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Theorem idlsrgtset 32305
Description: Topology component of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
Hypotheses
Ref Expression
idlsrgtset.1 𝑆 = (IDLsrgβ€˜π‘…)
idlsrgtset.2 𝐼 = (LIdealβ€˜π‘…)
idlsrgtset.3 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})
Assertion
Ref Expression
idlsrgtset (𝑅 ∈ 𝑉 β†’ 𝐽 = (TopSetβ€˜π‘†))
Distinct variable groups:   𝑖,𝐼,𝑗   𝑅,𝑖,𝑗   𝑖,𝑉
Allowed substitution hints:   𝑆(𝑖,𝑗)   𝐽(𝑖,𝑗)   𝑉(𝑗)

Proof of Theorem idlsrgtset
StepHypRef Expression
1 idlsrgtset.3 . 2 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})
2 idlsrgtset.2 . . . . . . 7 𝐼 = (LIdealβ€˜π‘…)
32fvexi 6860 . . . . . 6 𝐼 ∈ V
43mptex 7177 . . . . 5 (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) ∈ V
54rnex 7853 . . . 4 ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) ∈ V
6 eqid 2733 . . . . . 6 ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘…))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)}⟩}) = ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘…))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)}⟩})
76idlsrgstr 32299 . . . . 5 ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘…))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)}⟩}) Struct ⟨1, 10⟩
8 tsetid 17242 . . . . 5 TopSet = Slot (TopSetβ€˜ndx)
9 snsspr1 4778 . . . . . 6 {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩} βŠ† {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)}⟩}
10 ssun2 4137 . . . . . 6 {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐼 ∧ 𝑖 βŠ† 𝑗)}⟩} βŠ† ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘…))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)}⟩})
119, 10sstri 3957 . . . . 5 {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩} βŠ† ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘…))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)}⟩})
127, 8, 11strfv 17084 . . . 4 (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) ∈ V β†’ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (TopSetβ€˜({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘…))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)}⟩})))
135, 12ax-mp 5 . . 3 ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (TopSetβ€˜({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘…))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)}⟩}))
14 idlsrgtset.1 . . . . 5 𝑆 = (IDLsrgβ€˜π‘…)
15 eqid 2733 . . . . . 6 (LSSumβ€˜π‘…) = (LSSumβ€˜π‘…)
16 eqid 2733 . . . . . 6 (mulGrpβ€˜π‘…) = (mulGrpβ€˜π‘…)
17 eqid 2733 . . . . . 6 (LSSumβ€˜(mulGrpβ€˜π‘…)) = (LSSumβ€˜(mulGrpβ€˜π‘…))
182, 15, 16, 17idlsrgval 32300 . . . . 5 (𝑅 ∈ 𝑉 β†’ (IDLsrgβ€˜π‘…) = ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘…))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)}⟩}))
1914, 18eqtrid 2785 . . . 4 (𝑅 ∈ 𝑉 β†’ 𝑆 = ({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘…))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)}⟩}))
2019fveq2d 6850 . . 3 (𝑅 ∈ 𝑉 β†’ (TopSetβ€˜π‘†) = (TopSetβ€˜({⟨(Baseβ€˜ndx), 𝐼⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖(LSSumβ€˜(mulGrpβ€˜π‘…))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)}⟩})))
2113, 20eqtr4id 2792 . 2 (𝑅 ∈ 𝑉 β†’ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (TopSetβ€˜π‘†))
221, 21eqtrid 2785 1 (𝑅 ∈ 𝑉 β†’ 𝐽 = (TopSetβ€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3447   βˆͺ cun 3912   βŠ† wss 3914  {csn 4590  {cpr 4592  {ctp 4594  βŸ¨cop 4596  {copab 5171   ↦ cmpt 5192  ran crn 5638  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  0cc0 11059  1c1 11060  cdc 12626  ndxcnx 17073  Basecbs 17091  +gcplusg 17141  .rcmulr 17142  TopSetcts 17147  lecple 17148  LSSumclsm 19424  mulGrpcmgp 19904  LIdealclidl 20676  RSpancrsp 20677  IDLsrgcidlsrg 32297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-fz 13434  df-struct 17027  df-slot 17062  df-ndx 17074  df-base 17092  df-plusg 17154  df-mulr 17155  df-tset 17160  df-ple 17161  df-idlsrg 32298
This theorem is referenced by:  rspectset  32511  rspectopn  32512
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