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Mirrors > Home > MPE Home > Th. List > lo1o12 | Structured version Visualization version GIF version |
Description: A function is eventually bounded iff its absolute value is eventually upper bounded. (This function is useful for converting theorems about β€π(1) to π(1).) (Contributed by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
lo1o12.1 | β’ ((π β§ π₯ β π΄) β π΅ β β) |
Ref | Expression |
---|---|
lo1o12 | β’ (π β ((π₯ β π΄ β¦ π΅) β π(1) β (π₯ β π΄ β¦ (absβπ΅)) β β€π(1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lo1o12.1 | . . . 4 β’ ((π β§ π₯ β π΄) β π΅ β β) | |
2 | 1 | fmpttd 7128 | . . 3 β’ (π β (π₯ β π΄ β¦ π΅):π΄βΆβ) |
3 | lo1o1 15514 | . . 3 β’ ((π₯ β π΄ β¦ π΅):π΄βΆβ β ((π₯ β π΄ β¦ π΅) β π(1) β (abs β (π₯ β π΄ β¦ π΅)) β β€π(1))) | |
4 | 2, 3 | syl 17 | . 2 β’ (π β ((π₯ β π΄ β¦ π΅) β π(1) β (abs β (π₯ β π΄ β¦ π΅)) β β€π(1))) |
5 | absf 15322 | . . . . 5 β’ abs:ββΆβ | |
6 | 5 | a1i 11 | . . . 4 β’ (π β abs:ββΆβ) |
7 | 6, 1 | cofmpt 7145 | . . 3 β’ (π β (abs β (π₯ β π΄ β¦ π΅)) = (π₯ β π΄ β¦ (absβπ΅))) |
8 | 7 | eleq1d 2813 | . 2 β’ (π β ((abs β (π₯ β π΄ β¦ π΅)) β β€π(1) β (π₯ β π΄ β¦ (absβπ΅)) β β€π(1))) |
9 | 4, 8 | bitrd 278 | 1 β’ (π β ((π₯ β π΄ β¦ π΅) β π(1) β (π₯ β π΄ β¦ (absβπ΅)) β β€π(1))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β wcel 2098 β¦ cmpt 5233 β ccom 5684 βΆwf 6547 βcfv 6551 βcc 11142 βcr 11143 abscabs 15219 π(1)co1 15468 β€π(1)clo1 15469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-pm 8852 df-en 8969 df-dom 8970 df-sdom 8971 df-sup 9471 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-ico 13368 df-seq 14005 df-exp 14065 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-o1 15472 df-lo1 15473 |
This theorem is referenced by: elo1mpt 15516 elo1mpt2 15517 elo1d 15518 o1bdd2 15523 o1bddrp 15524 o1eq 15552 o1le 15637 pntrlog2bndlem1 27528 |
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