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Theorem idlsrgval 32067
Description: Lemma for idlsrgbas 32068 through idlsrgtset 32072. (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 3461 . 2 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
2 fvexd 6854 . . . 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 6843 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LIdealβ€˜π‘Ÿ) = (LIdealβ€˜π‘…))
63, 5eqtrd 2777 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ 𝑏 = (LIdealβ€˜π‘…))
7 idlsrgval.1 . . . . . . . 8 𝐼 = (LIdealβ€˜π‘…)
86, 7eqtr4di 2795 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ 𝑏 = 𝐼)
98opeq2d 4835 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ⟨(Baseβ€˜ndx), π‘βŸ© = ⟨(Baseβ€˜ndx), 𝐼⟩)
104fveq2d 6843 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LSSumβ€˜π‘Ÿ) = (LSSumβ€˜π‘…))
11 idlsrgval.2 . . . . . . . 8 βŠ• = (LSSumβ€˜π‘…)
1210, 11eqtr4di 2795 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LSSumβ€˜π‘Ÿ) = βŠ• )
1312opeq2d 4835 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ ⟨(+gβ€˜ndx), (LSSumβ€˜π‘Ÿ)⟩ = ⟨(+gβ€˜ndx), βŠ• ⟩)
144fveq2d 6843 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (RSpanβ€˜π‘Ÿ) = (RSpanβ€˜π‘…))
154fveq2d 6843 . . . . . . . . . . . . 13 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (mulGrpβ€˜π‘Ÿ) = (mulGrpβ€˜π‘…))
16 idlsrgval.3 . . . . . . . . . . . . 13 𝐺 = (mulGrpβ€˜π‘…)
1715, 16eqtr4di 2795 . . . . . . . . . . . 12 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (mulGrpβ€˜π‘Ÿ) = 𝐺)
1817fveq2d 6843 . . . . . . . . . . 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 2795 . . . . . . . . . 10 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (LSSumβ€˜(mulGrpβ€˜π‘Ÿ)) = βŠ— )
2120oveqd 7368 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ (𝑖(LSSumβ€˜(mulGrpβ€˜π‘Ÿ))𝑗) = (𝑖 βŠ— 𝑗))
2214, 21fveq12d 6846 . . . . . . . 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 7426 . . . . . . 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 4835 . . . . . 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 4707 . . . . 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 3420 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑏 = (LIdealβ€˜π‘Ÿ)) β†’ {𝑗 ∈ 𝑏 ∣ Β¬ 𝑖 βŠ† 𝑗} = {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})
278, 26mpteq12dv 5194 . . . . . . . 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 5891 . . . . . . 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 4835 . . . . . 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 3974 . . . . . . . . 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 5169 . . . . . . 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 4835 . . . . . 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 4700 . . . . 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 4122 . . . 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 3891 . . 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 32065 . . 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 7673 . . . 4 {⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), βŠ• ⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} ∈ V
39 prex 5387 . . . 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 7672 . . 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 6945 . 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 3405  Vcvv 3443  β¦‹csb 3853   βˆͺ cun 3906   βŠ† wss 3908  {cpr 4586  {ctp 4588  βŸ¨cop 4590  {copab 5165   ↦ cmpt 5186  ran crn 5632  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  ndxcnx 17025  Basecbs 17043  +gcplusg 17093  .rcmulr 17094  TopSetcts 17099  lecple 17100  LSSumclsm 19375  mulGrpcmgp 19855  LIdealclidl 20584  RSpancrsp 20585  IDLsrgcidlsrg 32064
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 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-idlsrg 32065
This theorem is referenced by:  idlsrgbas  32068  idlsrgplusg  32069  idlsrgmulr  32071  idlsrgtset  32072
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