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

Theorem letsr 17428
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 10384 . . 3 Rel ≤
2 lerelxr 10383 . . . . . . . . . . 11 ≤ ⊆ (ℝ* × ℝ*)
32brel 5365 . . . . . . . . . 10 (𝑥𝑦 → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*))
43adantr 468 . . . . . . . . 9 ((𝑥𝑦𝑦𝑧) → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*))
54simpld 484 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑥 ∈ ℝ*)
64simprd 485 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑦 ∈ ℝ*)
72brel 5365 . . . . . . . . . 10 (𝑦𝑧 → (𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
87simprd 485 . . . . . . . . 9 (𝑦𝑧𝑧 ∈ ℝ*)
98adantl 469 . . . . . . . 8 ((𝑥𝑦𝑦𝑧) → 𝑧 ∈ ℝ*)
105, 6, 93jca 1151 . . . . . . 7 ((𝑥𝑦𝑦𝑧) → (𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*))
11 xrletr 12203 . . . . . . 7 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1210, 11mpcom 38 . . . . . 6 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1312ax-gen 1880 . . . . 5 𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
1413gen2 1881 . . . 4 𝑥𝑦𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
15 cotr 5715 . . . 4 (( ≤ ∘ ≤ ) ⊆ ≤ ↔ ∀𝑥𝑦𝑧((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1614, 15mpbir 222 . . 3 ( ≤ ∘ ≤ ) ⊆ ≤
17 asymref 5720 . . . 4 (( ≤ ∩ ≤ ) = ( I ↾ ≤ ) ↔ ∀𝑥 ≤ ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
18 simpr 473 . . . . . . . . 9 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → (𝑥𝑦𝑦𝑥))
192brel 5365 . . . . . . . . . . . 12 (𝑦𝑥 → (𝑦 ∈ ℝ*𝑥 ∈ ℝ*))
2019simpld 484 . . . . . . . . . . 11 (𝑦𝑥𝑦 ∈ ℝ*)
2120adantl 469 . . . . . . . . . 10 ((𝑥𝑦𝑦𝑥) → 𝑦 ∈ ℝ*)
22 xrletri3 12199 . . . . . . . . . 10 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥 = 𝑦 ↔ (𝑥𝑦𝑦𝑥)))
2321, 22sylan2 582 . . . . . . . . 9 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → (𝑥 = 𝑦 ↔ (𝑥𝑦𝑦𝑥)))
2418, 23mpbird 248 . . . . . . . 8 ((𝑥 ∈ ℝ* ∧ (𝑥𝑦𝑦𝑥)) → 𝑥 = 𝑦)
2524ex 399 . . . . . . 7 (𝑥 ∈ ℝ* → ((𝑥𝑦𝑦𝑥) → 𝑥 = 𝑦))
26 xrleid 12196 . . . . . . . . 9 (𝑥 ∈ ℝ*𝑥𝑥)
2726, 26jca 503 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑥𝑥𝑥𝑥))
28 breq2 4844 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
29 breq1 4843 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
3028, 29anbi12d 618 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥𝑥𝑥𝑥) ↔ (𝑥𝑦𝑦𝑥)))
3127, 30syl5ibcom 236 . . . . . . 7 (𝑥 ∈ ℝ* → (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥)))
3225, 31impbid 203 . . . . . 6 (𝑥 ∈ ℝ* → ((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
3332alrimiv 2020 . . . . 5 (𝑥 ∈ ℝ* → ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
34 lefld 17427 . . . . . 6 * =
3534eqcomi 2814 . . . . 5 ≤ = ℝ*
3633, 35eleq2s 2902 . . . 4 (𝑥 ≤ → ∀𝑦((𝑥𝑦𝑦𝑥) ↔ 𝑥 = 𝑦))
3717, 36mprgbir 3114 . . 3 ( ≤ ∩ ≤ ) = ( I ↾ ≤ )
38 xrex 12039 . . . . . 6 * ∈ V
3938, 38xpex 7189 . . . . 5 (ℝ* × ℝ*) ∈ V
4039, 2ssexi 4995 . . . 4 ≤ ∈ V
41 isps 17403 . . . 4 ( ≤ ∈ V → ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ≤ ) = ( I ↾ ≤ ))))
4240, 41ax-mp 5 . . 3 ( ≤ ∈ PosetRel ↔ (Rel ≤ ∧ ( ≤ ∘ ≤ ) ⊆ ≤ ∧ ( ≤ ∩ ≤ ) = ( I ↾ ≤ )))
431, 16, 37, 42mpbir3an 1434 . 2 ≤ ∈ PosetRel
44 xrletri 12198 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥𝑦𝑦𝑥))
4544rgen2a 3164 . . 3 𝑥 ∈ ℝ*𝑦 ∈ ℝ* (𝑥𝑦𝑦𝑥)
46 qfto 5725 . . 3 ((ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ ) ↔ ∀𝑥 ∈ ℝ*𝑦 ∈ ℝ* (𝑥𝑦𝑦𝑥))
4745, 46mpbir 222 . 2 (ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ )
48 ledm 17425 . . 3 * = dom ≤
4948istsr 17418 . 2 ( ≤ ∈ TosetRel ↔ ( ≤ ∈ PosetRel ∧ (ℝ* × ℝ*) ⊆ ( ≤ ∪ ≤ )))
5043, 47, 49mpbir2an 693 1 ≤ ∈ TosetRel
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 865  w3a 1100  wal 1635   = wceq 1637  wcel 2158  wral 3095  Vcvv 3390  cun 3764  cin 3765  wss 3766   cuni 4626   class class class wbr 4840   I cid 5215   × cxp 5306  ccnv 5307  cres 5310  ccom 5312  Rel wrel 5313  *cxr 10355  cle 10357  PosetRelcps 17399   TosetRel ctsr 17400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176  ax-cnex 10274  ax-resscn 10275  ax-pre-lttri 10292  ax-pre-lttrn 10293
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-br 4841  df-opab 4903  df-mpt 4920  df-id 5216  df-po 5229  df-so 5230  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-er 7976  df-en 8190  df-dom 8191  df-sdom 8192  df-pnf 10358  df-mnf 10359  df-xr 10360  df-ltxr 10361  df-le 10362  df-ps 17401  df-tsr 17402
This theorem is referenced by:  cnfldle  19959  cnfldfun  19962  cnfldfunALT  19963  letopon  21219  leordtval2  21226  leordtval  21227  iccordt  21228  ordtrestixx  21236  xrge0tsms  22846  icopnfhmeo  22951  iccpnfhmeo  22953  xrhmeo  22954  xrhaus  29859  xrge0tsmsd  30107  cnvordtrestixx  30281  xrmulc1cn  30298  xrge0iifhmeo  30304  poimir  33752
  Copyright terms: Public domain W3C validator