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Theorem carsgmon 29509
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 4728 . 2 (𝜑𝐴 ∈ V)
4 id 22 . 2 (𝜑𝜑)
5 sseq1 3588 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
653anbi2d 1395 . . . . 5 (𝑥 = 𝐴 → ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) ↔ (𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂)))
7 fveq2 6088 . . . . . 6 (𝑥 = 𝐴 → (𝑀𝑥) = (𝑀𝐴))
87breq1d 4587 . . . . 5 (𝑥 = 𝐴 → ((𝑀𝑥) ≤ (𝑀𝑦) ↔ (𝑀𝐴) ≤ (𝑀𝑦)))
96, 8imbi12d 332 . . . 4 (𝑥 = 𝐴 → (((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦)) ↔ ((𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝐴) ≤ (𝑀𝑦))))
10 sseq2 3589 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
11 eleq1 2675 . . . . . 6 (𝑦 = 𝐵 → (𝑦 ∈ 𝒫 𝑂𝐵 ∈ 𝒫 𝑂))
1210, 113anbi23d 1393 . . . . 5 (𝑦 = 𝐵 → ((𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂) ↔ (𝜑𝐴𝐵𝐵 ∈ 𝒫 𝑂)))
13 fveq2 6088 . . . . . 6 (𝑦 = 𝐵 → (𝑀𝑦) = (𝑀𝐵))
1413breq2d 4589 . . . . 5 (𝑦 = 𝐵 → ((𝑀𝐴) ≤ (𝑀𝑦) ↔ (𝑀𝐴) ≤ (𝑀𝐵)))
1512, 14imbi12d 332 . . . 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 3242 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) → ((𝜑𝐴𝐵𝐵 ∈ 𝒫 𝑂) → (𝑀𝐴) ≤ (𝑀𝐵)))
1817imp 443 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) ∧ (𝜑𝐴𝐵𝐵 ∈ 𝒫 𝑂)) → (𝑀𝐴) ≤ (𝑀𝐵))
193, 1, 4, 2, 1, 18syl23anc 1324 1 (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3172  wss 3539  𝒫 cpw 4107   class class class wbr 4577  wf 5786  cfv 5790  (class class class)co 6527  0cc0 9792  +∞cpnf 9927  cle 9931  [,]cicc 12005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798
This theorem is referenced by:  carsggect  29513  carsgclctunlem2  29514
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