tgioo2 - Metamath Proof Explorer

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Theorem tgioo2 22414

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 2610	. 2 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
2		cnxmet 22386	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
3		ax-resscn 9872	. . 3 ⊢ ℝ ⊆ ℂ
4		tgioo2.1	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
5	4	cnfldtopn 22395	. . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
6		eqid 2610	. . . 4 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
7	1, 5, 6	metrest 22139	. . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ ℝ ⊆ ℂ) → (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))))
8	2, 3, 7	mp2an 704	. 2 ⊢ (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
9	1, 8	tgioo 22407	1 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)

Colors of variables: wff setvar class

Syntax hints: = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 × cxp 5036 ran crn 5039 ↾ cres 5040 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 − cmin 10145 (,)cioo 12046 abscabs 13822 ↾_t crest 15904 TopOpenctopn 15905 topGenctg 15921 ∞Metcxmt 19552 MetOpencmopn 19557 ℂ_fldccnfld 19567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523

This theorem is referenced by: rerest 22415 tgioo3 22416 zcld2 22426 metdcn 22451 ngnmcncn 22456 metdscn2 22468 abscncfALT 22531 cnrehmeo 22560 rellycmp 22564 evth 22566 evth2 22567 lebnumlem2 22569 resscdrg 22962 retopn 22975 cncombf 23231 cnmbf 23232 dvcjbr 23518 rolle 23557 cmvth 23558 mvth 23559 dvlip 23560 dvlipcn 23561 dvlip2 23562 c1liplem1 23563 dvgt0lem1 23569 dvle 23574 dvivthlem1 23575 dvne0 23578 lhop1lem 23580 lhop2 23582 lhop 23583 dvcnvrelem1 23584 dvcnvrelem2 23585 dvcnvre 23586 dvcvx 23587 dvfsumle 23588 dvfsumabs 23590 dvfsumlem2 23594 ftc1 23609 ftc1cn 23610 ftc2 23611 ftc2ditglem 23612 itgparts 23614 itgsubstlem 23615 taylthlem2 23932 efcvx 24007 pige3 24073 dvloglem 24194 logdmopn 24195 advlog 24200 advlogexp 24201 logccv 24209 loglesqrt 24299 lgamgulmlem2 24556 ftalem3 24601 log2sumbnd 25033 nmcnc 26935 ipasslem7 27075 rmulccn 29302 raddcn 29303 knoppcnlem10 31662 knoppcnlem11 31663 broucube 32613 ftc1cnnc 32654 ftc2nc 32664 dvasin 32666 dvacos 32667 dvreasin 32668 dvreacos 32669 areacirclem1 32670 areacirc 32675 itgpowd 36819 lhe4.4ex1a 37550 refsumcn 38212 climreeq 38680 limcresiooub 38709 limcresioolb 38710 lptioo2cn 38712 lptioo1cn 38713 limclner 38718 cncfiooicclem1 38779 jumpncnp 38784 fperdvper 38808 dvmptresicc 38809 dvresioo 38811 dvbdfbdioolem1 38818 itgsin0pilem1 38841 itgsinexplem1 38845 itgcoscmulx 38861 itgsubsticclem 38867 itgiccshift 38872 itgperiod 38873 itgsbtaddcnst 38874 dirkeritg 38995 dirkercncflem2 38997 dirkercncflem3 38998 dirkercncflem4 38999 dirkercncf 39000 fourierdlem28 39028 fourierdlem32 39032 fourierdlem33 39033 fourierdlem39 39039 fourierdlem56 39055 fourierdlem57 39056 fourierdlem58 39057 fourierdlem59 39058 fourierdlem60 39059 fourierdlem61 39060 fourierdlem62 39061 fourierdlem68 39067 fourierdlem72 39071 fourierdlem73 39072 fourierdlem74 39073 fourierdlem75 39074 fourierdlem80 39079 fourierdlem94 39093 fourierdlem103 39102 fourierdlem104 39103 fourierdlem113 39112 fouriercnp 39119 fouriersw 39124 fouriercn 39125 etransclem2 39129 etransclem23 39150 etransclem35 39162 etransclem38 39165 etransclem39 39166 etransclem44 39171 etransclem45 39172 etransclem46 39173 etransclem47 39174

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