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Mirrors > Home > MPE Home > Th. List > o1sub | Structured version Visualization version GIF version |
Description: The difference of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.) |
Ref | Expression |
---|---|
o1sub | ⊢ ((𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1)) → (𝐹 ∘f − 𝐺) ∈ 𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | readdcl 11193 | . 2 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ) | |
2 | subcl 11459 | . 2 ⊢ ((𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑚 − 𝑛) ∈ ℂ) | |
3 | simp2l 1200 | . . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → 𝑚 ∈ ℂ) | |
4 | simp2r 1201 | . . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → 𝑛 ∈ ℂ) | |
5 | 3, 4 | subcld 11571 | . . . . 5 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → (𝑚 − 𝑛) ∈ ℂ) |
6 | 5 | abscld 15383 | . . . 4 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → (abs‘(𝑚 − 𝑛)) ∈ ℝ) |
7 | 3 | abscld 15383 | . . . . 5 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → (abs‘𝑚) ∈ ℝ) |
8 | 4 | abscld 15383 | . . . . 5 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → (abs‘𝑛) ∈ ℝ) |
9 | 7, 8 | readdcld 11243 | . . . 4 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → ((abs‘𝑚) + (abs‘𝑛)) ∈ ℝ) |
10 | simp1l 1198 | . . . . 5 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → 𝑥 ∈ ℝ) | |
11 | simp1r 1199 | . . . . 5 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → 𝑦 ∈ ℝ) | |
12 | 10, 11 | readdcld 11243 | . . . 4 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → (𝑥 + 𝑦) ∈ ℝ) |
13 | 3, 4 | abs2dif2d 15405 | . . . 4 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → (abs‘(𝑚 − 𝑛)) ≤ ((abs‘𝑚) + (abs‘𝑛))) |
14 | simp3l 1202 | . . . . 5 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → (abs‘𝑚) ≤ 𝑥) | |
15 | simp3r 1203 | . . . . 5 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → (abs‘𝑛) ≤ 𝑦) | |
16 | 7, 8, 10, 11, 14, 15 | le2addd 11833 | . . . 4 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → ((abs‘𝑚) + (abs‘𝑛)) ≤ (𝑥 + 𝑦)) |
17 | 6, 9, 12, 13, 16 | letrd 11371 | . . 3 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ) ∧ ((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦)) → (abs‘(𝑚 − 𝑛)) ≤ (𝑥 + 𝑦)) |
18 | 17 | 3expia 1122 | . 2 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ)) → (((abs‘𝑚) ≤ 𝑥 ∧ (abs‘𝑛) ≤ 𝑦) → (abs‘(𝑚 − 𝑛)) ≤ (𝑥 + 𝑦))) |
19 | 1, 2, 18 | o1of2 15557 | 1 ⊢ ((𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1)) → (𝐹 ∘f − 𝐺) ∈ 𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ∘f cof 7668 ℂcc 11108 ℝcr 11109 + caddc 11113 ≤ cle 11249 − cmin 11444 abscabs 15181 𝑂(1)co1 15430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-ico 13330 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-o1 15434 |
This theorem is referenced by: o1sub2 15570 o1dif 15574 vmadivsum 26985 rpvmasumlem 26990 selberglem1 27048 selberg2 27054 pntrsumo1 27068 selbergr 27071 |
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