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Theorem rmulccn 24267
Description: Multiplication by a real constant is a continuous function (Contributed by Thierry Arnoux, 23-May-2017.)
Hypotheses
Ref Expression
rmulccn.1  |-  J  =  ( topGen `  ran  (,) )
rmulccn.2  |-  ( ph  ->  C  e.  RR )
Assertion
Ref Expression
rmulccn  |-  ( ph  ->  ( x  e.  RR  |->  ( x  x.  C
) )  e.  ( J  Cn  J ) )
Distinct variable groups:    x, C    ph, x
Allowed substitution hint:    J( x)

Proof of Theorem rmulccn
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldtopon 18770 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
32a1i 11 . . . . 5  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
43cnmptid 17646 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  x )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
5 rmulccn.2 . . . . . . 7  |-  ( ph  ->  C  e.  RR )
65recnd 9070 . . . . . 6  |-  ( ph  ->  C  e.  CC )
73, 3, 6cnmptc 17647 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  C )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
8 ax-mulf 9026 . . . . . . . . 9  |-  x.  :
( CC  X.  CC )
--> CC
9 ffn 5550 . . . . . . . . 9  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
108, 9ax-mp 8 . . . . . . . 8  |-  x.  Fn  ( CC  X.  CC )
11 fnov 6137 . . . . . . . 8  |-  (  x.  Fn  ( CC  X.  CC )  <->  x.  =  (
y  e.  CC , 
z  e.  CC  |->  ( y  x.  z ) ) )
1210, 11mpbi 200 . . . . . . 7  |-  x.  =  ( y  e.  CC ,  z  e.  CC  |->  ( y  x.  z
) )
131mulcn 18850 . . . . . . 7  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
1412, 13eqeltrri 2475 . . . . . 6  |-  ( y  e.  CC ,  z  e.  CC  |->  ( y  x.  z ) )  e.  ( ( (
TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
)
1514a1i 11 . . . . 5  |-  ( ph  ->  ( y  e.  CC ,  z  e.  CC  |->  ( y  x.  z
) )  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
16 oveq12 6049 . . . . 5  |-  ( ( y  =  x  /\  z  =  C )  ->  ( y  x.  z
)  =  ( x  x.  C ) )
173, 4, 7, 3, 3, 15, 16cnmpt12 17652 . . . 4  |-  ( ph  ->  ( x  e.  CC  |->  ( x  x.  C
) )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
18 ax-resscn 9003 . . . 4  |-  RR  C_  CC
192toponunii 16952 . . . . 5  |-  CC  =  U. ( TopOpen ` fld )
2019cnrest 17303 . . . 4  |-  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )  /\  RR  C_  CC )  ->  (
( x  e.  CC  |->  ( x  x.  C
) )  |`  RR )  e.  ( ( (
TopOpen ` fld )t  RR )  Cn  ( TopOpen
` fld
) ) )
2117, 18, 20sylancl 644 . . 3  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  (
( ( TopOpen ` fld )t  RR )  Cn  ( TopOpen
` fld
) ) )
22 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
236adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  C  e.  CC )
2422, 23mulcld 9064 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  C )  e.  CC )
2524ralrimiva 2749 . . . . . . 7  |-  ( ph  ->  A. x  e.  CC  ( x  x.  C
)  e.  CC )
26 eqid 2404 . . . . . . . 8  |-  ( x  e.  CC  |->  ( x  x.  C ) )  =  ( x  e.  CC  |->  ( x  x.  C ) )
2726fnmpt 5530 . . . . . . 7  |-  ( A. x  e.  CC  (
x  x.  C )  e.  CC  ->  (
x  e.  CC  |->  ( x  x.  C ) )  Fn  CC )
2825, 27syl 16 . . . . . 6  |-  ( ph  ->  ( x  e.  CC  |->  ( x  x.  C
) )  Fn  CC )
29 fnssres 5517 . . . . . 6  |-  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  Fn  CC  /\  RR  C_  CC )  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  Fn  RR )
3028, 18, 29sylancl 644 . . . . 5  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  Fn  RR )
31 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  w  e.  RR )  ->  w  e.  RR )
32 fvres 5704 . . . . . . . . 9  |-  ( w  e.  RR  ->  (
( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w )  =  ( ( x  e.  CC  |->  ( x  x.  C ) ) `
 w ) )
33 recn 9036 . . . . . . . . . 10  |-  ( w  e.  RR  ->  w  e.  CC )
34 oveq1 6047 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  x.  C )  =  ( w  x.  C ) )
35 ovex 6065 . . . . . . . . . . 11  |-  ( w  x.  C )  e. 
_V
3634, 26, 35fvmpt 5765 . . . . . . . . . 10  |-  ( w  e.  CC  ->  (
( x  e.  CC  |->  ( x  x.  C
) ) `  w
)  =  ( w  x.  C ) )
3733, 36syl 16 . . . . . . . . 9  |-  ( w  e.  RR  ->  (
( x  e.  CC  |->  ( x  x.  C
) ) `  w
)  =  ( w  x.  C ) )
3832, 37eqtrd 2436 . . . . . . . 8  |-  ( w  e.  RR  ->  (
( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w )  =  ( w  x.  C ) )
3931, 38syl 16 . . . . . . 7  |-  ( (
ph  /\  w  e.  RR )  ->  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  |`  RR ) `
 w )  =  ( w  x.  C
) )
405adantr 452 . . . . . . . 8  |-  ( (
ph  /\  w  e.  RR )  ->  C  e.  RR )
4131, 40remulcld 9072 . . . . . . 7  |-  ( (
ph  /\  w  e.  RR )  ->  ( w  x.  C )  e.  RR )
4239, 41eqeltrd 2478 . . . . . 6  |-  ( (
ph  /\  w  e.  RR )  ->  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  |`  RR ) `
 w )  e.  RR )
4342ralrimiva 2749 . . . . 5  |-  ( ph  ->  A. w  e.  RR  ( ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w
)  e.  RR )
44 fnfvrnss 5855 . . . . 5  |-  ( ( ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  Fn  RR  /\ 
A. w  e.  RR  ( ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w
)  e.  RR )  ->  ran  ( (
x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  C_  RR )
4530, 43, 44syl2anc 643 . . . 4  |-  ( ph  ->  ran  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  C_  RR )
4618a1i 11 . . . 4  |-  ( ph  ->  RR  C_  CC )
47 cnrest2 17304 . . . 4  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ran  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  C_  RR  /\  RR  C_  CC )  -> 
( ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  ( ( ( TopOpen ` fld )t  RR )  Cn  ( TopOpen
` fld
) )  <->  ( (
x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  ( ( ( TopOpen ` fld )t  RR )  Cn  ( ( TopOpen ` fld )t  RR ) ) ) )
483, 45, 46, 47syl3anc 1184 . . 3  |-  ( ph  ->  ( ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  ( ( ( TopOpen ` fld )t  RR )  Cn  ( TopOpen
` fld
) )  <->  ( (
x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  ( ( ( TopOpen ` fld )t  RR )  Cn  ( ( TopOpen ` fld )t  RR ) ) ) )
4921, 48mpbid 202 . 2  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  (
( ( TopOpen ` fld )t  RR )  Cn  (
( TopOpen ` fld )t  RR ) ) )
50 resmpt 5150 . . 3  |-  ( RR  C_  CC  ->  ( (
x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  =  ( x  e.  RR  |->  ( x  x.  C ) ) )
5118, 50ax-mp 8 . 2  |-  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  =  ( x  e.  RR  |->  ( x  x.  C ) )
52 rmulccn.1 . . . . 5  |-  J  =  ( topGen `  ran  (,) )
531tgioo2 18787 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
5452, 53eqtri 2424 . . . 4  |-  J  =  ( ( TopOpen ` fld )t  RR )
5554, 54oveq12i 6052 . . 3  |-  ( J  Cn  J )  =  ( ( ( TopOpen ` fld )t  RR )  Cn  ( ( TopOpen ` fld )t  RR ) )
5655eqcomi 2408 . 2  |-  ( ( ( TopOpen ` fld )t  RR )  Cn  (
( TopOpen ` fld )t  RR ) )  =  ( J  Cn  J
)
5749, 51, 563eltr3g 2486 1  |-  ( ph  ->  ( x  e.  RR  |->  ( x  x.  C
) )  e.  ( J  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280    e. cmpt 4226    X. cxp 4835   ran crn 4838    |` cres 4839    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   CCcc 8944   RRcr 8945    x. cmul 8951   (,)cioo 10872   ↾t crest 13603   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658  TopOnctopon 16914    Cn ccn 17242    tX ctx 17545
This theorem is referenced by:  rrvmulc  24664
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cn 17245  df-cnp 17246  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305
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