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Theorem evlslem2 22128
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 · = (.r𝑆)
evlslem2.z 0 = (0g𝑅)
evlslem2.d 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
evlslem2.i (𝜑𝐼𝑊)
evlslem2.r (𝜑𝑅 ∈ CRing)
evlslem2.s (𝜑𝑆 ∈ CRing)
evlslem2.e1 (𝜑𝐸 ∈ (𝑃 GrpHom 𝑆))
evlslem2.e2 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑗𝐷𝑖𝐷))) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = (𝑗f + 𝑖), ((𝑥𝑗)(.r𝑅)(𝑦𝑖)), 0 ))) = ((𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) · (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
Assertion
Ref Expression
evlslem2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸‘(𝑥(.r𝑃)𝑦)) = ((𝐸𝑥) · (𝐸𝑦)))
Distinct variable groups:   𝜑,𝑖,𝑗,𝑘,𝑦   𝐵,𝑖,𝑗,𝑘,𝑥,𝑦   𝐷,𝑖,𝑗,𝑘,𝑥,𝑦   𝑖,𝐸,𝑗   ,𝐼,𝑖,𝑗,𝑘   · ,𝑖,𝑗   𝑃,𝑖,𝑗,𝑘,𝑥,𝑦   𝑅,,𝑖,𝑗,𝑘   𝑆,𝑖,𝑗   𝑖,𝑊,𝑗,𝑘   0 ,,𝑖,𝑗,𝑘,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,)   𝐵()   𝐷()   𝑃()   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦,,𝑘)   · (𝑥,𝑦,,𝑘)   𝐸(𝑥,𝑦,,𝑘)   𝐼(𝑥,𝑦)   𝑊(𝑥,𝑦,)

Proof of Theorem evlslem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 evlslem2.b . . . . 5 𝐵 = (Base‘𝑃)
2 eqid 2740 . . . . 5 (.r𝑃) = (.r𝑃)
3 eqid 2740 . . . . 5 (0g𝑃) = (0g𝑃)
4 evlslem2.d . . . . . . 7 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
5 ovex 7483 . . . . . . 7 (ℕ0m 𝐼) ∈ V
64, 5rabex2 5359 . . . . . 6 𝐷 ∈ V
76a1i 11 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐷 ∈ V)
8 evlslem2.p . . . . . . 7 𝑃 = (𝐼 mPoly 𝑅)
9 evlslem2.i . . . . . . 7 (𝜑𝐼𝑊)
10 evlslem2.r . . . . . . . 8 (𝜑𝑅 ∈ CRing)
11 crngring 20274 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1210, 11syl 17 . . . . . . 7 (𝜑𝑅 ∈ Ring)
138, 9, 12mplringd 22068 . . . . . 6 (𝜑𝑃 ∈ Ring)
1413adantr 480 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑃 ∈ Ring)
15 evlslem2.z . . . . . 6 0 = (0g𝑅)
16 eqid 2740 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
179ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝐼𝑊)
1812ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝑅 ∈ Ring)
19 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
208, 16, 1, 4, 19mplelf 22043 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥:𝐷⟶(Base‘𝑅))
2120ffvelcdmda 7120 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝑥𝑗) ∈ (Base‘𝑅))
22 simpr 484 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝑗𝐷)
238, 4, 15, 16, 17, 18, 1, 21, 22mplmon2cl 22117 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ 𝐵)
249ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝐼𝑊)
2512ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝑅 ∈ Ring)
26 simprr 772 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
278, 16, 1, 4, 26mplelf 22043 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦:𝐷⟶(Base‘𝑅))
2827ffvelcdmda 7120 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝑦𝑖) ∈ (Base‘𝑅))
29 simpr 484 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝑖𝐷)
308, 4, 15, 16, 24, 25, 1, 28, 29mplmon2cl 22117 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ 𝐵)
316mptex 7262 . . . . . . . . . . . 12 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∈ V
32 funmpt 6618 . . . . . . . . . . . 12 Fun (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
33 fvex 6935 . . . . . . . . . . . 12 (0g𝑃) ∈ V
3431, 32, 333pm3.2i 1339 . . . . . . . . . . 11 ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∈ V ∧ Fun (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∧ (0g𝑃) ∈ V)
3534a1i 11 . . . . . . . . . 10 ((𝜑𝑦𝐵) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∈ V ∧ Fun (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∧ (0g𝑃) ∈ V))
36 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑦𝐵) → 𝑦𝐵)
3710adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦𝐵) → 𝑅 ∈ CRing)
388, 1, 15, 36, 37mplelsfi 22040 . . . . . . . . . . 11 ((𝜑𝑦𝐵) → 𝑦 finSupp 0 )
3938fsuppimpd 9441 . . . . . . . . . 10 ((𝜑𝑦𝐵) → (𝑦 supp 0 ) ∈ Fin)
408, 16, 1, 4, 36mplelf 22043 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 𝑦:𝐷⟶(Base‘𝑅))
41 ssidd 4032 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 ))
426a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 𝐷 ∈ V)
4315fvexi 6936 . . . . . . . . . . . . . . . . 17 0 ∈ V
4443a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 0 ∈ V)
4540, 41, 42, 44suppssr 8238 . . . . . . . . . . . . . . 15 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑦𝑗) = 0 )
4645ifeq1d 4567 . . . . . . . . . . . . . 14 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = if(𝑘 = 𝑗, 0 , 0 ))
47 ifid 4588 . . . . . . . . . . . . . 14 if(𝑘 = 𝑗, 0 , 0 ) = 0
4846, 47eqtrdi 2796 . . . . . . . . . . . . 13 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = 0 )
4948mpteq2dv 5268 . . . . . . . . . . . 12 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (𝑘𝐷0 ))
50 ringgrp 20267 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
5112, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑅 ∈ Grp)
528, 4, 15, 3, 9, 51mpl0 22051 . . . . . . . . . . . . . 14 (𝜑 → (0g𝑃) = (𝐷 × { 0 }))
53 fconstmpt 5762 . . . . . . . . . . . . . 14 (𝐷 × { 0 }) = (𝑘𝐷0 )
5452, 53eqtrdi 2796 . . . . . . . . . . . . 13 (𝜑 → (0g𝑃) = (𝑘𝐷0 ))
5554ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (0g𝑃) = (𝑘𝐷0 ))
5649, 55eqtr4d 2783 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (0g𝑃))
5756, 42suppss2 8243 . . . . . . . . . 10 ((𝜑𝑦𝐵) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) supp (0g𝑃)) ⊆ (𝑦 supp 0 ))
58 suppssfifsupp 9451 . . . . . . . . . 10 ((((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∈ V ∧ Fun (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∧ (0g𝑃) ∈ V) ∧ ((𝑦 supp 0 ) ∈ Fin ∧ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) supp (0g𝑃)) ⊆ (𝑦 supp 0 ))) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
5935, 39, 57, 58syl12anc 836 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
6059ralrimiva 3152 . . . . . . . 8 (𝜑 → ∀𝑦𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
61 fveq1 6921 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦𝑗) = (𝑥𝑗))
6261ifeq1d 4567 . . . . . . . . . . . 12 (𝑦 = 𝑥 → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = if(𝑘 = 𝑗, (𝑥𝑗), 0 ))
6362mpteq2dv 5268 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))
6463mpteq2dv 5268 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
6564breq1d 5176 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃) ↔ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃)))
6665cbvralvw 3243 . . . . . . . 8 (∀𝑦𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃) ↔ ∀𝑥𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
6760, 66sylib 218 . . . . . . 7 (𝜑 → ∀𝑥𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
6867r19.21bi 3257 . . . . . 6 ((𝜑𝑥𝐵) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
6968adantrr 716 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
70 equequ2 2025 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑘 = 𝑖𝑘 = 𝑗))
71 fveq2 6922 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑦𝑖) = (𝑦𝑗))
7270, 71ifbieq1d 4572 . . . . . . . 8 (𝑖 = 𝑗 → if(𝑘 = 𝑖, (𝑦𝑖), 0 ) = if(𝑘 = 𝑗, (𝑦𝑗), 0 ))
7372mpteq2dv 5268 . . . . . . 7 (𝑖 = 𝑗 → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
7473cbvmptv 5279 . . . . . 6 (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
7559adantrl 715 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
7674, 75eqbrtrid 5201 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) finSupp (0g𝑃))
771, 2, 3, 7, 7, 14, 23, 30, 69, 76gsumdixp 20344 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))(.r𝑃)(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))) = (𝑃 Σg (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
7877fveq2d 6926 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸‘((𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))(.r𝑃)(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))) = (𝐸‘(𝑃 Σg (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))))
79 ringcmn 20307 . . . . . 6 (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
8013, 79syl 17 . . . . 5 (𝜑𝑃 ∈ CMnd)
8180adantr 480 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑃 ∈ CMnd)
82 evlslem2.s . . . . . . 7 (𝜑𝑆 ∈ CRing)
83 crngring 20274 . . . . . . 7 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
8482, 83syl 17 . . . . . 6 (𝜑𝑆 ∈ Ring)
8584adantr 480 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Ring)
86 ringmnd 20272 . . . . 5 (𝑆 ∈ Ring → 𝑆 ∈ Mnd)
8785, 86syl 17 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Mnd)
886, 6xpex 7790 . . . . 5 (𝐷 × 𝐷) ∈ V
8988a1i 11 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐷 × 𝐷) ∈ V)
90 evlslem2.e1 . . . . . 6 (𝜑𝐸 ∈ (𝑃 GrpHom 𝑆))
91 ghmmhm 19268 . . . . . 6 (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸 ∈ (𝑃 MndHom 𝑆))
9290, 91syl 17 . . . . 5 (𝜑𝐸 ∈ (𝑃 MndHom 𝑆))
9392adantr 480 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐸 ∈ (𝑃 MndHom 𝑆))
9413ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑃 ∈ Ring)
9523adantrr 716 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ 𝐵)
9630adantrl 715 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ 𝐵)
971, 2ringcl 20279 . . . . . . 7 ((𝑃 ∈ Ring ∧ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ 𝐵 ∧ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ 𝐵) → ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ 𝐵)
9894, 95, 96, 97syl3anc 1371 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ 𝐵)
9998ralrimivva 3208 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑗𝐷𝑖𝐷 ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ 𝐵)
100 eqid 2740 . . . . . 6 (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) = (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
101100fmpo 8111 . . . . 5 (∀𝑗𝐷𝑖𝐷 ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ 𝐵 ↔ (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))):(𝐷 × 𝐷)⟶𝐵)
10299, 101sylib 218 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))):(𝐷 × 𝐷)⟶𝐵)
1036, 6mpoex 8122 . . . . . . 7 (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V
104100mpofun 7576 . . . . . . 7 Fun (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
105103, 104, 333pm3.2i 1339 . . . . . 6 ((𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V ∧ Fun (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∧ (0g𝑃) ∈ V)
106105a1i 11 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V ∧ Fun (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∧ (0g𝑃) ∈ V))
10769fsuppimpd 9441 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ∈ Fin)
10876fsuppimpd 9441 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ∈ Fin)
109 xpfi 9388 . . . . . 6 ((((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ∈ Fin ∧ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ∈ Fin) → (((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) × ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃))) ∈ Fin)
110107, 108, 109syl2anc 583 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) × ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃))) ∈ Fin)
1111, 3, 2, 14, 23, 30, 7, 7evlslem4 22125 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) supp (0g𝑃)) ⊆ (((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) × ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃))))
112 suppssfifsupp 9451 . . . . 5 ((((𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V ∧ Fun (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∧ (0g𝑃) ∈ V) ∧ ((((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) × ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃))) ∈ Fin ∧ ((𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) supp (0g𝑃)) ⊆ (((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) × ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃))))) → (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) finSupp (0g𝑃))
113106, 110, 111, 112syl12anc 836 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) finSupp (0g𝑃))
1141, 3, 81, 87, 89, 93, 102, 113gsummhm 19982 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))) = (𝐸‘(𝑃 Σg (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))))
1159ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝐼𝑊)
11610ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑅 ∈ CRing)
117 eqid 2740 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
118 simprl 770 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑗𝐷)
119 simprr 772 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑖𝐷)
12021adantrr 716 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑥𝑗) ∈ (Base‘𝑅))
12128adantrl 715 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑦𝑖) ∈ (Base‘𝑅))
1228, 4, 15, 16, 115, 116, 2, 117, 118, 119, 120, 121mplmon2mul 22118 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) = (𝑘𝐷 ↦ if(𝑘 = (𝑗f + 𝑖), ((𝑥𝑗)(.r𝑅)(𝑦𝑖)), 0 )))
123122fveq2d 6926 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝐸‘((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) = (𝐸‘(𝑘𝐷 ↦ if(𝑘 = (𝑗f + 𝑖), ((𝑥𝑗)(.r𝑅)(𝑦𝑖)), 0 ))))
124 evlslem2.e2 . . . . . . . . 9 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑗𝐷𝑖𝐷))) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = (𝑗f + 𝑖), ((𝑥𝑗)(.r𝑅)(𝑦𝑖)), 0 ))) = ((𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) · (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
125124anassrs 467 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = (𝑗f + 𝑖), ((𝑥𝑗)(.r𝑅)(𝑦𝑖)), 0 ))) = ((𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) · (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
126123, 125eqtrd 2780 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝐸‘((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) = ((𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) · (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
1271263impb 1115 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷𝑖𝐷) → (𝐸‘((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) = ((𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) · (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
128127mpoeq3dva 7529 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷, 𝑖𝐷 ↦ (𝐸‘((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))) = (𝑗𝐷, 𝑖𝐷 ↦ ((𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) · (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
129128oveq2d 7466 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝑗𝐷, 𝑖𝐷 ↦ (𝐸‘((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))) = (𝑆 Σg (𝑗𝐷, 𝑖𝐷 ↦ ((𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) · (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))))
130 eqidd 2741 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) = (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
131 eqid 2740 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
1321, 131ghmf 19262 . . . . . . . . 9 (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸:𝐵⟶(Base‘𝑆))
13390, 132syl 17 . . . . . . . 8 (𝜑𝐸:𝐵⟶(Base‘𝑆))
134133feqmptd 6992 . . . . . . 7 (𝜑𝐸 = (𝑧𝐵 ↦ (𝐸𝑧)))
135134adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐸 = (𝑧𝐵 ↦ (𝐸𝑧)))
136 fveq2 6922 . . . . . 6 (𝑧 = ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) → (𝐸𝑧) = (𝐸‘((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
13798, 130, 135, 136fmpoco 8138 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸 ∘ (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))) = (𝑗𝐷, 𝑖𝐷 ↦ (𝐸‘((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
138137oveq2d 7466 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))) = (𝑆 Σg (𝑗𝐷, 𝑖𝐷 ↦ (𝐸‘((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))))
139 eqidd 2741 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
140 fveq2 6922 . . . . . . . 8 (𝑧 = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) → (𝐸𝑧) = (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
14123, 139, 135, 140fmptco 7165 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) = (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))))
142141oveq2d 7466 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))) = (𝑆 Σg (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
143 eqidd 2741 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) = (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
144 fveq2 6922 . . . . . . . 8 (𝑧 = (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) → (𝐸𝑧) = (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
14530, 143, 135, 144fmptco 7165 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) = (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
146145oveq2d 7466 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))) = (𝑆 Σg (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
147142, 146oveq12d 7468 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))) · (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))) = ((𝑆 Σg (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))) · (𝑆 Σg (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))))
148 evlslem2.m . . . . . 6 · = (.r𝑆)
149 eqid 2740 . . . . . 6 (0g𝑆) = (0g𝑆)
150133ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
151150, 23ffvelcdmd 7121 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) ∈ (Base‘𝑆))
152133ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
153152, 30ffvelcdmd 7121 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ (Base‘𝑆))
1546mptex 7262 . . . . . . . . 9 (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∈ V
155 funmpt 6618 . . . . . . . . 9 Fun (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
156 fvex 6935 . . . . . . . . 9 (0g𝑆) ∈ V
157154, 155, 1563pm3.2i 1339 . . . . . . . 8 ((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∈ V ∧ Fun (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∧ (0g𝑆) ∈ V)
158157a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∈ V ∧ Fun (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∧ (0g𝑆) ∈ V))
159 ssidd 4032 . . . . . . . . 9 (𝜑 → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))
1603, 149ghmid 19264 . . . . . . . . . 10 (𝐸 ∈ (𝑃 GrpHom 𝑆) → (𝐸‘(0g𝑃)) = (0g𝑆))
16190, 160syl 17 . . . . . . . . 9 (𝜑 → (𝐸‘(0g𝑃)) = (0g𝑆))
1626mptex 7262 . . . . . . . . . 10 (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ V
163162a1i 11 . . . . . . . . 9 ((𝜑𝑗𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ V)
16433a1i 11 . . . . . . . . 9 (𝜑 → (0g𝑃) ∈ V)
165159, 161, 163, 164suppssfv 8245 . . . . . . . 8 (𝜑 → ((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) supp (0g𝑆)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))
166165adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) supp (0g𝑆)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))
167 suppssfifsupp 9451 . . . . . . 7 ((((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∈ V ∧ Fun (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∧ (0g𝑆) ∈ V) ∧ (((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ∈ Fin ∧ ((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) supp (0g𝑆)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))) → (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) finSupp (0g𝑆))
168158, 107, 166, 167syl12anc 836 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) finSupp (0g𝑆))
1696mptex 7262 . . . . . . . . 9 (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V
170 funmpt 6618 . . . . . . . . 9 Fun (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
171169, 170, 1563pm3.2i 1339 . . . . . . . 8 ((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V ∧ Fun (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∧ (0g𝑆) ∈ V)
172171a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V ∧ Fun (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∧ (0g𝑆) ∈ V))
173 ssidd 4032 . . . . . . . . 9 (𝜑 → ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))
1746mptex 7262 . . . . . . . . . 10 (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ V
175174a1i 11 . . . . . . . . 9 ((𝜑𝑖𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ V)
176173, 161, 175, 164suppssfv 8245 . . . . . . . 8 (𝜑 → ((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) supp (0g𝑆)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))
177176adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) supp (0g𝑆)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))
178 suppssfifsupp 9451 . . . . . . 7 ((((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V ∧ Fun (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∧ (0g𝑆) ∈ V) ∧ (((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ∈ Fin ∧ ((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) supp (0g𝑆)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))) → (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) finSupp (0g𝑆))
179172, 108, 177, 178syl12anc 836 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) finSupp (0g𝑆))
180131, 148, 149, 7, 7, 85, 151, 153, 168, 179gsumdixp 20344 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑆 Σg (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))) · (𝑆 Σg (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))) = (𝑆 Σg (𝑗𝐷, 𝑖𝐷 ↦ ((𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) · (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))))
181147, 180eqtrd 2780 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))) · (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))) = (𝑆 Σg (𝑗𝐷, 𝑖𝐷 ↦ ((𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) · (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))))
182129, 138, 1813eqtr4d 2790 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))) = ((𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))) · (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))))
18378, 114, 1823eqtr2d 2786 . 2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸‘((𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))(.r𝑃)(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))) = ((𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))) · (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))))
1849adantr 480 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐼𝑊)
18512adantr 480 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
1868, 4, 15, 1, 184, 185, 19mplcoe4 22120 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 = (𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))))
1878, 4, 15, 1, 184, 185, 26mplcoe4 22120 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 = (𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
188186, 187oveq12d 7468 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝑃)𝑦) = ((𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))(.r𝑃)(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
189188fveq2d 6926 . 2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸‘(𝑥(.r𝑃)𝑦)) = (𝐸‘((𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))(.r𝑃)(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))))
190186fveq2d 6926 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑥) = (𝐸‘(𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
19123fmpttd 7151 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))):𝐷𝐵)
1921, 3, 81, 87, 7, 93, 191, 69gsummhm 19982 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))) = (𝐸‘(𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
193190, 192eqtr4d 2783 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑥) = (𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
194187fveq2d 6926 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑦) = (𝐸‘(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
19530fmpttd 7151 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))):𝐷𝐵)
1961, 3, 81, 87, 7, 93, 195, 76gsummhm 19982 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))) = (𝐸‘(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
197194, 196eqtr4d 2783 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑦) = (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
198193, 197oveq12d 7468 . 2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐸𝑥) · (𝐸𝑦)) = ((𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))) · (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))))
199183, 189, 1983eqtr4d 2790 1 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸‘(𝑥(.r𝑃)𝑦)) = ((𝐸𝑥) · (𝐸𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1537  wcel 2108  wral 3067  {crab 3443  Vcvv 3488  cdif 3973  wss 3976  ifcif 4548  {csn 4648   class class class wbr 5166  cmpt 5249   × cxp 5698  ccnv 5699  cima 5703  ccom 5704  Fun wfun 6569  wf 6571  cfv 6575  (class class class)co 7450  cmpo 7452  f cof 7714   supp csupp 8203  m cmap 8886  Fincfn 9005   finSupp cfsupp 9433   + caddc 11189  cn 12295  0cn0 12555  Basecbs 17260  .rcmulr 17314  0gc0g 17501   Σg cgsu 17502  Mndcmnd 18774   MndHom cmhm 18818  Grpcgrp 18975   GrpHom cghm 19254  CMndccmn 19824  Ringcrg 20262  CRingccrg 20263   mPoly cmpl 21951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-cnex 11242  ax-resscn 11243  ax-1cn 11244  ax-icn 11245  ax-addcl 11246  ax-addrcl 11247  ax-mulcl 11248  ax-mulrcl 11249  ax-mulcom 11250  ax-addass 11251  ax-mulass 11252  ax-distr 11253  ax-i2m1 11254  ax-1ne0 11255  ax-1rid 11256  ax-rnegex 11257  ax-rrecex 11258  ax-cnre 11259  ax-pre-lttri 11260  ax-pre-lttrn 11261  ax-pre-ltadd 11262  ax-pre-mulgt0 11263
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-isom 6584  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-of 7716  df-ofr 7717  df-om 7906  df-1st 8032  df-2nd 8033  df-supp 8204  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-1o 8524  df-2o 8525  df-er 8765  df-map 8888  df-pm 8889  df-ixp 8958  df-en 9006  df-dom 9007  df-sdom 9008  df-fin 9009  df-fsupp 9434  df-sup 9513  df-oi 9581  df-card 10010  df-pnf 11328  df-mnf 11329  df-xr 11330  df-ltxr 11331  df-le 11332  df-sub 11524  df-neg 11525  df-nn 12296  df-2 12358  df-3 12359  df-4 12360  df-5 12361  df-6 12362  df-7 12363  df-8 12364  df-9 12365  df-n0 12556  df-z 12642  df-dec 12761  df-uz 12906  df-fz 13570  df-fzo 13714  df-seq 14055  df-hash 14382  df-struct 17196  df-sets 17213  df-slot 17231  df-ndx 17243  df-base 17261  df-ress 17290  df-plusg 17326  df-mulr 17327  df-sca 17329  df-vsca 17330  df-ip 17331  df-tset 17332  df-ple 17333  df-ds 17335  df-hom 17337  df-cco 17338  df-0g 17503  df-gsum 17504  df-prds 17509  df-pws 17511  df-mre 17646  df-mrc 17647  df-acs 17649  df-mgm 18680  df-sgrp 18759  df-mnd 18775  df-mhm 18820  df-submnd 18821  df-grp 18978  df-minusg 18979  df-sbg 18980  df-mulg 19110  df-subg 19165  df-ghm 19255  df-cntz 19359  df-cmn 19826  df-abl 19827  df-mgp 20164  df-rng 20182  df-ur 20211  df-ring 20264  df-cring 20265  df-subrng 20574  df-subrg 20599  df-lmod 20884  df-lss 20955  df-assa 21898  df-psr 21954  df-mpl 21956
This theorem is referenced by:  evlslem1  22131
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