![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > retopsconn | Structured version Visualization version GIF version |
Description: The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
retopsconn | ⊢ (topGen‘ran (,)) ∈ SConn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retop 23053 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | ioomax 12661 | . . . . 5 ⊢ (-∞(,)+∞) = ℝ | |
3 | uniretop 23054 | . . . . 5 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
4 | 2, 3 | eqtri 2819 | . . . 4 ⊢ (-∞(,)+∞) = ∪ (topGen‘ran (,)) |
5 | 4 | restid 16536 | . . 3 ⊢ ((topGen‘ran (,)) ∈ Top → ((topGen‘ran (,)) ↾t (-∞(,)+∞)) = (topGen‘ran (,))) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ ((topGen‘ran (,)) ↾t (-∞(,)+∞)) = (topGen‘ran (,)) |
7 | ioosconn 32102 | . 2 ⊢ ((topGen‘ran (,)) ↾t (-∞(,)+∞)) ∈ SConn | |
8 | 6, 7 | eqeltrri 2880 | 1 ⊢ (topGen‘ran (,)) ∈ SConn |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∈ wcel 2081 ∪ cuni 4745 ran crn 5444 ‘cfv 6225 (class class class)co 7016 ℝcr 10382 +∞cpnf 10518 -∞cmnf 10519 (,)cioo 12588 ↾t crest 16523 topGenctg 16540 Topctop 21185 SConncsconn 32075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-fi 8721 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-seq 13220 df-exp 13280 df-hash 13541 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-mulg 17982 df-cntz 18188 df-cmn 18635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-cn 21519 df-cnp 21520 df-conn 21704 df-tx 21854 df-hmeo 22047 df-xms 22613 df-ms 22614 df-tms 22615 df-ii 23168 df-htpy 23257 df-phtpy 23258 df-phtpc 23279 df-pconn 32076 df-sconn 32077 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |