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Theorem o1compt 14166
 Description: Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)
Hypotheses
Ref Expression
o1compt.1 (𝜑𝐹:𝐴⟶ℂ)
o1compt.2 (𝜑𝐹 ∈ 𝑂(1))
o1compt.3 ((𝜑𝑦𝐵) → 𝐶𝐴)
o1compt.4 (𝜑𝐵 ⊆ ℝ)
o1compt.5 ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶))
Assertion
Ref Expression
o1compt (𝜑 → (𝐹 ∘ (𝑦𝐵𝐶)) ∈ 𝑂(1))
Distinct variable groups:   𝑥,𝑚,𝑦,𝐴   𝐵,𝑚,𝑥,𝑦   𝐶,𝑚,𝑥   𝜑,𝑚,𝑥,𝑦   𝑚,𝐹,𝑥
Allowed substitution hints:   𝐶(𝑦)   𝐹(𝑦)

Proof of Theorem o1compt
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 o1compt.1 . 2 (𝜑𝐹:𝐴⟶ℂ)
2 o1compt.2 . 2 (𝜑𝐹 ∈ 𝑂(1))
3 o1compt.3 . . 3 ((𝜑𝑦𝐵) → 𝐶𝐴)
4 eqid 2610 . . 3 (𝑦𝐵𝐶) = (𝑦𝐵𝐶)
53, 4fmptd 6292 . 2 (𝜑 → (𝑦𝐵𝐶):𝐵𝐴)
6 o1compt.4 . 2 (𝜑𝐵 ⊆ ℝ)
7 o1compt.5 . . 3 ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶))
8 nfv 1830 . . . . . . . 8 𝑦 𝑥𝑧
9 nfcv 2751 . . . . . . . . 9 𝑦𝑚
10 nfcv 2751 . . . . . . . . 9 𝑦
11 nffvmpt1 6111 . . . . . . . . 9 𝑦((𝑦𝐵𝐶)‘𝑧)
129, 10, 11nfbr 4629 . . . . . . . 8 𝑦 𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)
138, 12nfim 1813 . . . . . . 7 𝑦(𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧))
14 nfv 1830 . . . . . . 7 𝑧(𝑥𝑦𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦))
15 breq2 4587 . . . . . . . 8 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
16 fveq2 6103 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑦𝐵𝐶)‘𝑧) = ((𝑦𝐵𝐶)‘𝑦))
1716breq2d 4595 . . . . . . . 8 (𝑧 = 𝑦 → (𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧) ↔ 𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦)))
1815, 17imbi12d 333 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)) ↔ (𝑥𝑦𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦))))
1913, 14, 18cbvral 3143 . . . . . 6 (∀𝑧𝐵 (𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)) ↔ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦)))
20 simpr 476 . . . . . . . . . 10 ((𝜑𝑦𝐵) → 𝑦𝐵)
214fvmpt2 6200 . . . . . . . . . 10 ((𝑦𝐵𝐶𝐴) → ((𝑦𝐵𝐶)‘𝑦) = 𝐶)
2220, 3, 21syl2anc 691 . . . . . . . . 9 ((𝜑𝑦𝐵) → ((𝑦𝐵𝐶)‘𝑦) = 𝐶)
2322breq2d 4595 . . . . . . . 8 ((𝜑𝑦𝐵) → (𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦) ↔ 𝑚𝐶))
2423imbi2d 329 . . . . . . 7 ((𝜑𝑦𝐵) → ((𝑥𝑦𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦)) ↔ (𝑥𝑦𝑚𝐶)))
2524ralbidva 2968 . . . . . 6 (𝜑 → (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶)))
2619, 25syl5bb 271 . . . . 5 (𝜑 → (∀𝑧𝐵 (𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)) ↔ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶)))
2726rexbidv 3034 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧𝐵 (𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)) ↔ ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶)))
2827adantr 480 . . 3 ((𝜑𝑚 ∈ ℝ) → (∃𝑥 ∈ ℝ ∀𝑧𝐵 (𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)) ↔ ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶)))
297, 28mpbird 246 . 2 ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧𝐵 (𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)))
301, 2, 5, 6, 29o1co 14165 1 (𝜑 → (𝐹 ∘ (𝑦𝐵𝐶)) ∈ 𝑂(1))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  ℂcc 9813  ℝcr 9814   ≤ cle 9954  𝑂(1)co1 14065 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-ico 12052  df-o1 14069 This theorem is referenced by:  dchrisum0  25009
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