Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idlsrgval Structured version   Visualization version   GIF version
Theorem idlsrgval 32888
Description: Lemma for idlsrgbas 32889 through idlsrgtset 32893. (Contributed by Thierry Arnoux, 1-Jun-2024.)
Hypotheses
Ref Expression
idlsrgval.1 𝐼 = (LIdealβ€˜π‘…)
idlsrgval.2 βŠ• = (LSSumβ€˜π‘…)
idlsrgval.3 𝐺 = (mulGrpβ€˜π‘…)
idlsrgval.4 βŠ— = (LSSumβ€˜πΊ)
Assertion
Ref Expression
idlsrgval (𝑅 ∈ 𝑉 β†’ (IDLsrgβ€˜π‘…) = ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), βŠ• ⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)}⟩}))
Distinct variable groups:   𝑖,𝐼,𝑗   𝑅,𝑖,𝑗
Allowed substitution hints:   βŠ• (𝑖,𝑗)   βŠ— (𝑖,𝑗)   𝐺(𝑖,𝑗)   𝑉(𝑖,𝑗)
Proof of Theorem idlsrgval
Dummy variables 𝑏 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
2 fvexd 6907 . . . 4 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜π‘Ÿ) ∈ V)
3 simpr 484 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ 𝑏 = (LIdealβ€˜π‘Ÿ))
4 simpl 482 . . . . . . . . . 10 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ π‘Ÿ = 𝑅)
54fveq2d 6896 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LIdealβ€˜π‘Ÿ) = (LIdealβ€˜π‘…))
63, 5eqtrd 2771 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ 𝑏 = (LIdealβ€˜π‘…))
7 idlsrgval.1 . . . . . . . 8 𝐼 = (LIdealβ€˜π‘…)
86, 7eqtr4di 2789 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ 𝑏 = 𝐼)
98opeq2d 4881 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ⟨(Baseβ€˜ndx), π‘βŸ© = ⟨(Baseβ€˜ndx), 𝐼⟩)
104fveq2d 6896 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LSSumβ€˜π‘Ÿ) = (LSSumβ€˜π‘…))
11 idlsrgval.2 . . . . . . . 8 βŠ• = (LSSumβ€˜π‘…)
1210, 11eqtr4di 2789 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LSSumβ€˜π‘Ÿ) = βŠ• )
1312opeq2d 4881 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ⟨(+gβ€˜ndx), (LSSumβ€˜π‘Ÿ)⟩ = ⟨(+gβ€˜ndx), βŠ• ⟩)
144fveq2d 6896 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (RSpanβ€˜π‘Ÿ) = (RSpanβ€˜π‘…))
154fveq2d 6896 . . . . . . . . . . . . 13 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (mulGrpβ€˜π‘Ÿ) = (mulGrpβ€˜π‘…))
16 idlsrgval.3 . . . . . . . . . . . . 13 𝐺 = (mulGrpβ€˜π‘…)
1715, 16eqtr4di 2789 . . . . . . . . . . . 12 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (mulGrpβ€˜π‘Ÿ) = 𝐺)
1817fveq2d 6896 . . . . . . . . . . 11 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LSSumβ€˜(mulGrpβ€˜π‘Ÿ)) = (LSSumβ€˜πΊ))
19 idlsrgval.4 . . . . . . . . . . 11 βŠ— = (LSSumβ€˜πΊ)
2018, 19eqtr4di 2789 . . . . . . . . . 10 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LSSumβ€˜(mulGrpβ€˜π‘Ÿ)) = βŠ— )
2120oveqd 7429 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (𝑖(LSSumβ€˜(mulGrpβ€˜π‘Ÿ))𝑗) = (𝑖 βŠ— 𝑗))
2214, 21fveq12d 6899 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ((RSpanβ€˜π‘Ÿ)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘Ÿ))𝑗)) = ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))
238, 8, 22mpoeq123dv 7487 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpanβ€˜π‘Ÿ)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘Ÿ))𝑗))) = (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗))))
2423opeq2d 4881 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ⟨(.rβ€˜ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpanβ€˜π‘Ÿ)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘Ÿ))𝑗)))⟩ = ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩)
259, 13, 24tpeq123d 4753 . . . . 5 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ {⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘Ÿ)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpanβ€˜π‘Ÿ)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘Ÿ))𝑗)))⟩} = {⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), βŠ• ⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩})
268rabeqdv 3446 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ {𝑗 ∈ 𝑏 ∣ Β¬ 𝑖 βŠ† 𝑗} = {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})
278, 26mpteq12dv 5240 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}))
2827rneqd 5938 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ Β¬ 𝑖 βŠ† 𝑗}) = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}))
2928opeq2d 4881 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩ = ⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩)
308sseq2d 4015 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ({𝑖, 𝑗} βŠ† 𝑏 ↔ {𝑖, 𝑗} βŠ† 𝐼))
3130anbi1d 629 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (({𝑖, 𝑗} βŠ† 𝑏 ∧ 𝑖 βŠ† 𝑗) ↔ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)))
3231opabbidv 5215 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝑏 ∧ 𝑖 βŠ† 𝑗)} = {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)})
3332opeq2d 4881 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝑏 ∧ 𝑖 βŠ† 𝑗)}⟩ = ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)}⟩)
3429, 33preq12d 4746 . . . . 5 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝑏 ∧ 𝑖 βŠ† 𝑗)}⟩} = {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)}⟩})
3525, 34uneq12d 4165 . . . 4 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ({⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘Ÿ)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpanβ€˜π‘Ÿ)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘Ÿ))𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝑏 ∧ 𝑖 βŠ† 𝑗)}⟩}) = ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), βŠ• ⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)}⟩}))
362, 35csbied 3932 . . 3 (π‘Ÿ = 𝑅 β†’ ⦋(LIdealβ€˜π‘Ÿ) / π‘β¦Œ({⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘Ÿ)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpanβ€˜π‘Ÿ)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘Ÿ))𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝑏 ∧ 𝑖 βŠ† 𝑗)}⟩}) = ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), βŠ• ⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)}⟩}))
37 df-idlsrg 32886 . . 3 IDLsrg = (π‘Ÿ ∈ V ↦ ⦋(LIdealβ€˜π‘Ÿ) / π‘β¦Œ({⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘Ÿ)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpanβ€˜π‘Ÿ)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘Ÿ))𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝑏 ∧ 𝑖 βŠ† 𝑗)}⟩}))
38 tpex 7737 . . . 4 {⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), βŠ• ⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} ∈ V
39 prex 5433 . . . 4 {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)}⟩} ∈ V
4038, 39unex 7736 . . 3 ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), βŠ• ⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)}⟩}) ∈ V
4136, 37, 40fvmpt 6999 . 2 (𝑅 ∈ V β†’ (IDLsrgβ€˜π‘…) = ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), βŠ• ⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)}⟩}))
421, 41syl 17 1 (𝑅 ∈ 𝑉 β†’ (IDLsrgβ€˜π‘…) = ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), βŠ• ⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)}⟩}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {crab 3431  Vcvv 3473  β¦‹csb 3894   βˆͺ cun 3947   βŠ† wss 3949  {cpr 4631  {ctp 4633  βŸ¨cop 4635  {copab 5211   ↦ cmpt 5232  ran crn 5678  β€˜cfv 6544  (class class class)co 7412   ∈ cmpo 7414  ndxcnx 17131  Basecbs 17149  +gcplusg 17202  .rcmulr 17203  TopSetcts 17208  lecple 17209  LSSumclsm 19544  mulGrpcmgp 20029  LIdealclidl 20929  RSpancrsp 20930  IDLsrgcidlsrg 32885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-idlsrg 32886
This theorem is referenced by:  idlsrgbas  32889  idlsrgplusg  32890  idlsrgmulr  32892  idlsrgtset  32893
  Copyright terms: Public domain W3C validator