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

Theorem letsr 18546
Description: The "less than or equal to" relationship on the extended reals is a toset. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
letsr ≤ ∈ TosetRel

Proof of Theorem letsr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lerel 11278 . . 3 Rel ≤
2 lerelxr 11277 . . . . . . . . . . 11 ≤ ⊆ (ℝ* × ℝ*)
32brel 5742 . . . . . . . . . 10 (𝑥𝑦 → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*))
43adantr 482 . . . . . . . . 9 ((𝑥𝑦𝑦𝑧) → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*))
54simpld 496 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑥 ∈ ℝ*)
64simprd 497 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑦 ∈ ℝ*)
72brel 5742 . . . . . . . . . 10 (𝑦𝑧 → (𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
87simprd 497 . . . . . . . . 9 (𝑦𝑧𝑧 ∈ ℝ*)
98adantl 483 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑧 ∈ ℝ*)
105, 6, 93jca 1129 . . . . . . 7 ((𝑥𝑦𝑦𝑧) → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
11 xrletr 13137 . . . . . . 7 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1210, 11mpcom 38 . . . . . 6 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1312ax-gen 1798 . . . . 5 𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1413gen2 1799 . . . 4 𝑥𝑦𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
15 cotr 6112 . . . 4 (( ≤ ∘ ≤ ) ⊆ ≤ ↔ ∀𝑥𝑦𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1614, 15mpbir 230 . . 3 ( ≤ ∘ ≤ ) ⊆ ≤
17 asymref 6118 . . . 4 (( ≤ ∩ ≤ ) = ( I ↾ ≤ ) ↔ ∀𝑥 ≤ ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
18 simpr 486 . . . . . . . . 9 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → (𝑥𝑦𝑦𝑥))
192brel 5742 . . . . . . . . . . . 12 (𝑦𝑥 → (𝑦 ∈ ℝ*𝑥 ∈ ℝ*))
2019simpld 496 . . . . . . . . . . 11 (𝑦𝑥𝑦 ∈ ℝ*)
2120adantl 483 . . . . . . . . . 10 ((𝑥𝑦𝑦𝑥) → 𝑦 ∈ ℝ*)
22 xrletri3 13133 . . . . . . . . . 10 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥𝑦𝑦𝑥)))
2321, 22sylan2 594 . . . . . . . . 9 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → (𝑥 = 𝑦 ↔ (𝑥𝑦𝑦𝑥)))
2418, 23mpbird 257 . . . . . . . 8 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → 𝑥 = 𝑦)
2524ex 414 . . . . . . 7 (𝑥 ∈ ℝ* → ((𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
26 xrleid 13130 . . . . . . . . 9 (𝑥 ∈ ℝ*𝑥𝑥)
2726, 26jca 513 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑥𝑥𝑥𝑥))
28 breq2 5153 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
29 breq1 5152 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
3028, 29anbi12d 632 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥𝑥𝑥𝑥) ↔ (𝑥𝑦𝑦𝑥)))
3127, 30syl5ibcom 244 . . . . . . 7 (𝑥 ∈ ℝ* → (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥)))
3225, 31impbid 211 . . . . . 6 (𝑥 ∈ ℝ* → ((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
3332alrimiv 1931 . . . . 5 (𝑥 ∈ ℝ* → ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
34 lefld 18545 . . . . . 6 * =
3534eqcomi 2742 . . . . 5 ≤ = ℝ*
3633, 35eleq2s 2852 . . . 4 (𝑥 ≤ → ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
3717, 36mprgbir 3069 . . 3 ( ≤ ∩ ≤ ) = ( I ↾ ≤ )
38 xrex 12971 . . . . . 6 * ∈ V
3938, 38xpex 7740 . . . . 5 (ℝ* × ℝ*) ∈ V
4039, 2ssexi 5323 . . . 4 ≤ ∈ V
41 isps 18521 . . . 4 ( ≤ ∈ V → ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ≤ ) = ( I ↾ ≤ ))))
4240, 41ax-mp 5 . . 3 ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ≤ ) = ( I ↾ ≤ )))
431, 16, 37, 42mpbir3an 1342 . 2 ≤ ∈ PosetRel
44 xrletri 13132 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥𝑦𝑦𝑥))
4544rgen2 3198 . . 3 𝑥 ∈ ℝ*𝑦 ∈ ℝ* (𝑥𝑦𝑦𝑥)
46 qfto 6123 . . 3 ((ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ ) ↔ ∀𝑥 ∈ ℝ*𝑦 ∈ ℝ* (𝑥𝑦𝑦𝑥))
4745, 46mpbir 230 . 2 (ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ )
48 ledm 18543 . . 3 * = dom ≤
4948istsr 18536 . 2 ( ≤ ∈ TosetRel ↔ ( ≤ ∈ PosetRel ∧ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ )))
5043, 47, 49mpbir2an 710 1 ≤ ∈ TosetRel
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  w3a 1088  wal 1540   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cun 3947  cin 3948  wss 3949   cuni 4909   class class class wbr 5149   I cid 5574   × cxp 5675  ccnv 5676  cres 5679  ccom 5681  Rel wrel 5682  *cxr 11247  cle 11249  PosetRelcps 18517   TosetRel ctsr 18518
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-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-pre-lttri 11184  ax-pre-lttrn 11185
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-ps 18519  df-tsr 18520
This theorem is referenced by:  cnfldle  20953  cnfldfun  20956  cnfldfunALT  20957  cnfldfunALTOLD  20958  letopon  22709  leordtval2  22716  leordtval  22717  iccordt  22718  ordtrestixx  22726  xrhaus  22889  xrge0tsms  24350  icopnfhmeo  24459  iccpnfhmeo  24461  xrhmeo  24462  xrge0tsmsd  32209  cnvordtrestixx  32893  xrmulc1cn  32910  xrge0iifhmeo  32916  poimir  36521
  Copyright terms: Public domain W3C validator