MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tuslemOLD Structured version   Visualization version   GIF version

Theorem tuslemOLD 23635
Description: Obsolete proof of tuslem 23634 as of 28-Oct-2024. Lemma for tusbas 23636, tusunif 23637, and tustopn 23639. (Contributed by Thierry Arnoux, 5-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
tuslem.k 𝐾 = (toUnifSp‘𝑈)
Assertion
Ref Expression
tuslemOLD (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))

Proof of Theorem tuslemOLD
StepHypRef Expression
1 baseid 17091 . . . 4 Base = Slot (Base‘ndx)
2 1re 11160 . . . . . 6 1 ∈ ℝ
3 1lt9 12364 . . . . . 6 1 < 9
42, 3ltneii 11273 . . . . 5 1 ≠ 9
5 basendx 17097 . . . . . 6 (Base‘ndx) = 1
6 tsetndx 17238 . . . . . 6 (TopSet‘ndx) = 9
75, 6neeq12i 3007 . . . . 5 ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9)
84, 7mpbir 230 . . . 4 (Base‘ndx) ≠ (TopSet‘ndx)
91, 8setsnid 17086 . . 3 (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
10 ustbas2 23593 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
11 uniexg 7678 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ V)
12 dmexg 7841 . . . . 5 ( 𝑈 ∈ V → dom 𝑈 ∈ V)
13 eqid 2733 . . . . . 6 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
14 df-unif 17161 . . . . . 6 UnifSet = Slot 13
15 1nn 12169 . . . . . . 7 1 ∈ ℕ
16 3nn0 12436 . . . . . . 7 3 ∈ ℕ0
17 1nn0 12434 . . . . . . 7 1 ∈ ℕ0
18 1lt10 12762 . . . . . . 7 1 < 10
1915, 16, 17, 18declti 12661 . . . . . 6 1 < 13
20 3nn 12237 . . . . . . 7 3 ∈ ℕ
2117, 20decnncl 12643 . . . . . 6 13 ∈ ℕ
2213, 14, 19, 212strbas 17111 . . . . 5 (dom 𝑈 ∈ V → dom 𝑈 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
2311, 12, 223syl 18 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → dom 𝑈 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
2410, 23eqtrd 2773 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
25 tuslem.k . . . . 5 𝐾 = (toUnifSp‘𝑈)
26 tusval 23633 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
2725, 26eqtrid 2785 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
2827fveq2d 6847 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
299, 24, 283eqtr4a 2799 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
30 unifid 17282 . . . 4 UnifSet = Slot (UnifSet‘ndx)
31 9re 12257 . . . . . 6 9 ∈ ℝ
32 9nn0 12442 . . . . . . 7 9 ∈ ℕ0
33 9lt10 12754 . . . . . . 7 9 < 10
3415, 16, 32, 33declti 12661 . . . . . 6 9 < 13
3531, 34gtneii 11272 . . . . 5 13 ≠ 9
36 unifndx 17281 . . . . . 6 (UnifSet‘ndx) = 13
3736, 6neeq12i 3007 . . . . 5 ((UnifSet‘ndx) ≠ (TopSet‘ndx) ↔ 13 ≠ 9)
3835, 37mpbir 230 . . . 4 (UnifSet‘ndx) ≠ (TopSet‘ndx)
3930, 38setsnid 17086 . . 3 (UnifSet‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (UnifSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
4013, 14, 19, 212strop 17112 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
4127fveq2d 6847 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
4239, 40, 413eqtr4a 2799 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
43 prex 5390 . . . . 5 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
44 fvex 6856 . . . . 5 (unifTop‘𝑈) ∈ V
45 tsetid 17239 . . . . . 6 TopSet = Slot (TopSet‘ndx)
4645setsid 17085 . . . . 5 (({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V ∧ (unifTop‘𝑈) ∈ V) → (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
4743, 44, 46mp2an 691 . . . 4 (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
4827fveq2d 6847 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
4947, 48eqtr4id 2792 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
50 utopbas 23603 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
5149unieqd 4880 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
5250, 29, 513eqtr3rd 2782 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (Base‘𝐾))
5352oveq2d 7374 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
54 fvex 6856 . . . . 5 (TopSet‘𝐾) ∈ V
55 eqid 2733 . . . . . 6 (TopSet‘𝐾) = (TopSet‘𝐾)
5655restid 17320 . . . . 5 ((TopSet‘𝐾) ∈ V → ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = (TopSet‘𝐾))
5754, 56ax-mp 5 . . . 4 ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = (TopSet‘𝐾)
58 eqid 2733 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
59 eqid 2733 . . . . 5 (TopSet‘𝐾) = (TopSet‘𝐾)
6058, 59topnval 17321 . . . 4 ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
6153, 57, 603eqtr3g 2796 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
6249, 61eqtrd 2773 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
6329, 42, 623jca 1129 1 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  wne 2940  Vcvv 3444  {cpr 4589  cop 4593   cuni 4866  dom cdm 5634  cfv 6497  (class class class)co 7358  1c1 11057  3c3 12214  9c9 12220  cdc 12623   sSet csts 17040  ndxcnx 17070  Basecbs 17088  TopSetcts 17144  UnifSetcunif 17148  t crest 17307  TopOpenctopn 17308  UnifOncust 23567  unifTopcutop 23598  toUnifSpctus 23623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-tset 17157  df-unif 17161  df-rest 17309  df-topn 17310  df-ust 23568  df-utop 23599  df-tus 23626
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator