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Theorem o1compt 14247
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 2626 . . 3 (𝑦𝐵𝐶) = (𝑦𝐵𝐶)
53, 4fmptd 6341 . 2 (𝜑 → (𝑦𝐵𝐶):𝐵𝐴)
6 o1compt.4 . 2 (𝜑𝐵 ⊆ ℝ)
7 o1compt.5 . . 3 ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶))
8 nfv 1845 . . . . . . . 8 𝑦 𝑥𝑧
9 nfcv 2767 . . . . . . . . 9 𝑦𝑚
10 nfcv 2767 . . . . . . . . 9 𝑦
11 nffvmpt1 6158 . . . . . . . . 9 𝑦((𝑦𝐵𝐶)‘𝑧)
129, 10, 11nfbr 4664 . . . . . . . 8 𝑦 𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)
138, 12nfim 1827 . . . . . . 7 𝑦(𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧))
14 nfv 1845 . . . . . . 7 𝑧(𝑥𝑦𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦))
15 breq2 4622 . . . . . . . 8 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
16 fveq2 6150 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑦𝐵𝐶)‘𝑧) = ((𝑦𝐵𝐶)‘𝑦))
1716breq2d 4630 . . . . . . . 8 (𝑧 = 𝑦 → (𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧) ↔ 𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦)))
1815, 17imbi12d 334 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)) ↔ (𝑥𝑦𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦))))
1913, 14, 18cbvral 3160 . . . . . 6 (∀𝑧𝐵 (𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)) ↔ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦)))
20 simpr 477 . . . . . . . . . 10 ((𝜑𝑦𝐵) → 𝑦𝐵)
214fvmpt2 6249 . . . . . . . . . 10 ((𝑦𝐵𝐶𝐴) → ((𝑦𝐵𝐶)‘𝑦) = 𝐶)
2220, 3, 21syl2anc 692 . . . . . . . . 9 ((𝜑𝑦𝐵) → ((𝑦𝐵𝐶)‘𝑦) = 𝐶)
2322breq2d 4630 . . . . . . . 8 ((𝜑𝑦𝐵) → (𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦) ↔ 𝑚𝐶))
2423imbi2d 330 . . . . . . 7 ((𝜑𝑦𝐵) → ((𝑥𝑦𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦)) ↔ (𝑥𝑦𝑚𝐶)))
2524ralbidva 2984 . . . . . 6 (𝜑 → (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ ((𝑦𝐵𝐶)‘𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶)))
2619, 25syl5bb 272 . . . . 5 (𝜑 → (∀𝑧𝐵 (𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)) ↔ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶)))
2726rexbidv 3050 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧𝐵 (𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)) ↔ ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶)))
2827adantr 481 . . 3 ((𝜑𝑚 ∈ ℝ) → (∃𝑥 ∈ ℝ ∀𝑧𝐵 (𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)) ↔ ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚𝐶)))
297, 28mpbird 247 . 2 ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧𝐵 (𝑥𝑧𝑚 ≤ ((𝑦𝐵𝐶)‘𝑧)))
301, 2, 5, 6, 29o1co 14246 1 (𝜑 → (𝐹 ∘ (𝑦𝐵𝐶)) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  wss 3560   class class class wbr 4618  cmpt 4678  ccom 5083  wf 5846  cfv 5850  cc 9879  cr 9880  cle 10020  𝑂(1)co1 14146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-pre-lttri 9955  ax-pre-lttrn 9956
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-er 7688  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-ico 12120  df-o1 14150
This theorem is referenced by:  dchrisum0  25104
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