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Mirrors > Home > HSE Home > Th. List > lnfnconi | Structured version Visualization version GIF version |
Description: A condition equivalent to "π is continuous" when π is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnfncon.1 | β’ π β LinFn |
Ref | Expression |
---|---|
lnfnconi | β’ (π β ContFn β βπ₯ β β βπ¦ β β (absβ(πβπ¦)) β€ (π₯ Β· (normββπ¦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnfncon.1 | . . 3 β’ π β LinFn | |
2 | nmcfnex 31856 | . . 3 β’ ((π β LinFn β§ π β ContFn) β (normfnβπ) β β) | |
3 | 1, 2 | mpan 689 | . 2 β’ (π β ContFn β (normfnβπ) β β) |
4 | nmcfnlb 31857 | . . 3 β’ ((π β LinFn β§ π β ContFn β§ π¦ β β) β (absβ(πβπ¦)) β€ ((normfnβπ) Β· (normββπ¦))) | |
5 | 1, 4 | mp3an1 1445 | . 2 β’ ((π β ContFn β§ π¦ β β) β (absβ(πβπ¦)) β€ ((normfnβπ) Β· (normββπ¦))) |
6 | 1 | lnfnfi 31844 | . . 3 β’ π: ββΆβ |
7 | elcnfn 31685 | . . 3 β’ (π β ContFn β (π: ββΆβ β§ βπ₯ β β βπ§ β β+ βπ¦ β β+ βπ€ β β ((normββ(π€ ββ π₯)) < π¦ β (absβ((πβπ€) β (πβπ₯))) < π§))) | |
8 | 6, 7 | mpbiran 708 | . 2 β’ (π β ContFn β βπ₯ β β βπ§ β β+ βπ¦ β β+ βπ€ β β ((normββ(π€ ββ π₯)) < π¦ β (absβ((πβπ€) β (πβπ₯))) < π§)) |
9 | 6 | ffvelcdmi 7087 | . . 3 β’ (π¦ β β β (πβπ¦) β β) |
10 | 9 | abscld 15409 | . 2 β’ (π¦ β β β (absβ(πβπ¦)) β β) |
11 | 1 | lnfnsubi 31849 | . 2 β’ ((π€ β β β§ π₯ β β) β (πβ(π€ ββ π₯)) = ((πβπ€) β (πβπ₯))) |
12 | 3, 5, 8, 10, 11 | lnconi 31836 | 1 β’ (π β ContFn β βπ₯ β β βπ¦ β β (absβ(πβπ¦)) β€ (π₯ Β· (normββπ¦))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β wcel 2099 βwral 3057 βwrex 3066 class class class wbr 5142 βΆwf 6538 βcfv 6542 (class class class)co 7414 βcc 11130 βcr 11131 Β· cmul 11137 < clt 11272 β€ cle 11273 β cmin 11468 β+crp 13000 abscabs 15207 βchba 30722 normβcno 30726 ββ cmv 30728 normfncnmf 30754 ContFnccnfn 30756 LinFnclf 30757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-hilex 30802 ax-hfvadd 30803 ax-hv0cl 30806 ax-hvaddid 30807 ax-hfvmul 30808 ax-hvmulid 30809 ax-hvmulass 30810 ax-hvmul0 30813 ax-hfi 30882 ax-his1 30885 ax-his3 30887 ax-his4 30888 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-hnorm 30771 df-hvsub 30774 df-nmfn 31648 df-cnfn 31650 df-lnfn 31651 |
This theorem is referenced by: lnfncon 31859 |
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