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

Theorem prcdnq 10409
Description: A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
prcdnq ((𝐴P𝐵𝐴) → (𝐶 <Q 𝐵𝐶𝐴))

Proof of Theorem prcdnq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 10342 . . . . . . 7 <Q ⊆ (Q × Q)
2 relxp 5572 . . . . . . 7 Rel (Q × Q)
3 relss 5655 . . . . . . 7 ( <Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel <Q ))
41, 2, 3mp2 9 . . . . . 6 Rel <Q
54brrelex1i 5607 . . . . 5 (𝐶 <Q 𝐵𝐶 ∈ V)
6 eleq1 2905 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
76anbi2d 628 . . . . . . . 8 (𝑥 = 𝐵 → ((𝐴P𝑥𝐴) ↔ (𝐴P𝐵𝐴)))
8 breq2 5067 . . . . . . . 8 (𝑥 = 𝐵 → (𝑦 <Q 𝑥𝑦 <Q 𝐵))
97, 8anbi12d 630 . . . . . . 7 (𝑥 = 𝐵 → (((𝐴P𝑥𝐴) ∧ 𝑦 <Q 𝑥) ↔ ((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵)))
109imbi1d 343 . . . . . 6 (𝑥 = 𝐵 → ((((𝐴P𝑥𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦𝐴) ↔ (((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦𝐴)))
11 breq1 5066 . . . . . . . 8 (𝑦 = 𝐶 → (𝑦 <Q 𝐵𝐶 <Q 𝐵))
1211anbi2d 628 . . . . . . 7 (𝑦 = 𝐶 → (((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵) ↔ ((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵)))
13 eleq1 2905 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝐴𝐶𝐴))
1412, 13imbi12d 346 . . . . . 6 (𝑦 = 𝐶 → ((((𝐴P𝐵𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦𝐴) ↔ (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴)))
15 elnpi 10404 . . . . . . . . . . 11 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
1615simprbi 497 . . . . . . . . . 10 (𝐴P → ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))
1716r19.21bi 3213 . . . . . . . . 9 ((𝐴P𝑥𝐴) → (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))
1817simpld 495 . . . . . . . 8 ((𝐴P𝑥𝐴) → ∀𝑦(𝑦 <Q 𝑥𝑦𝐴))
191819.21bi 2180 . . . . . . 7 ((𝐴P𝑥𝐴) → (𝑦 <Q 𝑥𝑦𝐴))
2019imp 407 . . . . . 6 (((𝐴P𝑥𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦𝐴)
2110, 14, 20vtocl2g 3577 . . . . 5 ((𝐵𝐴𝐶 ∈ V) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
225, 21sylan2 592 . . . 4 ((𝐵𝐴𝐶 <Q 𝐵) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
2322adantll 710 . . 3 (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴))
2423pm2.43i 52 . 2 (((𝐴P𝐵𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶𝐴)
2524ex 413 1 ((𝐴P𝐵𝐴) → (𝐶 <Q 𝐵𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081  wal 1528   = wceq 1530  wcel 2107  wral 3143  wrex 3144  Vcvv 3500  wss 3940  wpss 3941  c0 4295   class class class wbr 5063   × cxp 5552  Rel wrel 5559  Qcnq 10268   <Q cltq 10274  Pcnp 10275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5560  df-rel 5561  df-ltnq 10334  df-np 10397
This theorem is referenced by:  prub  10410  addclprlem1  10432  mulclprlem  10435  distrlem4pr  10442  1idpr  10445  psslinpr  10447  prlem934  10449  ltaddpr  10450  ltexprlem2  10453  ltexprlem3  10454  ltexprlem6  10457  prlem936  10463  reclem2pr  10464  suplem1pr  10468
  Copyright terms: Public domain W3C validator