Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  carsgmon Structured version   Visualization version   GIF version

Theorem carsgmon 31471
Description: Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.)
Hypotheses
Ref Expression
carsgval.1 (𝜑𝑂𝑉)
carsgval.2 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
carsgmon.1 (𝜑𝐴𝐵)
carsgmon.2 (𝜑𝐵 ∈ 𝒫 𝑂)
carsgmon.3 ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
Assertion
Ref Expression
carsgmon (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝑀,𝑦   𝑥,𝑂,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem carsgmon
StepHypRef Expression
1 carsgmon.2 . . 3 (𝜑𝐵 ∈ 𝒫 𝑂)
2 carsgmon.1 . . 3 (𝜑𝐴𝐵)
31, 2ssexd 5219 . 2 (𝜑𝐴 ∈ V)
4 id 22 . 2 (𝜑𝜑)
5 sseq1 3989 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
653anbi2d 1432 . . . . 5 (𝑥 = 𝐴 → ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) ↔ (𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂)))
7 fveq2 6663 . . . . . 6 (𝑥 = 𝐴 → (𝑀𝑥) = (𝑀𝐴))
87breq1d 5067 . . . . 5 (𝑥 = 𝐴 → ((𝑀𝑥) ≤ (𝑀𝑦) ↔ (𝑀𝐴) ≤ (𝑀𝑦)))
96, 8imbi12d 346 . . . 4 (𝑥 = 𝐴 → (((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦)) ↔ ((𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝐴) ≤ (𝑀𝑦))))
10 sseq2 3990 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
11 eleq1 2897 . . . . . 6 (𝑦 = 𝐵 → (𝑦 ∈ 𝒫 𝑂𝐵 ∈ 𝒫 𝑂))
1210, 113anbi23d 1430 . . . . 5 (𝑦 = 𝐵 → ((𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂) ↔ (𝜑𝐴𝐵𝐵 ∈ 𝒫 𝑂)))
13 fveq2 6663 . . . . . 6 (𝑦 = 𝐵 → (𝑀𝑦) = (𝑀𝐵))
1413breq2d 5069 . . . . 5 (𝑦 = 𝐵 → ((𝑀𝐴) ≤ (𝑀𝑦) ↔ (𝑀𝐴) ≤ (𝑀𝐵)))
1512, 14imbi12d 346 . . . 4 (𝑦 = 𝐵 → (((𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝐴) ≤ (𝑀𝑦)) ↔ ((𝜑𝐴𝐵𝐵 ∈ 𝒫 𝑂) → (𝑀𝐴) ≤ (𝑀𝐵))))
16 carsgmon.3 . . . 4 ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
179, 15, 16vtocl2g 3569 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) → ((𝜑𝐴𝐵𝐵 ∈ 𝒫 𝑂) → (𝑀𝐴) ≤ (𝑀𝐵)))
1817imp 407 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) ∧ (𝜑𝐴𝐵𝐵 ∈ 𝒫 𝑂)) → (𝑀𝐴) ≤ (𝑀𝐵))
193, 1, 4, 2, 1, 18syl23anc 1369 1 (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  𝒫 cpw 4535   class class class wbr 5057  wf 6344  cfv 6348  (class class class)co 7145  0cc0 10525  +∞cpnf 10660  cle 10664  [,]cicc 12729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by:  carsggect  31475  carsgclctunlem2  31476
  Copyright terms: Public domain W3C validator