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Theorem evlslem2 20220
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.)
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 (𝜑𝐼 ∈ V)
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   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)

Proof of Theorem evlslem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 evlslem2.b . . . . 5 𝐵 = (Base‘𝑃)
2 eqid 2818 . . . . 5 (.r𝑃) = (.r𝑃)
3 eqid 2818 . . . . 5 (0g𝑃) = (0g𝑃)
4 evlslem2.d . . . . . . 7 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
5 ovex 7178 . . . . . . 7 (ℕ0m 𝐼) ∈ V
64, 5rabex2 5228 . . . . . 6 𝐷 ∈ V
76a1i 11 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐷 ∈ V)
8 evlslem2.i . . . . . . 7 (𝜑𝐼 ∈ V)
9 evlslem2.r . . . . . . . 8 (𝜑𝑅 ∈ CRing)
10 crngring 19237 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
119, 10syl 17 . . . . . . 7 (𝜑𝑅 ∈ Ring)
12 evlslem2.p . . . . . . . 8 𝑃 = (𝐼 mPoly 𝑅)
1312mplring 20160 . . . . . . 7 ((𝐼 ∈ V ∧ 𝑅 ∈ 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 2818 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
188ad2antrr 722 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝐼 ∈ V)
1911ad2antrr 722 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝑅 ∈ Ring)
20 simprl 767 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
2112, 17, 1, 4, 20mplelf 20141 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥:𝐷⟶(Base‘𝑅))
2221ffvelrnda 6843 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝑥𝑗) ∈ (Base‘𝑅))
23 simpr 485 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝑗𝐷)
2412, 4, 16, 17, 18, 19, 1, 22, 23mplmon2cl 20208 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ 𝐵)
258ad2antrr 722 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝐼 ∈ V)
2611ad2antrr 722 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝑅 ∈ Ring)
27 simprr 769 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
2812, 17, 1, 4, 27mplelf 20141 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦:𝐷⟶(Base‘𝑅))
2928ffvelrnda 6843 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝑦𝑖) ∈ (Base‘𝑅))
30 simpr 485 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝑖𝐷)
3112, 4, 16, 17, 25, 26, 1, 29, 30mplmon2cl 20208 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ 𝐵)
326mptex 6977 . . . . . . . . . . . 12 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) ∈ V
33 funmpt 6386 . . . . . . . . . . . 12 Fun (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
34 fvex 6676 . . . . . . . . . . . 12 (0g𝑃) ∈ V
3532, 33, 343pm3.2i 1331 . . . . . . . . . . 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 20199 . . . . . . . . . . 11 ((𝜑𝑦𝐵) → 𝑦 finSupp 0 )
4039fsuppimpd 8828 . . . . . . . . . 10 ((𝜑𝑦𝐵) → (𝑦 supp 0 ) ∈ Fin)
4112, 17, 1, 4, 37mplelf 20141 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 𝑦:𝐷⟶(Base‘𝑅))
42 ssidd 3987 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 ))
436a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 𝐷 ∈ V)
4416fvexi 6677 . . . . . . . . . . . . . . . . 17 0 ∈ V
4544a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐵) → 0 ∈ V)
4641, 42, 43, 45suppssr 7850 . . . . . . . . . . . . . . 15 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑦𝑗) = 0 )
4746ifeq1d 4481 . . . . . . . . . . . . . 14 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = if(𝑘 = 𝑗, 0 , 0 ))
48 ifid 4502 . . . . . . . . . . . . . 14 if(𝑘 = 𝑗, 0 , 0 ) = 0
4947, 48syl6eq 2869 . . . . . . . . . . . . 13 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = 0 )
5049mpteq2dv 5153 . . . . . . . . . . . 12 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (𝑘𝐷0 ))
51 ringgrp 19231 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
5211, 51syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑅 ∈ Grp)
5312, 4, 16, 3, 8, 52mpl0 20149 . . . . . . . . . . . . . 14 (𝜑 → (0g𝑃) = (𝐷 × { 0 }))
54 fconstmpt 5607 . . . . . . . . . . . . . 14 (𝐷 × { 0 }) = (𝑘𝐷0 )
5553, 54syl6eq 2869 . . . . . . . . . . . . 13 (𝜑 → (0g𝑃) = (𝑘𝐷0 ))
5655ad2antrr 722 . . . . . . . . . . . 12 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (0g𝑃) = (𝑘𝐷0 ))
5750, 56eqtr4d 2856 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑗 ∈ (𝐷 ∖ (𝑦 supp 0 ))) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (0g𝑃))
5857, 43suppss2 7853 . . . . . . . . . 10 ((𝜑𝑦𝐵) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) supp (0g𝑃)) ⊆ (𝑦 supp 0 ))
59 suppssfifsupp 8836 . . . . . . . . . 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 832 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
6160ralrimiva 3179 . . . . . . . 8 (𝜑 → ∀𝑦𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
62 fveq1 6662 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦𝑗) = (𝑥𝑗))
6362ifeq1d 4481 . . . . . . . . . . . 12 (𝑦 = 𝑥 → if(𝑘 = 𝑗, (𝑦𝑗), 0 ) = if(𝑘 = 𝑗, (𝑥𝑗), 0 ))
6463mpteq2dv 5153 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )) = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))
6564mpteq2dv 5153 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
6665breq1d 5067 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃) ↔ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃)))
6766cbvralvw 3447 . . . . . . . 8 (∀𝑦𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃) ↔ ∀𝑥𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
6861, 67sylib 219 . . . . . . 7 (𝜑 → ∀𝑥𝐵 (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
6968r19.21bi 3205 . . . . . 6 ((𝜑𝑥𝐵) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
7069adantrr 713 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) finSupp (0g𝑃))
71 equequ2 2024 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑘 = 𝑖𝑘 = 𝑗))
72 fveq2 6663 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑦𝑖) = (𝑦𝑗))
7371, 72ifbieq1d 4486 . . . . . . . 8 (𝑖 = 𝑗 → if(𝑘 = 𝑖, (𝑦𝑖), 0 ) = if(𝑘 = 𝑗, (𝑦𝑗), 0 ))
7473mpteq2dv 5153 . . . . . . 7 (𝑖 = 𝑗 → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
7574cbvmptv 5160 . . . . . 6 (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 )))
7660adantrl 712 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑦𝑗), 0 ))) finSupp (0g𝑃))
7775, 76eqbrtrid 5092 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) finSupp (0g𝑃))
781, 2, 3, 7, 7, 15, 24, 31, 70, 77gsumdixp 19288 . . . 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 6667 . . 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 19260 . . . . . 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 19237 . . . . . . 7 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
8583, 84syl 17 . . . . . 6 (𝜑𝑆 ∈ Ring)
8685adantr 481 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Ring)
87 ringmnd 19235 . . . . 5 (𝑆 ∈ Ring → 𝑆 ∈ Mnd)
8886, 87syl 17 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Mnd)
896, 6xpex 7465 . . . . 5 (𝐷 × 𝐷) ∈ V
9089a1i 11 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐷 × 𝐷) ∈ V)
91 evlslem2.e1 . . . . . 6 (𝜑𝐸 ∈ (𝑃 GrpHom 𝑆))
92 ghmmhm 18306 . . . . . 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 722 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑃 ∈ Ring)
9624adantrr 713 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) ∈ 𝐵)
9731adantrl 712 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ 𝐵)
981, 2ringcl 19240 . . . . . . 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 1363 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ 𝐵)
10099ralrimivva 3188 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑗𝐷𝑖𝐷 ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ 𝐵)
101 eqid 2818 . . . . . 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 7755 . . . . 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 219 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))):(𝐷 × 𝐷)⟶𝐵)
1046, 6mpoex 7766 . . . . . . 7 (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V
105101mpofun 7265 . . . . . . 7 Fun (𝑗𝐷, 𝑖𝐷 ↦ ((𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))(.r𝑃)(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
106104, 105, 343pm3.2i 1331 . . . . . 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 8828 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ∈ Fin)
10977fsuppimpd 8828 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ∈ Fin)
110 xpfi 8777 . . . . . 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 20216 . . . . 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 8836 . . . . 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 832 . . . 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 18987 . . 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 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝐼 ∈ V)
1179ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑅 ∈ CRing)
118 eqid 2818 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
119 simprl 767 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑗𝐷)
120 simprr 769 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → 𝑖𝐷)
12122adantrr 713 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑥𝑗) ∈ (Base‘𝑅))
12229adantrl 712 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑗𝐷𝑖𝐷)) → (𝑦𝑖) ∈ (Base‘𝑅))
12312, 4, 16, 17, 116, 117, 2, 118, 119, 120, 121, 122mplmon2mul 20209 . . . . . . . . 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 6667 . . . . . . . 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 2853 . . . . . . 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 1107 . . . . . 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 7220 . . . . 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 7161 . . . 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 2819 . . . . . 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 2818 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
1331, 132ghmf 18300 . . . . . . . . 9 (𝐸 ∈ (𝑃 GrpHom 𝑆) → 𝐸:𝐵⟶(Base‘𝑆))
13491, 133syl 17 . . . . . . . 8 (𝜑𝐸:𝐵⟶(Base‘𝑆))
135134feqmptd 6726 . . . . . . 7 (𝜑𝐸 = (𝑧𝐵 ↦ (𝐸𝑧)))
136135adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐸 = (𝑧𝐵 ↦ (𝐸𝑧)))
137 fveq2 6663 . . . . . 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 7779 . . . . 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 7161 . . . 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 2819 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) = (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
141 fveq2 6663 . . . . . . . 8 (𝑧 = (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )) → (𝐸𝑧) = (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
14224, 140, 136, 141fmptco 6883 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) = (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))))
143142oveq2d 7161 . . . . . 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 2819 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) = (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
145 fveq2 6663 . . . . . . . 8 (𝑧 = (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) → (𝐸𝑧) = (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
14631, 144, 136, 145fmptco 6883 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) = (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
147146oveq2d 7161 . . . . . 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 7163 . . . . 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 2818 . . . . . 6 (0g𝑆) = (0g𝑆)
151134ad2antrr 722 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
152151, 24ffvelrnd 6844 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑗𝐷) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) ∈ (Base‘𝑆))
153134ad2antrr 722 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → 𝐸:𝐵⟶(Base‘𝑆))
154153, 31ffvelrnd 6844 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑖𝐷) → (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) ∈ (Base‘𝑆))
1556mptex 6977 . . . . . . . . 9 (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) ∈ V
156 funmpt 6386 . . . . . . . . 9 Fun (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))
157 fvex 6676 . . . . . . . . 9 (0g𝑆) ∈ V
158155, 156, 1573pm3.2i 1331 . . . . . . . 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 3987 . . . . . . . . 9 (𝜑 → ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)) ⊆ ((𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))) supp (0g𝑃)))
1613, 150ghmid 18302 . . . . . . . . . 10 (𝐸 ∈ (𝑃 GrpHom 𝑆) → (𝐸‘(0g𝑃)) = (0g𝑆))
16291, 161syl 17 . . . . . . . . 9 (𝜑 → (𝐸‘(0g𝑃)) = (0g𝑆))
1636mptex 6977 . . . . . . . . . 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 7855 . . . . . . . 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 8836 . . . . . . 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 832 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))) finSupp (0g𝑆))
1706mptex 6977 . . . . . . . . 9 (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) ∈ V
171 funmpt 6386 . . . . . . . . 9 Fun (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))
172170, 171, 1573pm3.2i 1331 . . . . . . . 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 3987 . . . . . . . . 9 (𝜑 → ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)) ⊆ ((𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))) supp (0g𝑃)))
1756mptex 6977 . . . . . . . . . 10 (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ V
176175a1i 11 . . . . . . . . 9 ((𝜑𝑖𝐷) → (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )) ∈ V)
177174, 162, 176, 165suppssfv 7855 . . . . . . . 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 8836 . . . . . . 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 832 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝐸‘(𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))) finSupp (0g𝑆))
181132, 149, 150, 7, 7, 86, 152, 154, 169, 180gsumdixp 19288 . . . . 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 2853 . . . 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 2863 . . 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 2859 . 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)) → 𝐼 ∈ V)
18611adantr 481 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
18712, 4, 16, 1, 185, 186, 20mplcoe4 20211 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 = (𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 )))))
18812, 4, 16, 1, 185, 186, 27mplcoe4 20211 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 = (𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 )))))
189187, 188oveq12d 7163 . . 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 6667 . 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 6667 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑥) = (𝐸‘(𝑃 Σg (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
19224fmpttd 6871 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))):𝐷𝐵)
1931, 3, 82, 88, 7, 94, 192, 70gsummhm 18987 . . . 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 2856 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑥) = (𝑆 Σg (𝐸 ∘ (𝑗𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑗, (𝑥𝑗), 0 ))))))
195188fveq2d 6667 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑦) = (𝐸‘(𝑃 Σg (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
19631fmpttd 6871 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))):𝐷𝐵)
1971, 3, 82, 88, 7, 94, 196, 77gsummhm 18987 . . . 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 2856 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸𝑦) = (𝑆 Σg (𝐸 ∘ (𝑖𝐷 ↦ (𝑘𝐷 ↦ if(𝑘 = 𝑖, (𝑦𝑖), 0 ))))))
199194, 198oveq12d 7163 . 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 2863 1 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐸‘(𝑥(.r𝑃)𝑦)) = ((𝐸𝑥) · (𝐸𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  cdif 3930  wss 3933  ifcif 4463  {csn 4557   class class class wbr 5057  cmpt 5137   × cxp 5546  ccnv 5547  cima 5551  ccom 5552  Fun wfun 6342  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  f cof 7396   supp csupp 7819  m cmap 8395  Fincfn 8497   finSupp cfsupp 8821   + caddc 10528  cn 11626  0cn0 11885  Basecbs 16471  .rcmulr 16554  0gc0g 16701   Σg cgsu 16702  Mndcmnd 17899   MndHom cmhm 17942  Grpcgrp 18041   GrpHom cghm 18293  CMndccmn 18835  Ringcrg 19226  CRingccrg 19227   mPoly cmpl 20061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-ofr 7399  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-sca 16569  df-vsca 16570  df-tset 16572  df-0g 16703  df-gsum 16704  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-mulg 18163  df-subg 18214  df-ghm 18294  df-cntz 18385  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-ring 19228  df-cring 19229  df-subrg 19462  df-lmod 19565  df-lss 19633  df-assa 20013  df-psr 20064  df-mpl 20066
This theorem is referenced by:  evlslem1  20223
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