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Theorem idlsrgval 32124
Description: Lemma for idlsrgbas 32125 through idlsrgtset 32129. (Contributed by Thierry Arnoux, 1-Jun-2024.)
Hypotheses
Ref Expression
idlsrgval.1 𝐼 = (LIdeal‘𝑅)
idlsrgval.2 = (LSSum‘𝑅)
idlsrgval.3 𝐺 = (mulGrp‘𝑅)
idlsrgval.4 = (LSSum‘𝐺)
Assertion
Ref Expression
idlsrgval (𝑅𝑉 → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩}))
Distinct variable groups:   𝑖,𝐼,𝑗   𝑅,𝑖,𝑗
Allowed substitution hints:   (𝑖,𝑗)   (𝑖,𝑗)   𝐺(𝑖,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem idlsrgval
Dummy variables 𝑏 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3461 . 2 (𝑅𝑉𝑅 ∈ V)
2 fvexd 6854 . . . 4 (𝑟 = 𝑅 → (LIdeal‘𝑟) ∈ V)
3 simpr 485 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → 𝑏 = (LIdeal‘𝑟))
4 simpl 483 . . . . . . . . . 10 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → 𝑟 = 𝑅)
54fveq2d 6843 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (LIdeal‘𝑟) = (LIdeal‘𝑅))
63, 5eqtrd 2776 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → 𝑏 = (LIdeal‘𝑅))
7 idlsrgval.1 . . . . . . . 8 𝐼 = (LIdeal‘𝑅)
86, 7eqtr4di 2794 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → 𝑏 = 𝐼)
98opeq2d 4835 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐼⟩)
104fveq2d 6843 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (LSSum‘𝑟) = (LSSum‘𝑅))
11 idlsrgval.2 . . . . . . . 8 = (LSSum‘𝑅)
1210, 11eqtr4di 2794 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (LSSum‘𝑟) = )
1312opeq2d 4835 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(+g‘ndx), (LSSum‘𝑟)⟩ = ⟨(+g‘ndx), ⟩)
144fveq2d 6843 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (RSpan‘𝑟) = (RSpan‘𝑅))
154fveq2d 6843 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (mulGrp‘𝑟) = (mulGrp‘𝑅))
16 idlsrgval.3 . . . . . . . . . . . . 13 𝐺 = (mulGrp‘𝑅)
1715, 16eqtr4di 2794 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (mulGrp‘𝑟) = 𝐺)
1817fveq2d 6843 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (LSSum‘(mulGrp‘𝑟)) = (LSSum‘𝐺))
19 idlsrgval.4 . . . . . . . . . . 11 = (LSSum‘𝐺)
2018, 19eqtr4di 2794 . . . . . . . . . 10 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (LSSum‘(mulGrp‘𝑟)) = )
2120oveqd 7370 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (𝑖(LSSum‘(mulGrp‘𝑟))𝑗) = (𝑖 𝑗))
2214, 21fveq12d 6846 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)) = ((RSpan‘𝑅)‘(𝑖 𝑗)))
238, 8, 22mpoeq123dv 7428 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (𝑖𝑏, 𝑗𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗))) = (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗))))
2423opeq2d 4835 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(.r‘ndx), (𝑖𝑏, 𝑗𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩ = ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩)
259, 13, 24tpeq123d 4707 . . . . 5 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖𝑏, 𝑗𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} = {⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩})
268rabeqdv 3420 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → {𝑗𝑏 ∣ ¬ 𝑖𝑗} = {𝑗𝐼 ∣ ¬ 𝑖𝑗})
278, 26mpteq12dv 5194 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗}) = (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}))
2827rneqd 5891 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗}) = ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}))
2928opeq2d 4835 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(TopSet‘ndx), ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗})⟩ = ⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩)
308sseq2d 3974 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ({𝑖, 𝑗} ⊆ 𝑏 ↔ {𝑖, 𝑗} ⊆ 𝐼))
3130anbi1d 630 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗) ↔ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)))
3231opabbidv 5169 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)})
3332opeq2d 4835 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)}⟩ = ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩)
3429, 33preq12d 4700 . . . . 5 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → {⟨(TopSet‘ndx), ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)}⟩} = {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩})
3525, 34uneq12d 4122 . . . 4 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖𝑏, 𝑗𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)}⟩}) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩}))
362, 35csbied 3891 . . 3 (𝑟 = 𝑅(LIdeal‘𝑟) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖𝑏, 𝑗𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)}⟩}) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩}))
37 df-idlsrg 32122 . . 3 IDLsrg = (𝑟 ∈ V ↦ (LIdeal‘𝑟) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖𝑏, 𝑗𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)}⟩}))
38 tpex 7677 . . . 4 {⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩} ∈ V
39 prex 5387 . . . 4 {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩} ∈ V
4038, 39unex 7676 . . 3 ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩}) ∈ V
4136, 37, 40fvmpt 6945 . 2 (𝑅 ∈ V → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩}))
421, 41syl 17 1 (𝑅𝑉 → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3405  Vcvv 3443  csb 3853  cun 3906  wss 3908  {cpr 4586  {ctp 4588  cop 4590  {copab 5165  cmpt 5186  ran crn 5632  cfv 6493  (class class class)co 7353  cmpo 7355  ndxcnx 17057  Basecbs 17075  +gcplusg 17125  .rcmulr 17126  TopSetcts 17131  lecple 17132  LSSumclsm 19407  mulGrpcmgp 19887  LIdealclidl 20616  RSpancrsp 20617  IDLsrgcidlsrg 32121
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 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  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 7356  df-oprab 7357  df-mpo 7358  df-idlsrg 32122
This theorem is referenced by:  idlsrgbas  32125  idlsrgplusg  32126  idlsrgmulr  32128  idlsrgtset  32129
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