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Theorem idlsrgmulr 33266
Description: Multiplicative operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
Hypotheses
Ref Expression
idlsrgmulr.1 𝑆 = (IDLsrgβ€˜π‘…)
idlsrgmulr.2 𝐡 = (LIdealβ€˜π‘…)
idlsrgmulr.3 𝐺 = (mulGrpβ€˜π‘…)
idlsrgmulr.4 βŠ— = (LSSumβ€˜πΊ)
Assertion
Ref Expression
idlsrgmulr (𝑅 ∈ 𝑉 β†’ (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗))) = (.rβ€˜π‘†))
Distinct variable groups:   𝐡,𝑖,𝑗   𝑅,𝑖,𝑗
Allowed substitution hints:   𝑆(𝑖,𝑗)   βŠ— (𝑖,𝑗)   𝐺(𝑖,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem idlsrgmulr
StepHypRef Expression
1 idlsrgmulr.2 . . . . 5 𝐡 = (LIdealβ€˜π‘…)
21fvexi 6905 . . . 4 𝐡 ∈ V
32, 2mpoex 8080 . . 3 (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗))) ∈ V
4 eqid 2725 . . . . 5 ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐡 ↦ {𝑗 ∈ 𝐡 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐡 ∧ 𝑖 βŠ† 𝑗)}⟩}) = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐡 ↦ {𝑗 ∈ 𝐡 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐡 ∧ 𝑖 βŠ† 𝑗)}⟩})
54idlsrgstr 33261 . . . 4 ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐡 ↦ {𝑗 ∈ 𝐡 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐡 ∧ 𝑖 βŠ† 𝑗)}⟩}) Struct ⟨1, 10⟩
6 mulridx 17272 . . . 4 .r = Slot (.rβ€˜ndx)
7 snsstp3 4817 . . . . 5 {⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βŠ† {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩}
8 ssun1 4166 . . . . 5 {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βŠ† ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐡 ↦ {𝑗 ∈ 𝐡 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐡 ∧ 𝑖 βŠ† 𝑗)}⟩})
97, 8sstri 3982 . . . 4 {⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βŠ† ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐡 ↦ {𝑗 ∈ 𝐡 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐡 ∧ 𝑖 βŠ† 𝑗)}⟩})
105, 6, 9strfv 17170 . . 3 ((𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗))) ∈ V β†’ (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗))) = (.rβ€˜({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐡 ↦ {𝑗 ∈ 𝐡 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐡 ∧ 𝑖 βŠ† 𝑗)}⟩})))
113, 10ax-mp 5 . 2 (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗))) = (.rβ€˜({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐡 ↦ {𝑗 ∈ 𝐡 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐡 ∧ 𝑖 βŠ† 𝑗)}⟩}))
12 idlsrgmulr.1 . . . 4 𝑆 = (IDLsrgβ€˜π‘…)
13 eqid 2725 . . . . 5 (LSSumβ€˜π‘…) = (LSSumβ€˜π‘…)
14 idlsrgmulr.3 . . . . 5 𝐺 = (mulGrpβ€˜π‘…)
15 idlsrgmulr.4 . . . . 5 βŠ— = (LSSumβ€˜πΊ)
161, 13, 14, 15idlsrgval 33262 . . . 4 (𝑅 ∈ 𝑉 β†’ (IDLsrgβ€˜π‘…) = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐡 ↦ {𝑗 ∈ 𝐡 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐡 ∧ 𝑖 βŠ† 𝑗)}⟩}))
1712, 16eqtrid 2777 . . 3 (𝑅 ∈ 𝑉 β†’ 𝑆 = ({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐡 ↦ {𝑗 ∈ 𝐡 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐡 ∧ 𝑖 βŠ† 𝑗)}⟩}))
1817fveq2d 6895 . 2 (𝑅 ∈ 𝑉 β†’ (.rβ€˜π‘†) = (.rβ€˜({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (LSSumβ€˜π‘…)⟩, ⟨(.rβ€˜ndx), (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗)))⟩} βˆͺ {⟨(TopSetβ€˜ndx), ran (𝑖 ∈ 𝐡 ↦ {𝑗 ∈ 𝐡 ∣ Β¬ 𝑖 βŠ† 𝑗})⟩, ⟨(leβ€˜ndx), {βŸ¨π‘–, π‘—βŸ© ∣ ({𝑖, 𝑗} βŠ† 𝐡 ∧ 𝑖 βŠ† 𝑗)}⟩})))
1911, 18eqtr4id 2784 1 (𝑅 ∈ 𝑉 β†’ (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗))) = (.rβ€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3419  Vcvv 3463   βˆͺ cun 3938   βŠ† wss 3940  {csn 4624  {cpr 4626  {ctp 4628  βŸ¨cop 4630  {copab 5205   ↦ cmpt 5226  ran crn 5673  β€˜cfv 6542  (class class class)co 7415   ∈ cmpo 7417  0cc0 11136  1c1 11137  cdc 12705  ndxcnx 17159  Basecbs 17177  +gcplusg 17230  .rcmulr 17231  TopSetcts 17236  lecple 17237  LSSumclsm 19591  mulGrpcmgp 20076  LIdealclidl 21104  RSpancrsp 21105  IDLsrgcidlsrg 33259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12501  df-z 12587  df-dec 12706  df-uz 12851  df-fz 13515  df-struct 17113  df-slot 17148  df-ndx 17160  df-base 17178  df-plusg 17243  df-mulr 17244  df-tset 17249  df-ple 17250  df-idlsrg 33260
This theorem is referenced by:  idlsrgmulrval  33268
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