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Theorem evlslem2 22120
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 2734 . . . . 5 (.r𝑃) = (.r𝑃)
3 eqid 2734 . . . . 5 (0g𝑃) = (0g𝑃)
4 evlslem2.d . . . . . . 7 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
5 ovex 7478 . . . . . . 7 (ℕ0m 𝐼) ∈ V
64, 5rabex2 5362 . . . . . 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 20267 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1210, 11syl 17 . . . . . . 7 (𝜑𝑅 ∈ Ring)
138, 9, 12mplringd 22060 . . . . . 6 (𝜑𝑃 ∈ Ring)
1413adantr 480 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑃 ∈ Ring)
15 evlslem2.z . . . . . 6 0 = (0g𝑅)
16 eqid 2734 . . . . . 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 22035 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥:𝐷⟶(Base‘𝑅))
2120ffvelcdmda 7116 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝑥𝑗) ∈ (Base‘𝑅))
22 simpr 484 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝑗𝐷)
238, 4, 15, 16, 17, 18, 1, 21, 22mplmon2cl 22109 . . . . 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 22035 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦:𝐷⟶(Base‘𝑅))
2827ffvelcdmda 7116 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝑦𝑖) ∈ (Base‘𝑅))
29 simpr 484 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝑖𝐷)
308, 4, 15, 16, 24, 25, 1, 28, 29mplmon2cl 22109 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ 𝐵)
316mptex 7258 . . . . . . . . . . . 12 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∈ V
32 funmpt 6615 . . . . . . . . . . . 12 Fun (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
33 fvex 6932 . . . . . . . . . . . 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 22032 . . . . . . . . . . 11 ((𝜑𝑦𝐵) → 𝑦 finSupp 0 )
3938fsuppimpd 9435 . . . . . . . . . 10 ((𝜑𝑦𝐵) → (𝑦 supp 0 ) ∈ Fin)
408, 16, 1, 4, 36mplelf 22035 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 𝑦:𝐷⟶(Base‘𝑅))
41 ssidd 4026 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 ))
426a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 𝐷 ∈ V)
4315fvexi 6933 . . . . . . . . . . . . . . . . 17 0 ∈ V
4443a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 0 ∈ V)
4540, 41, 42, 44suppssr 8232 . . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . . 13 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = 0 )
4948mpteq2dv 5271 . . . . . . . . . . . 12 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (𝑘𝐷0 ))
50 ringgrp 20260 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
5112, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑅 ∈ Grp)
528, 4, 15, 3, 9, 51mpl0 22043 . . . . . . . . . . . . . 14 (𝜑 → (0g𝑃) = (𝐷 × { 0 }))
53 fconstmpt 5761 . . . . . . . . . . . . . 14 (𝐷 × { 0 }) = (𝑘𝐷0 )
5452, 53eqtrdi 2790 . . . . . . . . . . . . 13 (𝜑 → (0g𝑃) = (𝑘𝐷0 ))
5554ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (0g𝑃) = (𝑘𝐷0 ))
5649, 55eqtr4d 2777 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (0g𝑃))
5756, 42suppss2 8237 . . . . . . . . . 10 ((𝜑𝑦𝐵) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) supp (0g𝑃)) ⊆ (𝑦 supp 0 ))
58 suppssfifsupp 9445 . . . . . . . . . 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 3148 . . . . . . . 8 (𝜑 → ∀𝑦𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
61 fveq1 6918 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦𝑗) = (𝑥𝑗))
6261ifeq1d 4567 . . . . . . . . . . . 12 (𝑦 = 𝑥 → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = if(𝑘 = 𝑗, (𝑥𝑗), 0 ))
6362mpteq2dv 5271 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))
6463mpteq2dv 5271 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
6564breq1d 5179 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃) ↔ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃)))
6665cbvralvw 3238 . . . . . . . 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 3252 . . . . . 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 6919 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑦𝑖) = (𝑦𝑗))
7270, 71ifbieq1d 4572 . . . . . . . 8 (𝑖 = 𝑗 → if(𝑘 = 𝑖, (𝑦𝑖), 0 ) = if(𝑘 = 𝑗, (𝑦𝑗), 0 ))
7372mpteq2dv 5271 . . . . . . 7 (𝑖 = 𝑗 → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
7473cbvmptv 5282 . . . . . 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 5204 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) finSupp (0g𝑃))
771, 2, 3, 7, 7, 14, 23, 30, 69, 76gsumdixp 20337 . . . 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 6923 . . 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 20300 . . . . . 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 20267 . . . . . . 7 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
8482, 83syl 17 . . . . . 6 (𝜑𝑆 ∈ Ring)
8584adantr 480 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Ring)
86 ringmnd 20265 . . . . 5 (𝑆 ∈ Ring → 𝑆 ∈ Mnd)
8785, 86syl 17 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Mnd)
886, 6xpex 7784 . . . . 5 (𝐷 × 𝐷) ∈ V
8988a1i 11 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐷 × 𝐷) ∈ V)
90 evlslem2.e1 . . . . . 6 (𝜑𝐸 ∈ (𝑃 GrpHom 𝑆))
91 ghmmhm 19261 . . . . . 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 20272 . . . . . . 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 3204 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑗𝐷𝑖𝐷 ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ 𝐵)
100 eqid 2734 . . . . . 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 8105 . . . . 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 8116 . . . . . . 7 (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V
104100mpofun 7570 . . . . . . 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 9435 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ∈ Fin)
10876fsuppimpd 9435 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ∈ Fin)
109 xpfi 9382 . . . . . 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 22117 . . . . 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 9445 . . . . 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 19975 . . 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 2734 . . . . . . . . . 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 22110 . . . . . . . . 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 6923 . . . . . . . 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 2774 . . . . . . 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 7523 . . . . 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 7461 . . . 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 2735 . . . . . 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 2734 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
1321, 131ghmf 19255 . . . . . . . . 9 (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸:𝐵⟶(Base‘𝑆))
13390, 132syl 17 . . . . . . . 8 (𝜑𝐸:𝐵⟶(Base‘𝑆))
134133feqmptd 6989 . . . . . . 7 (𝜑𝐸 = (𝑧𝐵 ↦ (𝐸𝑧)))
135134adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐸 = (𝑧𝐵 ↦ (𝐸𝑧)))
136 fveq2 6919 . . . . . 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 8132 . . . . 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 7461 . . . 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 2735 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
140 fveq2 6919 . . . . . . . 8 (𝑧 = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) → (𝐸𝑧) = (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
14123, 139, 135, 140fmptco 7161 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) = (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))))
142141oveq2d 7461 . . . . . 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 2735 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) = (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
144 fveq2 6919 . . . . . . . 8 (𝑧 = (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) → (𝐸𝑧) = (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
14530, 143, 135, 144fmptco 7161 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) = (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
146145oveq2d 7461 . . . . . 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 7463 . . . . 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 2734 . . . . . 6 (0g𝑆) = (0g𝑆)
150133ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
151150, 23ffvelcdmd 7117 . . . . . 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 7117 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ (Base‘𝑆))
1546mptex 7258 . . . . . . . . 9 (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∈ V
155 funmpt 6615 . . . . . . . . 9 Fun (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
156 fvex 6932 . . . . . . . . 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 4026 . . . . . . . . 9 (𝜑 → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))
1603, 149ghmid 19257 . . . . . . . . . 10 (𝐸 ∈ (𝑃 GrpHom 𝑆) → (𝐸‘(0g𝑃)) = (0g𝑆))
16190, 160syl 17 . . . . . . . . 9 (𝜑 → (𝐸‘(0g𝑃)) = (0g𝑆))
1626mptex 7258 . . . . . . . . . 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 8239 . . . . . . . 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 9445 . . . . . . 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 7258 . . . . . . . . 9 (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V
170 funmpt 6615 . . . . . . . . 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 4026 . . . . . . . . 9 (𝜑 → ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))
1746mptex 7258 . . . . . . . . . 10 (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ V
175174a1i 11 . . . . . . . . 9 ((𝜑𝑖𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ V)
176173, 161, 175, 164suppssfv 8239 . . . . . . . 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 9445 . . . . . . 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 20337 . . . . 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 2774 . . . 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 2784 . . 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 2780 . 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 22112 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 = (𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))))
1878, 4, 15, 1, 184, 185, 26mplcoe4 22112 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 = (𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
188186, 187oveq12d 7463 . . 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 6923 . 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 6923 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑥) = (𝐸‘(𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
19123fmpttd 7147 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))):𝐷𝐵)
1921, 3, 81, 87, 7, 93, 191, 69gsummhm 19975 . . . 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 2777 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑥) = (𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
194187fveq2d 6923 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑦) = (𝐸‘(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
19530fmpttd 7147 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))):𝐷𝐵)
1961, 3, 81, 87, 7, 93, 195, 76gsummhm 19975 . . . 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 2777 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑦) = (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
198193, 197oveq12d 7463 . 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 2784 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 2103  wral 3063  {crab 3438  Vcvv 3482  cdif 3967  wss 3970  ifcif 4548  {csn 4648   class class class wbr 5169  cmpt 5252   × cxp 5697  ccnv 5698  cima 5702  ccom 5703  Fun wfun 6566  wf 6568  cfv 6572  (class class class)co 7445  cmpo 7447  f cof 7708   supp csupp 8197  m cmap 8880  Fincfn 8999   finSupp cfsupp 9427   + caddc 11183  cn 12289  0cn0 12549  Basecbs 17253  .rcmulr 17307  0gc0g 17494   Σg cgsu 17495  Mndcmnd 18767   MndHom cmhm 18811  Grpcgrp 18968   GrpHom cghm 19247  CMndccmn 19817  Ringcrg 20255  CRingccrg 20256   mPoly cmpl 21943
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-addrcl 11241  ax-mulcl 11242  ax-mulrcl 11243  ax-mulcom 11244  ax-addass 11245  ax-mulass 11246  ax-distr 11247  ax-i2m1 11248  ax-1ne0 11249  ax-1rid 11250  ax-rnegex 11251  ax-rrecex 11252  ax-cnre 11253  ax-pre-lttri 11254  ax-pre-lttrn 11255  ax-pre-ltadd 11256  ax-pre-mulgt0 11257
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-iin 5022  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-se 5655  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-isom 6581  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-of 7710  df-ofr 7711  df-om 7900  df-1st 8026  df-2nd 8027  df-supp 8198  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-2o 8519  df-er 8759  df-map 8882  df-pm 8883  df-ixp 8952  df-en 9000  df-dom 9001  df-sdom 9002  df-fin 9003  df-fsupp 9428  df-sup 9507  df-oi 9575  df-card 10004  df-pnf 11322  df-mnf 11323  df-xr 11324  df-ltxr 11325  df-le 11326  df-sub 11518  df-neg 11519  df-nn 12290  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-7 12357  df-8 12358  df-9 12359  df-n0 12550  df-z 12636  df-dec 12755  df-uz 12900  df-fz 13564  df-fzo 13708  df-seq 14049  df-hash 14376  df-struct 17189  df-sets 17206  df-slot 17224  df-ndx 17236  df-base 17254  df-ress 17283  df-plusg 17319  df-mulr 17320  df-sca 17322  df-vsca 17323  df-ip 17324  df-tset 17325  df-ple 17326  df-ds 17328  df-hom 17330  df-cco 17331  df-0g 17496  df-gsum 17497  df-prds 17502  df-pws 17504  df-mre 17639  df-mrc 17640  df-acs 17642  df-mgm 18673  df-sgrp 18752  df-mnd 18768  df-mhm 18813  df-submnd 18814  df-grp 18971  df-minusg 18972  df-sbg 18973  df-mulg 19103  df-subg 19158  df-ghm 19248  df-cntz 19352  df-cmn 19819  df-abl 19820  df-mgp 20157  df-rng 20175  df-ur 20204  df-ring 20257  df-cring 20258  df-subrng 20567  df-subrg 20592  df-lmod 20877  df-lss 20948  df-assa 21891  df-psr 21946  df-mpl 21948
This theorem is referenced by:  evlslem1  22123
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