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Theorem idlsrgval 32662
Description: Lemma for idlsrgbas 32663 through idlsrgtset 32667. (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 6906 . . . 4 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜π‘Ÿ) ∈ V)
3 simpr 485 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ 𝑏 = (LIdealβ€˜π‘Ÿ))
4 simpl 483 . . . . . . . . . 10 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ π‘Ÿ = 𝑅)
54fveq2d 6895 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LIdealβ€˜π‘Ÿ) = (LIdealβ€˜π‘…))
63, 5eqtrd 2772 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ 𝑏 = (LIdealβ€˜π‘…))
7 idlsrgval.1 . . . . . . . 8 𝐼 = (LIdealβ€˜π‘…)
86, 7eqtr4di 2790 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ 𝑏 = 𝐼)
98opeq2d 4880 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ⟨(Baseβ€˜ndx), π‘βŸ© = ⟨(Baseβ€˜ndx), 𝐼⟩)
104fveq2d 6895 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LSSumβ€˜π‘Ÿ) = (LSSumβ€˜π‘…))
11 idlsrgval.2 . . . . . . . 8 βŠ• = (LSSumβ€˜π‘…)
1210, 11eqtr4di 2790 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LSSumβ€˜π‘Ÿ) = βŠ• )
1312opeq2d 4880 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ⟨(+gβ€˜ndx), (LSSumβ€˜π‘Ÿ)⟩ = ⟨(+gβ€˜ndx), βŠ• ⟩)
144fveq2d 6895 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (RSpanβ€˜π‘Ÿ) = (RSpanβ€˜π‘…))
154fveq2d 6895 . . . . . . . . . . . . 13 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (mulGrpβ€˜π‘Ÿ) = (mulGrpβ€˜π‘…))
16 idlsrgval.3 . . . . . . . . . . . . 13 𝐺 = (mulGrpβ€˜π‘…)
1715, 16eqtr4di 2790 . . . . . . . . . . . 12 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (mulGrpβ€˜π‘Ÿ) = 𝐺)
1817fveq2d 6895 . . . . . . . . . . 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 2790 . . . . . . . . . 10 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LSSumβ€˜(mulGrpβ€˜π‘Ÿ)) = βŠ— )
2120oveqd 7428 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (𝑖(LSSumβ€˜(mulGrpβ€˜π‘Ÿ))𝑗) = (𝑖 βŠ— 𝑗))
2214, 21fveq12d 6898 . . . . . . . 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 7486 . . . . . . 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 4880 . . . . . 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 4752 . . . . 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 3447 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ {𝑗 ∈ 𝑏 ∣ Β¬ 𝑖 βŠ† 𝑗} = {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})
278, 26mpteq12dv 5239 . . . . . . . 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 5937 . . . . . . 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 4880 . . . . . 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 4014 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ({𝑖, 𝑗} βŠ† 𝑏 ↔ {𝑖, 𝑗} βŠ† 𝐼))
3130anbi1d 630 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (({𝑖, 𝑗} βŠ† 𝑏 ∧ 𝑖 βŠ† 𝑗) ↔ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)))
3231opabbidv 5214 . . . . . . 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 4880 . . . . . 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 4745 . . . . 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 4164 . . . 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 3931 . . 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 32660 . . 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 7736 . . . 4 {⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), βŠ• ⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} ∈ V
39 prex 5432 . . . 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 7735 . . 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 6998 . 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 396   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474  β¦‹csb 3893   βˆͺ cun 3946   βŠ† wss 3948  {cpr 4630  {ctp 4632  βŸ¨cop 4634  {copab 5210   ↦ cmpt 5231  ran crn 5677  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413  ndxcnx 17128  Basecbs 17146  +gcplusg 17199  .rcmulr 17200  TopSetcts 17205  lecple 17206  LSSumclsm 19504  mulGrpcmgp 19989  LIdealclidl 20789  RSpancrsp 20790  IDLsrgcidlsrg 32659
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-idlsrg 32660
This theorem is referenced by:  idlsrgbas  32663  idlsrgplusg  32664  idlsrgmulr  32666  idlsrgtset  32667
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