MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  evlslem2 Structured version   Visualization version   GIF version

Theorem evlslem2 21489
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 2736 . . . . 5 (.r𝑃) = (.r𝑃)
3 eqid 2736 . . . . 5 (0g𝑃) = (0g𝑃)
4 evlslem2.d . . . . . . 7 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
5 ovex 7390 . . . . . . 7 (ℕ0m 𝐼) ∈ V
64, 5rabex2 5291 . . . . . 6 𝐷 ∈ V
76a1i 11 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐷 ∈ V)
8 evlslem2.i . . . . . . 7 (𝜑𝐼𝑊)
9 evlslem2.r . . . . . . . 8 (𝜑𝑅 ∈ CRing)
10 crngring 19976 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
119, 10syl 17 . . . . . . 7 (𝜑𝑅 ∈ Ring)
12 evlslem2.p . . . . . . . 8 𝑃 = (𝐼 mPoly 𝑅)
1312mplring 21424 . . . . . . 7 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
148, 11, 13syl2anc 584 . . . . . 6 (𝜑𝑃 ∈ Ring)
1514adantr 481 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑃 ∈ Ring)
16 evlslem2.z . . . . . 6 0 = (0g𝑅)
17 eqid 2736 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
188ad2antrr 724 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝐼𝑊)
1911ad2antrr 724 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝑅 ∈ Ring)
20 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
2112, 17, 1, 4, 20mplelf 21404 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥:𝐷⟶(Base‘𝑅))
2221ffvelcdmda 7035 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝑥𝑗) ∈ (Base‘𝑅))
23 simpr 485 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝑗𝐷)
2412, 4, 16, 17, 18, 19, 1, 22, 23mplmon2cl 21476 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ 𝐵)
258ad2antrr 724 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝐼𝑊)
2611ad2antrr 724 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝑅 ∈ Ring)
27 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
2812, 17, 1, 4, 27mplelf 21404 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦:𝐷⟶(Base‘𝑅))
2928ffvelcdmda 7035 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝑦𝑖) ∈ (Base‘𝑅))
30 simpr 485 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝑖𝐷)
3112, 4, 16, 17, 25, 26, 1, 29, 30mplmon2cl 21476 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ 𝐵)
326mptex 7173 . . . . . . . . . . . 12 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∈ V
33 funmpt 6539 . . . . . . . . . . . 12 Fun (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
34 fvex 6855 . . . . . . . . . . . 12 (0g𝑃) ∈ V
3532, 33, 343pm3.2i 1339 . . . . . . . . . . 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 485 . . . . . . . . . . . 12 ((𝜑𝑦𝐵) → 𝑦𝐵)
389adantr 481 . . . . . . . . . . . 12 ((𝜑𝑦𝐵) → 𝑅 ∈ CRing)
3912, 1, 16, 37, 38mplelsfi 21401 . . . . . . . . . . 11 ((𝜑𝑦𝐵) → 𝑦 finSupp 0 )
4039fsuppimpd 9312 . . . . . . . . . 10 ((𝜑𝑦𝐵) → (𝑦 supp 0 ) ∈ Fin)
4112, 17, 1, 4, 37mplelf 21404 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 𝑦:𝐷⟶(Base‘𝑅))
42 ssidd 3967 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 ))
436a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 𝐷 ∈ V)
4416fvexi 6856 . . . . . . . . . . . . . . . . 17 0 ∈ V
4544a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 0 ∈ V)
4641, 42, 43, 45suppssr 8127 . . . . . . . . . . . . . . 15 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑦𝑗) = 0 )
4746ifeq1d 4505 . . . . . . . . . . . . . 14 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = if(𝑘 = 𝑗, 0 , 0 ))
48 ifid 4526 . . . . . . . . . . . . . 14 if(𝑘 = 𝑗, 0 , 0 ) = 0
4947, 48eqtrdi 2792 . . . . . . . . . . . . 13 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = 0 )
5049mpteq2dv 5207 . . . . . . . . . . . 12 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (𝑘𝐷0 ))
51 ringgrp 19969 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
5211, 51syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑅 ∈ Grp)
5312, 4, 16, 3, 8, 52mpl0 21412 . . . . . . . . . . . . . 14 (𝜑 → (0g𝑃) = (𝐷 × { 0 }))
54 fconstmpt 5694 . . . . . . . . . . . . . 14 (𝐷 × { 0 }) = (𝑘𝐷0 )
5553, 54eqtrdi 2792 . . . . . . . . . . . . 13 (𝜑 → (0g𝑃) = (𝑘𝐷0 ))
5655ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (0g𝑃) = (𝑘𝐷0 ))
5750, 56eqtr4d 2779 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (0g𝑃))
5857, 43suppss2 8131 . . . . . . . . . 10 ((𝜑𝑦𝐵) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) supp (0g𝑃)) ⊆ (𝑦 supp 0 ))
59 suppssfifsupp 9320 . . . . . . . . . 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 835 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
6160ralrimiva 3143 . . . . . . . 8 (𝜑 → ∀𝑦𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
62 fveq1 6841 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦𝑗) = (𝑥𝑗))
6362ifeq1d 4505 . . . . . . . . . . . 12 (𝑦 = 𝑥 → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = if(𝑘 = 𝑗, (𝑥𝑗), 0 ))
6463mpteq2dv 5207 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))
6564mpteq2dv 5207 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
6665breq1d 5115 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃) ↔ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃)))
6766cbvralvw 3225 . . . . . . . 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 3234 . . . . . 6 ((𝜑𝑥𝐵) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
7069adantrr 715 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
71 equequ2 2029 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑘 = 𝑖𝑘 = 𝑗))
72 fveq2 6842 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑦𝑖) = (𝑦𝑗))
7371, 72ifbieq1d 4510 . . . . . . . 8 (𝑖 = 𝑗 → if(𝑘 = 𝑖, (𝑦𝑖), 0 ) = if(𝑘 = 𝑗, (𝑦𝑗), 0 ))
7473mpteq2dv 5207 . . . . . . 7 (𝑖 = 𝑗 → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
7574cbvmptv 5218 . . . . . 6 (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
7660adantrl 714 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
7775, 76eqbrtrid 5140 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) finSupp (0g𝑃))
781, 2, 3, 7, 7, 15, 24, 31, 70, 77gsumdixp 20033 . . . 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 6846 . . 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 20003 . . . . . 6 (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
8114, 80syl 17 . . . . 5 (𝜑𝑃 ∈ CMnd)
8281adantr 481 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑃 ∈ CMnd)
83 evlslem2.s . . . . . . 7 (𝜑𝑆 ∈ CRing)
84 crngring 19976 . . . . . . 7 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
8583, 84syl 17 . . . . . 6 (𝜑𝑆 ∈ Ring)
8685adantr 481 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Ring)
87 ringmnd 19974 . . . . 5 (𝑆 ∈ Ring → 𝑆 ∈ Mnd)
8886, 87syl 17 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Mnd)
896, 6xpex 7687 . . . . 5 (𝐷 × 𝐷) ∈ V
9089a1i 11 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐷 × 𝐷) ∈ V)
91 evlslem2.e1 . . . . . 6 (𝜑𝐸 ∈ (𝑃 GrpHom 𝑆))
92 ghmmhm 19018 . . . . . 6 (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸 ∈ (𝑃 MndHom 𝑆))
9391, 92syl 17 . . . . 5 (𝜑𝐸 ∈ (𝑃 MndHom 𝑆))
9493adantr 481 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐸 ∈ (𝑃 MndHom 𝑆))
9514ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑃 ∈ Ring)
9624adantrr 715 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ 𝐵)
9731adantrl 714 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ 𝐵)
981, 2ringcl 19981 . . . . . . 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 1371 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ 𝐵)
10099ralrimivva 3197 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑗𝐷𝑖𝐷 ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ 𝐵)
101 eqid 2736 . . . . . 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 8000 . . . . 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 8012 . . . . . . 7 (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V
105101mpofun 7480 . . . . . . 7 Fun (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
106104, 105, 343pm3.2i 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)
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 9312 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ∈ Fin)
10977fsuppimpd 9312 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ∈ Fin)
110 xpfi 9261 . . . . . 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 584 . . . . 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 21484 . . . . 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 9320 . . . . 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 835 . . . 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 19715 . . 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 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝐼𝑊)
1179ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑅 ∈ CRing)
118 eqid 2736 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
119 simprl 769 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑗𝐷)
120 simprr 771 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑖𝐷)
12122adantrr 715 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑥𝑗) ∈ (Base‘𝑅))
12229adantrl 714 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑦𝑖) ∈ (Base‘𝑅))
12312, 4, 16, 17, 116, 117, 2, 118, 119, 120, 121, 122mplmon2mul 21477 . . . . . . . . 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 6846 . . . . . . . 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 468 . . . . . . . 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 2776 . . . . . . 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 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 )))))
129128mpoeq3dva 7434 . . . . 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 7373 . . . 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 2737 . . . . . 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 2736 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
1331, 132ghmf 19012 . . . . . . . . 9 (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸:𝐵⟶(Base‘𝑆))
13491, 133syl 17 . . . . . . . 8 (𝜑𝐸:𝐵⟶(Base‘𝑆))
135134feqmptd 6910 . . . . . . 7 (𝜑𝐸 = (𝑧𝐵 ↦ (𝐸𝑧)))
136135adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐸 = (𝑧𝐵 ↦ (𝐸𝑧)))
137 fveq2 6842 . . . . . 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 8027 . . . . 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 7373 . . . 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 2737 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
141 fveq2 6842 . . . . . . . 8 (𝑧 = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) → (𝐸𝑧) = (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
14224, 140, 136, 141fmptco 7075 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) = (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))))
143142oveq2d 7373 . . . . . 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 2737 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) = (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
145 fveq2 6842 . . . . . . . 8 (𝑧 = (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) → (𝐸𝑧) = (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
14631, 144, 136, 145fmptco 7075 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) = (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
147146oveq2d 7373 . . . . . 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 7375 . . . . 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 2736 . . . . . 6 (0g𝑆) = (0g𝑆)
151134ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
152151, 24ffvelcdmd 7036 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) ∈ (Base‘𝑆))
153134ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
154153, 31ffvelcdmd 7036 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ (Base‘𝑆))
1556mptex 7173 . . . . . . . . 9 (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∈ V
156 funmpt 6539 . . . . . . . . 9 Fun (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
157 fvex 6855 . . . . . . . . 9 (0g𝑆) ∈ V
158155, 156, 1573pm3.2i 1339 . . . . . . . 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 3967 . . . . . . . . 9 (𝜑 → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))
1613, 150ghmid 19014 . . . . . . . . . 10 (𝐸 ∈ (𝑃 GrpHom 𝑆) → (𝐸‘(0g𝑃)) = (0g𝑆))
16291, 161syl 17 . . . . . . . . 9 (𝜑 → (𝐸‘(0g𝑃)) = (0g𝑆))
1636mptex 7173 . . . . . . . . . 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 8133 . . . . . . . 8 (𝜑 → ((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) supp (0g𝑆)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))
167166adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) supp (0g𝑆)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))
168 suppssfifsupp 9320 . . . . . . 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 835 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) finSupp (0g𝑆))
1706mptex 7173 . . . . . . . . 9 (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V
171 funmpt 6539 . . . . . . . . 9 Fun (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
172170, 171, 1573pm3.2i 1339 . . . . . . . 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 3967 . . . . . . . . 9 (𝜑 → ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))
1756mptex 7173 . . . . . . . . . 10 (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ V
176175a1i 11 . . . . . . . . 9 ((𝜑𝑖𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ V)
177174, 162, 176, 165suppssfv 8133 . . . . . . . 8 (𝜑 → ((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) supp (0g𝑆)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))
178177adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) supp (0g𝑆)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))
179 suppssfifsupp 9320 . . . . . . 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 835 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) finSupp (0g𝑆))
181132, 149, 150, 7, 7, 86, 152, 154, 169, 180gsumdixp 20033 . . . . 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 2776 . . . 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 2786 . . 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 2782 . 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 481 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐼𝑊)
18611adantr 481 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
18712, 4, 16, 1, 185, 186, 20mplcoe4 21479 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 = (𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))))
18812, 4, 16, 1, 185, 186, 27mplcoe4 21479 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 = (𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
189187, 188oveq12d 7375 . . 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 6846 . 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 6846 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑥) = (𝐸‘(𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
19224fmpttd 7063 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))):𝐷𝐵)
1931, 3, 82, 88, 7, 94, 192, 70gsummhm 19715 . . . 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 2779 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑥) = (𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
195188fveq2d 6846 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑦) = (𝐸‘(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
19631fmpttd 7063 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))):𝐷𝐵)
1971, 3, 82, 88, 7, 94, 196, 77gsummhm 19715 . . . 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 2779 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑦) = (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
199194, 198oveq12d 7375 . 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 2786 1 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸‘(𝑥(.r𝑃)𝑦)) = ((𝐸𝑥) · (𝐸𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  cdif 3907  wss 3910  ifcif 4486  {csn 4586   class class class wbr 5105  cmpt 5188   × cxp 5631  ccnv 5632  cima 5636  ccom 5637  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  f cof 7615   supp csupp 8092  m cmap 8765  Fincfn 8883   finSupp cfsupp 9305   + caddc 11054  cn 12153  0cn0 12413  Basecbs 17083  .rcmulr 17134  0gc0g 17321   Σg cgsu 17322  Mndcmnd 18556   MndHom cmhm 18599  Grpcgrp 18748   GrpHom cghm 19005  CMndccmn 19562  Ringcrg 19964  CRingccrg 19965   mPoly cmpl 21308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-lmod 20324  df-lss 20393  df-assa 21259  df-psr 21311  df-mpl 21313
This theorem is referenced by:  evlslem1  21492
  Copyright terms: Public domain W3C validator