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Theorem idlsrgmulr 33263
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 6904 . . . 4 𝐡 ∈ V
32, 2mpoex 8077 . . 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 33258 . . . 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 17269 . . . 4 .r = Slot (.rβ€˜ndx)
7 snsstp3 4818 . . . . 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 4167 . . . . 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 3983 . . . 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 17167 . . 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 33259 . . . 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 6894 . 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 3939   βŠ† wss 3941  {csn 4625  {cpr 4627  {ctp 4629  βŸ¨cop 4631  {copab 5206   ↦ cmpt 5227  ran crn 5674  β€˜cfv 6543  (class class class)co 7413   ∈ cmpo 7415  0cc0 11133  1c1 11134  cdc 12702  ndxcnx 17156  Basecbs 17174  +gcplusg 17227  .rcmulr 17228  TopSetcts 17233  lecple 17234  LSSumclsm 19588  mulGrpcmgp 20073  LIdealclidl 21101  RSpancrsp 21102  IDLsrgcidlsrg 33256
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 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
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 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-struct 17110  df-slot 17145  df-ndx 17157  df-base 17175  df-plusg 17240  df-mulr 17241  df-tset 17246  df-ple 17247  df-idlsrg 33257
This theorem is referenced by:  idlsrgmulrval  33265
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