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Theorem carsgmon 31477
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 5225 . 2 (𝜑𝐴 ∈ V)
4 id 22 . 2 (𝜑𝜑)
5 sseq1 3996 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
653anbi2d 1434 . . . . 5 (𝑥 = 𝐴 → ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) ↔ (𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂)))
7 fveq2 6669 . . . . . 6 (𝑥 = 𝐴 → (𝑀𝑥) = (𝑀𝐴))
87breq1d 5073 . . . . 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 3997 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
11 eleq1 2905 . . . . . 6 (𝑦 = 𝐵 → (𝑦 ∈ 𝒫 𝑂𝐵 ∈ 𝒫 𝑂))
1210, 113anbi23d 1432 . . . . 5 (𝑦 = 𝐵 → ((𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂) ↔ (𝜑𝐴𝐵𝐵 ∈ 𝒫 𝑂)))
13 fveq2 6669 . . . . . 6 (𝑦 = 𝐵 → (𝑀𝑦) = (𝑀𝐵))
1413breq2d 5075 . . . . 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 3577 . . 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 1371 1 (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  Vcvv 3500  wss 3940  𝒫 cpw 4542   class class class wbr 5063  wf 6350  cfv 6354  (class class class)co 7150  0cc0 10531  +∞cpnf 10666  cle 10670  [,]cicc 12736
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
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-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-iota 6313  df-fv 6362
This theorem is referenced by:  carsggect  31481  carsgclctunlem2  31482
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