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Mirrors > Home > MPE Home > Th. List > mopnm | Structured version Visualization version GIF version |
Description: The base set of a metric space is open. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
mopnval.1 | β’ π½ = (MetOpenβπ·) |
Ref | Expression |
---|---|
mopnm | β’ (π· β (βMetβπ) β π β π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mopnval.1 | . . 3 β’ π½ = (MetOpenβπ·) | |
2 | 1 | mopntopon 24490 | . 2 β’ (π· β (βMetβπ) β π½ β (TopOnβπ)) |
3 | toponmax 22973 | . 2 β’ (π½ β (TopOnβπ) β π β π½) | |
4 | 2, 3 | syl 17 | 1 β’ (π· β (βMetβπ) β π β π½) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1537 β wcel 2108 βcfv 6576 βMetcxmet 21392 MetOpencmopn 21397 TopOnctopon 22957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5318 ax-nul 5325 ax-pow 5384 ax-pr 5448 ax-un 7773 ax-cnex 11243 ax-resscn 11244 ax-1cn 11245 ax-icn 11246 ax-addcl 11247 ax-addrcl 11248 ax-mulcl 11249 ax-mulrcl 11250 ax-mulcom 11251 ax-addass 11252 ax-mulass 11253 ax-distr 11254 ax-i2m1 11255 ax-1ne0 11256 ax-1rid 11257 ax-rnegex 11258 ax-rrecex 11259 ax-cnre 11260 ax-pre-lttri 11261 ax-pre-lttrn 11262 ax-pre-ltadd 11263 ax-pre-mulgt0 11264 ax-pre-sup 11265 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4933 df-iun 5018 df-br 5168 df-opab 5230 df-mpt 5251 df-tr 5285 df-id 5594 df-eprel 5600 df-po 5608 df-so 5609 df-fr 5653 df-we 5655 df-xp 5707 df-rel 5708 df-cnv 5709 df-co 5710 df-dm 5711 df-rn 5712 df-res 5713 df-ima 5714 df-pred 6335 df-ord 6401 df-on 6402 df-lim 6403 df-suc 6404 df-iota 6528 df-fun 6578 df-fn 6579 df-f 6580 df-f1 6581 df-fo 6582 df-f1o 6583 df-fv 6584 df-riota 7407 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7907 df-1st 8033 df-2nd 8034 df-frecs 8325 df-wrecs 8356 df-recs 8430 df-rdg 8469 df-er 8766 df-map 8889 df-en 9007 df-dom 9008 df-sdom 9009 df-sup 9514 df-inf 9515 df-pnf 11329 df-mnf 11330 df-xr 11331 df-ltxr 11332 df-le 11333 df-sub 11526 df-neg 11527 df-div 11953 df-nn 12299 df-2 12361 df-n0 12559 df-z 12646 df-uz 12911 df-q 13023 df-rp 13067 df-xneg 13186 df-xadd 13187 df-xmul 13188 df-topgen 17523 df-psmet 21399 df-xmet 21400 df-bl 21402 df-mopn 21403 df-top 22941 df-topon 22958 df-bases 22994 |
This theorem is referenced by: metrest 24578 |
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