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Theorem idlsrgmulr 33127
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 6899 . . . 4 𝐡 ∈ V
32, 2mpoex 8065 . . 3 (𝑖 ∈ 𝐡, 𝑗 ∈ 𝐡 ↦ ((RSpanβ€˜π‘…)β€˜(𝑖 βŠ— 𝑗))) ∈ V
4 eqid 2726 . . . . 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 33122 . . . 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 17248 . . . 4 .r = Slot (.rβ€˜ndx)
7 snsstp3 4816 . . . . 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 3986 . . . 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 17146 . . 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 2726 . . . . 5 (LSSumβ€˜π‘…) = (LSSumβ€˜π‘…)
14 idlsrgmulr.3 . . . . 5 𝐺 = (mulGrpβ€˜π‘…)
15 idlsrgmulr.4 . . . . 5 βŠ— = (LSSumβ€˜πΊ)
161, 13, 14, 15idlsrgval 33123 . . . 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 2778 . . 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 6889 . 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 2785 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 395   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468   βˆͺ cun 3941   βŠ† wss 3943  {csn 4623  {cpr 4625  {ctp 4627  βŸ¨cop 4629  {copab 5203   ↦ cmpt 5224  ran crn 5670  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  0cc0 11112  1c1 11113  cdc 12681  ndxcnx 17135  Basecbs 17153  +gcplusg 17206  .rcmulr 17207  TopSetcts 17212  lecple 17213  LSSumclsm 19554  mulGrpcmgp 20039  LIdealclidl 21065  RSpancrsp 21066  IDLsrgcidlsrg 33120
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-struct 17089  df-slot 17124  df-ndx 17136  df-base 17154  df-plusg 17219  df-mulr 17220  df-tset 17225  df-ple 17226  df-idlsrg 33121
This theorem is referenced by:  idlsrgmulrval  33129
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