tgioo2 - Metamath Proof Explorer

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Theorem tgioo2 23406

Description: The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.)

**Hypothesis**
Ref	Expression
tgioo2.1	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
tgioo2	⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)

**Proof of Theorem tgioo2**
Step	Hyp	Ref	Expression
1		eqid 2820	. 2 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
2		cnxmet 23376	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
3		ax-resscn 10587	. . 3 ⊢ ℝ ⊆ ℂ
4		tgioo2.1	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
5	4	cnfldtopn 23385	. . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
6		eqid 2820	. . . 4 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
7	1, 5, 6	metrest 23129	. . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ ℝ ⊆ ℂ) → (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))))
8	2, 3, 7	mp2an 690	. 2 ⊢ (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
9	1, 8	tgioo 23399	1 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)

Colors of variables: wff setvar class

Syntax hints: = wceq 1536 ∈ wcel 2113 ⊆ wss 3929 × cxp 5546 ran crn 5549 ↾ cres 5550 ∘ ccom 5552 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 ℝcr 10529 − cmin 10863 (,)cioo 12732 abscabs 14588 ↾_t crest 16689 TopOpenctopn 16690 topGenctg 16706 ∞Metcxmet 20525 MetOpencmopn 20530 ℂ_fldccnfld 20540

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-fz 12890 df-seq 13367 df-exp 13427 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-plusg 16573 df-mulr 16574 df-starv 16575 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-rest 16691 df-topn 16692 df-topgen 16712 df-psmet 20532 df-xmet 20533 df-met 20534 df-bl 20535 df-mopn 20536 df-cnfld 20541 df-top 21497 df-topon 21514 df-bases 21549

This theorem is referenced by: rerest 23407 tgioo3 23408 zcld2 23418 metdcn 23443 ngnmcncn 23448 metdscn2 23460 abscncfALT 23523 cnrehmeo 23552 rellycmp 23556 evth 23558 evth2 23559 lebnumlem2 23561 resscdrg 23956 retopn 23977 cncombf 24254 cnmbf 24255 dvcjbr 24543 rolle 24584 cmvth 24585 mvth 24586 dvlip 24587 dvlipcn 24588 dvlip2 24589 c1liplem1 24590 dvgt0lem1 24596 dvle 24601 dvivthlem1 24602 dvne0 24605 lhop1lem 24607 lhop2 24609 lhop 24610 dvcnvrelem1 24611 dvcnvrelem2 24612 dvcnvre 24613 dvcvx 24614 dvfsumle 24615 dvfsumabs 24617 dvfsumlem2 24621 ftc1 24636 ftc1cn 24637 ftc2 24638 ftc2ditglem 24639 itgpowd 24641 itgparts 24642 itgsubstlem 24643 taylthlem2 24960 efcvx 25035 pige3ALT 25103 dvloglem 25229 logdmopn 25230 advlog 25235 advlogexp 25236 logccv 25244 loglesqrt 25337 lgamgulmlem2 25605 ftalem3 25650 log2sumbnd 26118 nmcnc 28471 ipasslem7 28611 rmulccn 31192 raddcn 31193 ftc2re 31890 knoppcnlem10 33862 knoppcnlem11 33863 broucube 34961 ftc1cnnc 34999 ftc2nc 35009 dvasin 35011 dvacos 35012 dvreasin 35013 dvreacos 35014 areacirclem1 35015 areacirc 35020 lhe4.4ex1a 40735 refsumcn 41361 xrtgcntopre 41829 tgioo4 41923 climreeq 41968 limcresiooub 41997 limcresioolb 41998 lptioo2cn 42000 lptioo1cn 42001 limclner 42006 cncfiooicclem1 42250 jumpncnp 42255 dvmptresicc 42278 dvresioo 42280 dvbdfbdioolem1 42287 itgsin0pilem1 42309 itgsinexplem1 42313 itgcoscmulx 42328 itgsubsticclem 42334 itgiccshift 42339 itgperiod 42340 itgsbtaddcnst 42341 dirkeritg 42461 dirkercncflem2 42463 dirkercncflem3 42464 dirkercncflem4 42465 dirkercncf 42466 fourierdlem28 42494 fourierdlem32 42498 fourierdlem33 42499 fourierdlem39 42505 fourierdlem56 42521 fourierdlem57 42522 fourierdlem58 42523 fourierdlem59 42524 fourierdlem62 42527 fourierdlem68 42533 fourierdlem72 42537 fourierdlem73 42538 fourierdlem74 42539 fourierdlem75 42540 fourierdlem80 42545 fourierdlem94 42559 fourierdlem103 42568 fourierdlem104 42569 fourierdlem113 42578 fouriercnp 42585 fouriersw 42590 fouriercn 42591 etransclem2 42595 etransclem23 42616 etransclem35 42628 etransclem38 42631 etransclem39 42632 etransclem44 42637 etransclem45 42638 etransclem46 42639 etransclem47 42640

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