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Theorem o1co 14525
Description: Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)
Hypotheses
Ref Expression
o1co.1 (𝜑𝐹:𝐴⟶ℂ)
o1co.2 (𝜑𝐹 ∈ 𝑂(1))
o1co.3 (𝜑𝐺:𝐵𝐴)
o1co.4 (𝜑𝐵 ⊆ ℝ)
o1co.5 ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)))
Assertion
Ref Expression
o1co (𝜑 → (𝐹𝐺) ∈ 𝑂(1))
Distinct variable groups:   𝑥,𝑚,𝑦,𝐴   𝑚,𝐹,𝑥,𝑦   𝑚,𝐺,𝑥,𝑦   𝜑,𝑚,𝑥,𝑦   𝐵,𝑚,𝑥,𝑦

Proof of Theorem o1co
Dummy variables 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 o1co.2 . . . 4 (𝜑𝐹 ∈ 𝑂(1))
2 o1co.1 . . . . 5 (𝜑𝐹:𝐴⟶ℂ)
32fdmd 6193 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
4 o1dm 14469 . . . . . . 7 (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ)
51, 4syl 17 . . . . . 6 (𝜑 → dom 𝐹 ⊆ ℝ)
63, 5eqsstr3d 3789 . . . . 5 (𝜑𝐴 ⊆ ℝ)
7 elo12 14466 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)))
82, 6, 7syl2anc 573 . . . 4 (𝜑 → (𝐹 ∈ 𝑂(1) ↔ ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)))
91, 8mpbid 222 . . 3 (𝜑 → ∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛))
10 o1co.5 . . . . 5 ((𝜑𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)))
11 reeanv 3255 . . . . . 6 (∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ↔ (∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)))
12 o1co.3 . . . . . . . . . . . . . . . . 17 (𝜑𝐺:𝐵𝐴)
1312ad3antrrr 709 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → 𝐺:𝐵𝐴)
1413ffvelrnda 6504 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦𝐵) → (𝐺𝑦) ∈ 𝐴)
15 breq2 4791 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐺𝑦) → (𝑚𝑧𝑚 ≤ (𝐺𝑦)))
16 fveq2 6333 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐺𝑦) → (𝐹𝑧) = (𝐹‘(𝐺𝑦)))
1716fveq2d 6337 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐺𝑦) → (abs‘(𝐹𝑧)) = (abs‘(𝐹‘(𝐺𝑦))))
1817breq1d 4797 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐺𝑦) → ((abs‘(𝐹𝑧)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
1915, 18imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐺𝑦) → ((𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) ↔ (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛)))
2019rspcva 3458 . . . . . . . . . . . . . . 15 (((𝐺𝑦) ∈ 𝐴 ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2114, 20sylan 569 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ 𝑦𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2221an32s 631 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → (𝑚 ≤ (𝐺𝑦) → (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2313adantr 466 . . . . . . . . . . . . . . . 16 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → 𝐺:𝐵𝐴)
24 fvco3 6419 . . . . . . . . . . . . . . . 16 ((𝐺:𝐵𝐴𝑦𝐵) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
2523, 24sylan 569 . . . . . . . . . . . . . . 15 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
2625fveq2d 6337 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → (abs‘((𝐹𝐺)‘𝑦)) = (abs‘(𝐹‘(𝐺𝑦))))
2726breq1d 4797 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → ((abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛 ↔ (abs‘(𝐹‘(𝐺𝑦))) ≤ 𝑛))
2822, 27sylibrd 249 . . . . . . . . . . . 12 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → (𝑚 ≤ (𝐺𝑦) → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛))
2928imim2d 57 . . . . . . . . . . 11 ((((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) ∧ 𝑦𝐵) → ((𝑥𝑦𝑚 ≤ (𝐺𝑦)) → (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3029ralimdva 3111 . . . . . . . . . 10 (((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3130expimpd 441 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → ((∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) ∧ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦))) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3231ancomsd 451 . . . . . . . 8 ((((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℝ) → ((∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3332reximdva 3165 . . . . . . 7 (((𝜑𝑚 ∈ ℝ) ∧ 𝑥 ∈ ℝ) → (∃𝑛 ∈ ℝ (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3433reximdva 3165 . . . . . 6 ((𝜑𝑚 ∈ ℝ) → (∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ (∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3511, 34syl5bir 233 . . . . 5 ((𝜑𝑚 ∈ ℝ) → ((∃𝑥 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦𝑚 ≤ (𝐺𝑦)) ∧ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛)) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3610, 35mpand 675 . . . 4 ((𝜑𝑚 ∈ ℝ) → (∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
3736rexlimdva 3179 . . 3 (𝜑 → (∃𝑚 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑧𝐴 (𝑚𝑧 → (abs‘(𝐹𝑧)) ≤ 𝑛) → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
389, 37mpd 15 . 2 (𝜑 → ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛))
39 fco 6199 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐵𝐴) → (𝐹𝐺):𝐵⟶ℂ)
402, 12, 39syl2anc 573 . . 3 (𝜑 → (𝐹𝐺):𝐵⟶ℂ)
41 o1co.4 . . 3 (𝜑𝐵 ⊆ ℝ)
42 elo12 14466 . . 3 (((𝐹𝐺):𝐵⟶ℂ ∧ 𝐵 ⊆ ℝ) → ((𝐹𝐺) ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
4340, 41, 42syl2anc 573 . 2 (𝜑 → ((𝐹𝐺) ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑛 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (abs‘((𝐹𝐺)‘𝑦)) ≤ 𝑛)))
4438, 43mpbird 247 1 (𝜑 → (𝐹𝐺) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  wss 3723   class class class wbr 4787  dom cdm 5250  ccom 5254  wf 6026  cfv 6030  cc 10140  cr 10141  cle 10281  abscabs 14182  𝑂(1)co1 14425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-pre-lttri 10216  ax-pre-lttrn 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-po 5171  df-so 5172  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-er 7900  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-ico 12386  df-o1 14429
This theorem is referenced by:  o1compt  14526
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