	170, 171, 157	3pm3.2i 1337	. . . . . . . 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 )))) ∧ (0_g‘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 )))) ∧ (0_g‘𝑆) ∈ V))
174		ssidd 4004	. . . . . . . . 9 ⊢ (𝜑 → ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃)) ⊆ ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃)))
175	6	mptex 7226	. . . . . . . . . 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 8189	. . . . . . . 8 ⊢ (𝜑 → ((𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) supp (0_g‘𝑆)) ⊆ ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃)))
178	177	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) supp (0_g‘𝑆)) ⊆ ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃)))
179		suppssfifsupp 9380	. . . . . . 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 )))) ∧ (0_g‘𝑆) ∈ V) ∧ (((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃)) ∈ Fin ∧ ((𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) supp (0_g‘𝑆)) ⊆ ((𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))) supp (0_g‘𝑃)))) → (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) finSupp (0_g‘𝑆))
180	173, 109, 178, 179	syl12anc 833	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))) finSupp (0_g‘𝑆))
181	132, 149, 150, 7, 7, 86, 152, 154, 169, 180	gsumdixp 20207	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑆 Σ_g (𝑗 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σ_g (𝑖 ∈ 𝐷 ↦ (𝐸‘(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = (𝑆 Σ_g (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 2770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑆 Σ_g (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σ_g (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = (𝑆 Σ_g (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 2780	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σ_g (𝐸 ∘ (𝑗 ∈ 𝐷, 𝑖 ∈ 𝐷 ↦ ((𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))(._r‘𝑃)(𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = ((𝑆 Σ_g (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σ_g (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
184	79, 115, 183	3eqtr2d 2776	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘((𝑃 Σ_g (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))(._r‘𝑃)(𝑃 Σ_g (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))) = ((𝑆 Σ_g (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σ_g (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
185	8	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐼 ∈ 𝑊)
186	11	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ Ring)
187	12, 4, 16, 1, 185, 186, 20	mplcoe4 21851	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = (𝑃 Σ_g (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 )))))
188	12, 4, 16, 1, 185, 186, 27	mplcoe4 21851	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 = (𝑃 Σ_g (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))
189	187, 188	oveq12d 7429	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝑃)𝑦) = ((𝑃 Σ_g (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))(._r‘𝑃)(𝑃 Σ_g (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
190	189	fveq2d 6894	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘(𝑥(._r‘𝑃)𝑦)) = (𝐸‘((𝑃 Σ_g (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))(._r‘𝑃)(𝑃 Σ_g (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
191	187	fveq2d 6894	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑥) = (𝐸‘(𝑃 Σ_g (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
192	24	fmpttd 7115	. . . . 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 19847	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σ_g (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) = (𝐸‘(𝑃 Σ_g (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
194	191, 193	eqtr4d 2773	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑥) = (𝑆 Σ_g (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))))
195	188	fveq2d 6894	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑦) = (𝐸‘(𝑃 Σ_g (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
196	31	fmpttd 7115	. . . . 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 19847	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑆 Σ_g (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))) = (𝐸‘(𝑃 Σ_g (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
198	195, 197	eqtr4d 2773	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘𝑦) = (𝑆 Σ_g (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 ))))))
199	194, 198	oveq12d 7429	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐸‘𝑥) · (𝐸‘𝑦)) = ((𝑆 Σ_g (𝐸 ∘ (𝑗 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑗, (𝑥‘𝑗), 0 ))))) · (𝑆 Σ_g (𝐸 ∘ (𝑖 ∈ 𝐷 ↦ (𝑘 ∈ 𝐷 ↦ if(𝑘 = 𝑖, (𝑦‘𝑖), 0 )))))))
200	184, 190, 199	3eqtr4d 2780	1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐸‘(𝑥(._r‘𝑃)𝑦)) = ((𝐸‘𝑥) · (𝐸‘𝑦)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 {crab 3430 Vcvv 3472 ∖ cdif 3944 ⊆ wss 3947 ifcif 4527 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ^◡ccnv 5674 “ cima 5678 ∘ ccom 5679 Fun wfun 6536 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ∈ cmpo 7413 ∘_f cof 7670 supp csupp 8148 ↑_m cmap 8822 Fincfn 8941 finSupp cfsupp 9363 + caddc 11115 ℕcn 12216 ℕ₀cn0 12476 Basecbs 17148 ._rcmulr 17202 0_gc0g 17389 Σ_g cgsu 17390 Mndcmnd 18659 MndHom cmhm 18703 Grpcgrp 18855 GrpHom cghm 19127 CMndccmn 19689 Ringcrg 20127 CRingccrg 20128 mPoly cmpl 21678

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-subrng 20434 df-subrg 20459 df-lmod 20616 df-lss 20687 df-assa 21627 df-psr 21681 df-mpl 21683

This theorem is referenced by: evlslem1 21864

W3C validator