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Theorem idlsrgval 31117
 Description: Lemma for idlsrgbas 31118 through idlsrgtset 31122. (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 3460 . 2 (𝑅𝑉𝑅 ∈ V)
2 fvexd 6670 . . . 4 (𝑟 = 𝑅 → (LIdeal‘𝑟) ∈ V)
3 simpr 488 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → 𝑏 = (LIdeal‘𝑟))
4 simpl 486 . . . . . . . . . 10 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → 𝑟 = 𝑅)
54fveq2d 6659 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (LIdeal‘𝑟) = (LIdeal‘𝑅))
63, 5eqtrd 2833 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → 𝑏 = (LIdeal‘𝑅))
7 idlsrgval.1 . . . . . . . 8 𝐼 = (LIdeal‘𝑅)
86, 7eqtr4di 2851 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → 𝑏 = 𝐼)
98opeq2d 4776 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐼⟩)
104fveq2d 6659 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (LSSum‘𝑟) = (LSSum‘𝑅))
11 idlsrgval.2 . . . . . . . 8 = (LSSum‘𝑅)
1210, 11eqtr4di 2851 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (LSSum‘𝑟) = )
1312opeq2d 4776 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(+g‘ndx), (LSSum‘𝑟)⟩ = ⟨(+g‘ndx), ⟩)
144fveq2d 6659 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (RSpan‘𝑟) = (RSpan‘𝑅))
154fveq2d 6659 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (mulGrp‘𝑟) = (mulGrp‘𝑅))
16 idlsrgval.3 . . . . . . . . . . . . 13 𝐺 = (mulGrp‘𝑅)
1715, 16eqtr4di 2851 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (mulGrp‘𝑟) = 𝐺)
1817fveq2d 6659 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (LSSum‘(mulGrp‘𝑟)) = (LSSum‘𝐺))
19 idlsrgval.4 . . . . . . . . . . 11 = (LSSum‘𝐺)
2018, 19eqtr4di 2851 . . . . . . . . . 10 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (LSSum‘(mulGrp‘𝑟)) = )
2120oveqd 7162 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (𝑖(LSSum‘(mulGrp‘𝑟))𝑗) = (𝑖 𝑗))
2214, 21fveq12d 6662 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)) = ((RSpan‘𝑅)‘(𝑖 𝑗)))
238, 8, 22mpoeq123dv 7218 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (𝑖𝑏, 𝑗𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗))) = (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗))))
2423opeq2d 4776 . . . . . 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 4647 . . . . 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 3433 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → {𝑗𝑏 ∣ ¬ 𝑖𝑗} = {𝑗𝐼 ∣ ¬ 𝑖𝑗})
278, 26mpteq12dv 5119 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗}) = (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}))
2827rneqd 5778 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗}) = ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}))
2928opeq2d 4776 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(TopSet‘ndx), ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗})⟩ = ⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩)
308sseq2d 3949 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ({𝑖, 𝑗} ⊆ 𝑏 ↔ {𝑖, 𝑗} ⊆ 𝐼))
3130anbi1d 632 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗) ↔ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)))
3231opabbidv 5100 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)})
3332opeq2d 4776 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)}⟩ = ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩)
3429, 33preq12d 4640 . . . . 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 4094 . . . 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 3866 . . 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 31115 . . 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 7463 . . . 4 {⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩} ∈ V
39 prex 5302 . . . 4 {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩} ∈ V
4038, 39unex 7462 . . 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 6755 . 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 399   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3442  ⦋csb 3830   ∪ cun 3881   ⊆ wss 3883  {cpr 4530  {ctp 4532  ⟨cop 4534  {copab 5096   ↦ cmpt 5114  ran crn 5524  ‘cfv 6332  (class class class)co 7145   ∈ cmpo 7147  ndxcnx 16492  Basecbs 16495  +gcplusg 16577  .rcmulr 16578  TopSetcts 16583  lecple 16584  LSSumclsm 18772  mulGrpcmgp 19253  LIdealclidl 19956  RSpancrsp 19957  IDLsrgcidlsrg 31114 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6291  df-fun 6334  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-idlsrg 31115 This theorem is referenced by:  idlsrgbas  31118  idlsrgplusg  31119  idlsrgmulr  31121  idlsrgtset  31122
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