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Theorem evlslem2 21634
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 2733 . . . . 5 (.r𝑃) = (.r𝑃)
3 eqid 2733 . . . . 5 (0g𝑃) = (0g𝑃)
4 evlslem2.d . . . . . . 7 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
5 ovex 7439 . . . . . . 7 (ℕ0m 𝐼) ∈ V
64, 5rabex2 5334 . . . . . 6 𝐷 ∈ V
76a1i 11 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐷 ∈ V)
8 evlslem2.i . . . . . . 7 (𝜑𝐼𝑊)
9 evlslem2.r . . . . . . . 8 (𝜑𝑅 ∈ CRing)
10 crngring 20062 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
119, 10syl 17 . . . . . . 7 (𝜑𝑅 ∈ Ring)
12 evlslem2.p . . . . . . . 8 𝑃 = (𝐼 mPoly 𝑅)
1312mplring 21570 . . . . . . 7 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
148, 11, 13syl2anc 585 . . . . . 6 (𝜑𝑃 ∈ Ring)
1514adantr 482 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑃 ∈ Ring)
16 evlslem2.z . . . . . 6 0 = (0g𝑅)
17 eqid 2733 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
188ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝐼𝑊)
1911ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝑅 ∈ Ring)
20 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
2112, 17, 1, 4, 20mplelf 21549 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥:𝐷⟶(Base‘𝑅))
2221ffvelcdmda 7084 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝑥𝑗) ∈ (Base‘𝑅))
23 simpr 486 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝑗𝐷)
2412, 4, 16, 17, 18, 19, 1, 22, 23mplmon2cl 21621 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ 𝐵)
258ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝐼𝑊)
2611ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝑅 ∈ Ring)
27 simprr 772 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
2812, 17, 1, 4, 27mplelf 21549 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦:𝐷⟶(Base‘𝑅))
2928ffvelcdmda 7084 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝑦𝑖) ∈ (Base‘𝑅))
30 simpr 486 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝑖𝐷)
3112, 4, 16, 17, 25, 26, 1, 29, 30mplmon2cl 21621 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ 𝐵)
326mptex 7222 . . . . . . . . . . . 12 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∈ V
33 funmpt 6584 . . . . . . . . . . . 12 Fun (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
34 fvex 6902 . . . . . . . . . . . 12 (0g𝑃) ∈ V
3532, 33, 343pm3.2i 1340 . . . . . . . . . . 11 ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∈ V ∧ Fun (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∧ (0g𝑃) ∈ V)
3635a1i 11 . . . . . . . . . 10 ((𝜑𝑦𝐵) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∈ V ∧ Fun (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∧ (0g𝑃) ∈ V))
37 simpr 486 . . . . . . . . . . . 12 ((𝜑𝑦𝐵) → 𝑦𝐵)
389adantr 482 . . . . . . . . . . . 12 ((𝜑𝑦𝐵) → 𝑅 ∈ CRing)
3912, 1, 16, 37, 38mplelsfi 21546 . . . . . . . . . . 11 ((𝜑𝑦𝐵) → 𝑦 finSupp 0 )
4039fsuppimpd 9366 . . . . . . . . . 10 ((𝜑𝑦𝐵) → (𝑦 supp 0 ) ∈ Fin)
4112, 17, 1, 4, 37mplelf 21549 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 𝑦:𝐷⟶(Base‘𝑅))
42 ssidd 4005 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 ))
436a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 𝐷 ∈ V)
4416fvexi 6903 . . . . . . . . . . . . . . . . 17 0 ∈ V
4544a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 0 ∈ V)
4641, 42, 43, 45suppssr 8178 . . . . . . . . . . . . . . 15 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑦𝑗) = 0 )
4746ifeq1d 4547 . . . . . . . . . . . . . 14 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = if(𝑘 = 𝑗, 0 , 0 ))
48 ifid 4568 . . . . . . . . . . . . . 14 if(𝑘 = 𝑗, 0 , 0 ) = 0
4947, 48eqtrdi 2789 . . . . . . . . . . . . 13 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = 0 )
5049mpteq2dv 5250 . . . . . . . . . . . 12 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (𝑘𝐷0 ))
51 ringgrp 20055 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
5211, 51syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑅 ∈ Grp)
5312, 4, 16, 3, 8, 52mpl0 21557 . . . . . . . . . . . . . 14 (𝜑 → (0g𝑃) = (𝐷 × { 0 }))
54 fconstmpt 5737 . . . . . . . . . . . . . 14 (𝐷 × { 0 }) = (𝑘𝐷0 )
5553, 54eqtrdi 2789 . . . . . . . . . . . . 13 (𝜑 → (0g𝑃) = (𝑘𝐷0 ))
5655ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (0g𝑃) = (𝑘𝐷0 ))
5750, 56eqtr4d 2776 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (0g𝑃))
5857, 43suppss2 8182 . . . . . . . . . 10 ((𝜑𝑦𝐵) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) supp (0g𝑃)) ⊆ (𝑦 supp 0 ))
59 suppssfifsupp 9375 . . . . . . . . . 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))
6036, 40, 58, 59syl12anc 836 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
6160ralrimiva 3147 . . . . . . . 8 (𝜑 → ∀𝑦𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
62 fveq1 6888 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦𝑗) = (𝑥𝑗))
6362ifeq1d 4547 . . . . . . . . . . . 12 (𝑦 = 𝑥 → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = if(𝑘 = 𝑗, (𝑥𝑗), 0 ))
6463mpteq2dv 5250 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))
6564mpteq2dv 5250 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
6665breq1d 5158 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃) ↔ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃)))
6766cbvralvw 3235 . . . . . . . 8 (∀𝑦𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃) ↔ ∀𝑥𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
6861, 67sylib 217 . . . . . . 7 (𝜑 → ∀𝑥𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
6968r19.21bi 3249 . . . . . 6 ((𝜑𝑥𝐵) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
7069adantrr 716 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
71 equequ2 2030 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑘 = 𝑖𝑘 = 𝑗))
72 fveq2 6889 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑦𝑖) = (𝑦𝑗))
7371, 72ifbieq1d 4552 . . . . . . . 8 (𝑖 = 𝑗 → if(𝑘 = 𝑖, (𝑦𝑖), 0 ) = if(𝑘 = 𝑗, (𝑦𝑗), 0 ))
7473mpteq2dv 5250 . . . . . . 7 (𝑖 = 𝑗 → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
7574cbvmptv 5261 . . . . . 6 (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
7660adantrl 715 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
7775, 76eqbrtrid 5183 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) finSupp (0g𝑃))
781, 2, 3, 7, 7, 15, 24, 31, 70, 77gsumdixp 20125 . . . 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 ))))))
7978fveq2d 6893 . . 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 )))))))
80 ringcmn 20093 . . . . . 6 (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
8114, 80syl 17 . . . . 5 (𝜑𝑃 ∈ CMnd)
8281adantr 482 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑃 ∈ CMnd)
83 evlslem2.s . . . . . . 7 (𝜑𝑆 ∈ CRing)
84 crngring 20062 . . . . . . 7 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
8583, 84syl 17 . . . . . 6 (𝜑𝑆 ∈ Ring)
8685adantr 482 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Ring)
87 ringmnd 20060 . . . . 5 (𝑆 ∈ Ring → 𝑆 ∈ Mnd)
8886, 87syl 17 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Mnd)
896, 6xpex 7737 . . . . 5 (𝐷 × 𝐷) ∈ V
9089a1i 11 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐷 × 𝐷) ∈ V)
91 evlslem2.e1 . . . . . 6 (𝜑𝐸 ∈ (𝑃 GrpHom 𝑆))
92 ghmmhm 19097 . . . . . 6 (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸 ∈ (𝑃 MndHom 𝑆))
9391, 92syl 17 . . . . 5 (𝜑𝐸 ∈ (𝑃 MndHom 𝑆))
9493adantr 482 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐸 ∈ (𝑃 MndHom 𝑆))
9514ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑃 ∈ Ring)
9624adantrr 716 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ 𝐵)
9731adantrl 715 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ 𝐵)
981, 2ringcl 20067 . . . . . . 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)
9995, 96, 97, 98syl3anc 1372 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ 𝐵)
10099ralrimivva 3201 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑗𝐷𝑖𝐷 ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ 𝐵)
101 eqid 2733 . . . . . 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 ))))
102101fmpo 8051 . . . . 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)
103100, 102sylib 217 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))):(𝐷 × 𝐷)⟶𝐵)
1046, 6mpoex 8063 . . . . . . 7 (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V
105101mpofun 7529 . . . . . . 7 Fun (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
106104, 105, 343pm3.2i 1340 . . . . . 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)
107106a1i 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))
10870fsuppimpd 9366 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ∈ Fin)
10977fsuppimpd 9366 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ∈ Fin)
110 xpfi 9314 . . . . . 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)
111108, 109, 110syl2anc 585 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) × ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃))) ∈ Fin)
1121, 3, 2, 15, 24, 31, 7, 7evlslem4 21629 . . . . 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))))
113 suppssfifsupp 9375 . . . . 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))
114107, 111, 112, 113syl12anc 836 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) finSupp (0g𝑃))
1151, 3, 82, 88, 90, 94, 103, 114gsummhm 19801 . . 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 )))))))
1168ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝐼𝑊)
1179ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑅 ∈ CRing)
118 eqid 2733 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
119 simprl 770 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑗𝐷)
120 simprr 772 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑖𝐷)
12122adantrr 716 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑥𝑗) ∈ (Base‘𝑅))
12229adantrl 715 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑦𝑖) ∈ (Base‘𝑅))
12312, 4, 16, 17, 116, 117, 2, 118, 119, 120, 121, 122mplmon2mul 21622 . . . . . . . . 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 )))
124123fveq2d 6893 . . . . . . . 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 ))))
125 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 )))))
126125anassrs 469 . . . . . . . 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 )))))
127124, 126eqtrd 2773 . . . . . . 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 )))))
1281273impb 1116 . . . . . 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 )))))
129128mpoeq3dva 7483 . . . . 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 ))))))
130129oveq2d 7422 . . . 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 )))))))
131 eqidd 2734 . . . . . 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 )))))
132 eqid 2733 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
1331, 132ghmf 19091 . . . . . . . . 9 (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸:𝐵⟶(Base‘𝑆))
13491, 133syl 17 . . . . . . . 8 (𝜑𝐸:𝐵⟶(Base‘𝑆))
135134feqmptd 6958 . . . . . . 7 (𝜑𝐸 = (𝑧𝐵 ↦ (𝐸𝑧)))
136135adantr 482 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐸 = (𝑧𝐵 ↦ (𝐸𝑧)))
137 fveq2 6889 . . . . . 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 )))))
13899, 131, 136, 137fmpoco 8078 . . . . 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 ))))))
139138oveq2d 7422 . . . 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 )))))))
140 eqidd 2734 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
141 fveq2 6889 . . . . . . . 8 (𝑧 = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) → (𝐸𝑧) = (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
14224, 140, 136, 141fmptco 7124 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) = (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))))
143142oveq2d 7422 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))) = (𝑆 Σg (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
144 eqidd 2734 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) = (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
145 fveq2 6889 . . . . . . . 8 (𝑧 = (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) → (𝐸𝑧) = (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
14631, 144, 136, 145fmptco 7124 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) = (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
147146oveq2d 7422 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))) = (𝑆 Σg (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
148143, 147oveq12d 7424 . . . . 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 )))))))
149 evlslem2.m . . . . . 6 · = (.r𝑆)
150 eqid 2733 . . . . . 6 (0g𝑆) = (0g𝑆)
151134ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
152151, 24ffvelcdmd 7085 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) ∈ (Base‘𝑆))
153134ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
154153, 31ffvelcdmd 7085 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ (Base‘𝑆))
1556mptex 7222 . . . . . . . . 9 (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∈ V
156 funmpt 6584 . . . . . . . . 9 Fun (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
157 fvex 6902 . . . . . . . . 9 (0g𝑆) ∈ V
158155, 156, 1573pm3.2i 1340 . . . . . . . 8 ((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∈ V ∧ Fun (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∧ (0g𝑆) ∈ V)
159158a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∈ V ∧ Fun (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∧ (0g𝑆) ∈ V))
160 ssidd 4005 . . . . . . . . 9 (𝜑 → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))
1613, 150ghmid 19093 . . . . . . . . . 10 (𝐸 ∈ (𝑃 GrpHom 𝑆) → (𝐸‘(0g𝑃)) = (0g𝑆))
16291, 161syl 17 . . . . . . . . 9 (𝜑 → (𝐸‘(0g𝑃)) = (0g𝑆))
1636mptex 7222 . . . . . . . . . 10 (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ V
164163a1i 11 . . . . . . . . 9 ((𝜑𝑗𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ V)
16534a1i 11 . . . . . . . . 9 (𝜑 → (0g𝑃) ∈ V)
166160, 162, 164, 165suppssfv 8184 . . . . . . . 8 (𝜑 → ((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) supp (0g𝑆)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))
167166adantr 482 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) supp (0g𝑆)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))
168 suppssfifsupp 9375 . . . . . . 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))
169159, 108, 167, 168syl12anc 836 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) finSupp (0g𝑆))
1706mptex 7222 . . . . . . . . 9 (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V
171 funmpt 6584 . . . . . . . . 9 Fun (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
172170, 171, 1573pm3.2i 1340 . . . . . . . 8 ((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V ∧ Fun (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∧ (0g𝑆) ∈ V)
173172a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V ∧ Fun (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∧ (0g𝑆) ∈ V))
174 ssidd 4005 . . . . . . . . 9 (𝜑 → ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))
1756mptex 7222 . . . . . . . . . 10 (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ V
176175a1i 11 . . . . . . . . 9 ((𝜑𝑖𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ V)
177174, 162, 176, 165suppssfv 8184 . . . . . . . 8 (𝜑 → ((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) supp (0g𝑆)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))
178177adantr 482 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) supp (0g𝑆)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))
179 suppssfifsupp 9375 . . . . . . 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))
180173, 109, 178, 179syl12anc 836 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) finSupp (0g𝑆))
181132, 149, 150, 7, 7, 86, 152, 154, 169, 180gsumdixp 20125 . . . . 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 )))))))
182148, 181eqtrd 2773 . . . 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 )))))))
183130, 139, 1823eqtr4d 2783 . . 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 )))))))
18479, 115, 1833eqtr2d 2779 . 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 )))))))
1858adantr 482 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐼𝑊)
18611adantr 482 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
18712, 4, 16, 1, 185, 186, 20mplcoe4 21624 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 = (𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))))
18812, 4, 16, 1, 185, 186, 27mplcoe4 21624 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 = (𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
189187, 188oveq12d 7424 . . 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 ))))))
190189fveq2d 6893 . 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 )))))))
191187fveq2d 6893 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑥) = (𝐸‘(𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
19224fmpttd 7112 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))):𝐷𝐵)
1931, 3, 82, 88, 7, 94, 192, 70gsummhm 19801 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))) = (𝐸‘(𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
194191, 193eqtr4d 2776 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑥) = (𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
195188fveq2d 6893 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑦) = (𝐸‘(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
19631fmpttd 7112 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))):𝐷𝐵)
1971, 3, 82, 88, 7, 94, 196, 77gsummhm 19801 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))) = (𝐸‘(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
198195, 197eqtr4d 2776 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑦) = (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
199194, 198oveq12d 7424 . 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 )))))))
200184, 190, 1993eqtr4d 2783 1 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸‘(𝑥(.r𝑃)𝑦)) = ((𝐸𝑥) · (𝐸𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  {crab 3433  Vcvv 3475  cdif 3945  wss 3948  ifcif 4528  {csn 4628   class class class wbr 5148  cmpt 5231   × cxp 5674  ccnv 5675  cima 5679  ccom 5680  Fun wfun 6535  wf 6537  cfv 6541  (class class class)co 7406  cmpo 7408  f cof 7665   supp csupp 8143  m cmap 8817  Fincfn 8936   finSupp cfsupp 9358   + caddc 11110  cn 12209  0cn0 12469  Basecbs 17141  .rcmulr 17195  0gc0g 17382   Σg cgsu 17383  Mndcmnd 18622   MndHom cmhm 18666  Grpcgrp 18816   GrpHom cghm 19084  CMndccmn 19643  Ringcrg 20050  CRingccrg 20051   mPoly cmpl 21451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-ofr 7668  df-om 7853  df-1st 7972  df-2nd 7973  df-supp 8144  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-pm 8820  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fsupp 9359  df-sup 9434  df-oi 9502  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-fzo 13625  df-seq 13964  df-hash 14288  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-ress 17171  df-plusg 17207  df-mulr 17208  df-sca 17210  df-vsca 17211  df-ip 17212  df-tset 17213  df-ple 17214  df-ds 17216  df-hom 17218  df-cco 17219  df-0g 17384  df-gsum 17385  df-prds 17390  df-pws 17392  df-mre 17527  df-mrc 17528  df-acs 17530  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-mhm 18668  df-submnd 18669  df-grp 18819  df-minusg 18820  df-sbg 18821  df-mulg 18946  df-subg 18998  df-ghm 19085  df-cntz 19176  df-cmn 19645  df-abl 19646  df-mgp 19983  df-ur 20000  df-ring 20052  df-cring 20053  df-subrg 20354  df-lmod 20466  df-lss 20536  df-assa 21400  df-psr 21454  df-mpl 21456
This theorem is referenced by:  evlslem1  21637
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