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Theorem carsgmon 30685
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 4957 . 2 (𝜑𝐴 ∈ V)
4 id 22 . 2 (𝜑𝜑)
5 sseq1 3767 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
653anbi2d 1553 . . . . 5 (𝑥 = 𝐴 → ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) ↔ (𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂)))
7 fveq2 6352 . . . . . 6 (𝑥 = 𝐴 → (𝑀𝑥) = (𝑀𝐴))
87breq1d 4814 . . . . 5 (𝑥 = 𝐴 → ((𝑀𝑥) ≤ (𝑀𝑦) ↔ (𝑀𝐴) ≤ (𝑀𝑦)))
96, 8imbi12d 333 . . . 4 (𝑥 = 𝐴 → (((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦)) ↔ ((𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝐴) ≤ (𝑀𝑦))))
10 sseq2 3768 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
11 eleq1 2827 . . . . . 6 (𝑦 = 𝐵 → (𝑦 ∈ 𝒫 𝑂𝐵 ∈ 𝒫 𝑂))
1210, 113anbi23d 1551 . . . . 5 (𝑦 = 𝐵 → ((𝜑𝐴𝑦𝑦 ∈ 𝒫 𝑂) ↔ (𝜑𝐴𝐵𝐵 ∈ 𝒫 𝑂)))
13 fveq2 6352 . . . . . 6 (𝑦 = 𝐵 → (𝑀𝑦) = (𝑀𝐵))
1413breq2d 4816 . . . . 5 (𝑦 = 𝐵 → ((𝑀𝐴) ≤ (𝑀𝑦) ↔ (𝑀𝐴) ≤ (𝑀𝐵)))
1512, 14imbi12d 333 . . . 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 3410 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) → ((𝜑𝐴𝐵𝐵 ∈ 𝒫 𝑂) → (𝑀𝐴) ≤ (𝑀𝐵)))
1817imp 444 . 2 (((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) ∧ (𝜑𝐴𝐵𝐵 ∈ 𝒫 𝑂)) → (𝑀𝐴) ≤ (𝑀𝐵))
193, 1, 4, 2, 1, 18syl23anc 1484 1 (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340  wss 3715  𝒫 cpw 4302   class class class wbr 4804  wf 6045  cfv 6049  (class class class)co 6813  0cc0 10128  +∞cpnf 10263  cle 10267  [,]cicc 12371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057
This theorem is referenced by:  carsggect  30689  carsgclctunlem2  30690
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