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Theorem idlsrgval 32245
Description: Lemma for idlsrgbas 32246 through idlsrgtset 32250. (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 3463 . 2 (𝑅𝑉𝑅 ∈ V)
2 fvexd 6857 . . . 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 6846 . . . . . . . . 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 4837 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐼⟩)
104fveq2d 6846 . . . . . . . 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 4837 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(+g‘ndx), (LSSum‘𝑟)⟩ = ⟨(+g‘ndx), ⟩)
144fveq2d 6846 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (RSpan‘𝑟) = (RSpan‘𝑅))
154fveq2d 6846 . . . . . . . . . . . . 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 6846 . . . . . . . . . . 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 7374 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (𝑖(LSSum‘(mulGrp‘𝑟))𝑗) = (𝑖 𝑗))
2214, 21fveq12d 6849 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)) = ((RSpan‘𝑅)‘(𝑖 𝑗)))
238, 8, 22mpoeq123dv 7432 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (𝑖𝑏, 𝑗𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗))) = (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗))))
2423opeq2d 4837 . . . . . 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 4709 . . . . 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 3422 . . . . . . . . 9 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → {𝑗𝑏 ∣ ¬ 𝑖𝑗} = {𝑗𝐼 ∣ ¬ 𝑖𝑗})
278, 26mpteq12dv 5196 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗}) = (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}))
2827rneqd 5893 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗}) = ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}))
2928opeq2d 4837 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(TopSet‘ndx), ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗})⟩ = ⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩)
308sseq2d 3976 . . . . . . . . 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 5171 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)})
3332opeq2d 4837 . . . . . 6 ((𝑟 = 𝑅𝑏 = (LIdeal‘𝑟)) → ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)}⟩ = ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩)
3429, 33preq12d 4702 . . . . 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 4124 . . . 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 3893 . . 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 32243 . . 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 7681 . . . 4 {⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩} ∈ V
39 prex 5389 . . . 4 {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩} ∈ V
4038, 39unex 7680 . . 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 6948 . 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 3407  Vcvv 3445  csb 3855  cun 3908  wss 3910  {cpr 4588  {ctp 4590  cop 4592  {copab 5167  cmpt 5188  ran crn 5634  cfv 6496  (class class class)co 7357  cmpo 7359  ndxcnx 17065  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  TopSetcts 17139  lecple 17140  LSSumclsm 19416  mulGrpcmgp 19896  LIdealclidl 20631  RSpancrsp 20632  IDLsrgcidlsrg 32242
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 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
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 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-idlsrg 32243
This theorem is referenced by:  idlsrgbas  32246  idlsrgplusg  32247  idlsrgmulr  32249  idlsrgtset  32250
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