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Theorem raddcn 28737
Description: Addition in the real numbers is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.)
Hypothesis
Ref Expression
raddcn.1  |-  J  =  ( topGen `  ran  (,) )
Assertion
Ref Expression
raddcn  |-  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  y ) )  e.  ( ( J 
tX  J )  Cn  J )
Distinct variable group:    x, y
Allowed substitution hints:    J( x, y)

Proof of Theorem raddcn
StepHypRef Expression
1 eqid 2423 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21addcn 21889 . . . . 5  |-  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
3 ax-resscn 9598 . . . . . 6  |-  RR  C_  CC
4 xpss12 4957 . . . . . 6  |-  ( ( RR  C_  CC  /\  RR  C_  CC )  ->  ( RR  X.  RR )  C_  ( CC  X.  CC ) )
53, 3, 4mp2an 677 . . . . 5  |-  ( RR 
X.  RR )  C_  ( CC  X.  CC )
61cnfldtop 21796 . . . . . . 7  |-  ( TopOpen ` fld )  e.  Top
71cnfldtopon 21795 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
87toponunii 19939 . . . . . . 7  |-  CC  =  U. ( TopOpen ` fld )
96, 6, 8, 8txunii 20600 . . . . . 6  |-  ( CC 
X.  CC )  = 
U. ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )
109cnrest 20293 . . . . 5  |-  ( (  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
)  /\  ( RR  X.  RR )  C_  ( CC  X.  CC ) )  ->  (  +  |`  ( RR  X.  RR ) )  e.  ( ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)t  ( RR  X.  RR ) )  Cn  ( TopOpen
` fld
) ) )
112, 5, 10mp2an 677 . . . 4  |-  (  +  |`  ( RR  X.  RR ) )  e.  ( ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )t  ( RR  X.  RR ) )  Cn  ( TopOpen
` fld
) )
12 reex 9632 . . . . . . 7  |-  RR  e.  _V
13 txrest 20638 . . . . . . 7  |-  ( ( ( ( TopOpen ` fld )  e.  Top  /\  ( TopOpen ` fld )  e.  Top )  /\  ( RR  e.  _V  /\  RR  e.  _V ) )  ->  (
( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)t  ( RR  X.  RR ) )  =  ( ( ( TopOpen ` fld )t  RR )  tX  (
( TopOpen ` fld )t  RR ) ) )
146, 6, 12, 12, 13mp4an 678 . . . . . 6  |-  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)t  ( RR  X.  RR ) )  =  ( ( ( TopOpen ` fld )t  RR )  tX  (
( TopOpen ` fld )t  RR ) )
15 raddcn.1 . . . . . . . 8  |-  J  =  ( topGen `  ran  (,) )
161tgioo2 21813 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
1715, 16eqtr2i 2453 . . . . . . 7  |-  ( (
TopOpen ` fld )t  RR )  =  J
1817, 17oveq12i 6315 . . . . . 6  |-  ( ( ( TopOpen ` fld )t  RR )  tX  (
( TopOpen ` fld )t  RR ) )  =  ( J  tX  J
)
1914, 18eqtri 2452 . . . . 5  |-  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)t  ( RR  X.  RR ) )  =  ( J  tX  J )
2019oveq1i 6313 . . . 4  |-  ( ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)t  ( RR  X.  RR ) )  Cn  ( TopOpen
` fld
) )  =  ( ( J  tX  J
)  Cn  ( TopOpen ` fld )
)
2111, 20eleqtri 2509 . . 3  |-  (  +  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J
)  Cn  ( TopOpen ` fld )
)
22 ax-addf 9620 . . . . . . . . . 10  |-  +  :
( CC  X.  CC )
--> CC
23 ffn 5744 . . . . . . . . . 10  |-  (  +  : ( CC  X.  CC ) --> CC  ->  +  Fn  ( CC  X.  CC ) )
2422, 23ax-mp 5 . . . . . . . . 9  |-  +  Fn  ( CC  X.  CC )
25 fnssres 5705 . . . . . . . . 9  |-  ( (  +  Fn  ( CC 
X.  CC )  /\  ( RR  X.  RR )  C_  ( CC  X.  CC ) )  ->  (  +  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR ) )
2624, 5, 25mp2an 677 . . . . . . . 8  |-  (  +  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR )
27 fnov 6416 . . . . . . . 8  |-  ( (  +  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR ) 
<->  (  +  |`  ( RR  X.  RR ) )  =  ( x  e.  RR ,  y  e.  RR  |->  ( x (  +  |`  ( RR  X.  RR ) ) y ) ) )
2826, 27mpbi 212 . . . . . . 7  |-  (  +  |`  ( RR  X.  RR ) )  =  ( x  e.  RR , 
y  e.  RR  |->  ( x (  +  |`  ( RR  X.  RR ) ) y ) )
29 ovres 6448 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x (  +  |`  ( RR  X.  RR ) ) y )  =  ( x  +  y ) )
3029mpt2eq3ia 6368 . . . . . . 7  |-  ( x  e.  RR ,  y  e.  RR  |->  ( x (  +  |`  ( RR  X.  RR ) ) y ) )  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  y
) )
3128, 30eqtri 2452 . . . . . 6  |-  (  +  |`  ( RR  X.  RR ) )  =  ( x  e.  RR , 
y  e.  RR  |->  ( x  +  y ) )
3231rneqi 5078 . . . . 5  |-  ran  (  +  |`  ( RR  X.  RR ) )  =  ran  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  y
) )
33 readdcl 9624 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
3433rgen2a 2853 . . . . . 6  |-  A. x  e.  RR  A. y  e.  RR  ( x  +  y )  e.  RR
35 eqid 2423 . . . . . . 7  |-  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  y ) )  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  y ) )
3635rnmpt2ss 28272 . . . . . 6  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  +  y )  e.  RR  ->  ran  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  y ) )  C_  RR )
3734, 36ax-mp 5 . . . . 5  |-  ran  (
x  e.  RR , 
y  e.  RR  |->  ( x  +  y ) )  C_  RR
3832, 37eqsstri 3495 . . . 4  |-  ran  (  +  |`  ( RR  X.  RR ) )  C_  RR
39 cnrest2 20294 . . . 4  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ran  (  +  |`  ( RR 
X.  RR ) ) 
C_  RR  /\  RR  C_  CC )  ->  ( (  +  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J )  Cn  ( TopOpen
` fld
) )  <->  (  +  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J
)  Cn  ( (
TopOpen ` fld )t  RR ) ) ) )
407, 38, 3, 39mp3an 1361 . . 3  |-  ( (  +  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J )  Cn  ( TopOpen
` fld
) )  <->  (  +  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J
)  Cn  ( (
TopOpen ` fld )t  RR ) ) )
4121, 40mpbi 212 . 2  |-  (  +  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J
)  Cn  ( (
TopOpen ` fld )t  RR ) )
4217oveq2i 6314 . 2  |-  ( ( J  tX  J )  Cn  ( ( TopOpen ` fld )t  RR ) )  =  ( ( J  tX  J
)  Cn  J )
4341, 31, 423eltr3i 2523 1  |-  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  y ) )  e.  ( ( J 
tX  J )  Cn  J )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1438    e. wcel 1869   A.wral 2776   _Vcvv 3082    C_ wss 3437    X. cxp 4849   ran crn 4852    |` cres 4853    Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303    |-> cmpt2 6305   CCcc 9539   RRcr 9540    + caddc 9544   (,)cioo 11637   ↾t crest 15312   TopOpenctopn 15313   topGenctg 15329  ℂfldccnfld 18963   Topctop 19909  TopOnctopon 19910    Cn ccn 20232    tX ctx 20567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ioo 11641  df-icc 11644  df-fz 11787  df-fzo 11918  df-seq 12215  df-exp 12274  df-hash 12517  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cn 20235  df-cnp 20236  df-tx 20569  df-hmeo 20762  df-xms 21327  df-ms 21328  df-tms 21329
This theorem is referenced by:  rrvadd  29287
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