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Theorem rmulccn 26294
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 2441 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldtopon 20321 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
32a1i 11 . . . . 5  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
43cnmptid 19193 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  x )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
5 rmulccn.2 . . . . . . 7  |-  ( ph  ->  C  e.  RR )
65recnd 9408 . . . . . 6  |-  ( ph  ->  C  e.  CC )
73, 3, 6cnmptc 19194 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  C )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
8 ax-mulf 9358 . . . . . . . . 9  |-  x.  :
( CC  X.  CC )
--> CC
9 ffn 5556 . . . . . . . . 9  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
108, 9ax-mp 5 . . . . . . . 8  |-  x.  Fn  ( CC  X.  CC )
11 fnov 6197 . . . . . . . 8  |-  (  x.  Fn  ( CC  X.  CC )  <->  x.  =  (
y  e.  CC , 
z  e.  CC  |->  ( y  x.  z ) ) )
1210, 11mpbi 208 . . . . . . 7  |-  x.  =  ( y  e.  CC ,  z  e.  CC  |->  ( y  x.  z
) )
131mulcn 20402 . . . . . . 7  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
1412, 13eqeltrri 2512 . . . . . 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 6099 . . . . 5  |-  ( ( y  =  x  /\  z  =  C )  ->  ( y  x.  z
)  =  ( x  x.  C ) )
173, 4, 7, 3, 3, 15, 16cnmpt12 19199 . . . 4  |-  ( ph  ->  ( x  e.  CC  |->  ( x  x.  C
) )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
18 ax-resscn 9335 . . . 4  |-  RR  C_  CC
192toponunii 18496 . . . . 5  |-  CC  =  U. ( TopOpen ` fld )
2019cnrest 18848 . . . 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 657 . . 3  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  (
( ( TopOpen ` fld )t  RR )  Cn  ( TopOpen
` fld
) ) )
22 simpr 458 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
236adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  C  e.  CC )
2422, 23mulcld 9402 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  C )  e.  CC )
2524ralrimiva 2797 . . . . . . 7  |-  ( ph  ->  A. x  e.  CC  ( x  x.  C
)  e.  CC )
26 eqid 2441 . . . . . . . 8  |-  ( x  e.  CC  |->  ( x  x.  C ) )  =  ( x  e.  CC  |->  ( x  x.  C ) )
2726fnmpt 5534 . . . . . . 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 5521 . . . . . 6  |-  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  Fn  CC  /\  RR  C_  CC )  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  Fn  RR )
3028, 18, 29sylancl 657 . . . . 5  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  Fn  RR )
31 simpr 458 . . . . . . . 8  |-  ( (
ph  /\  w  e.  RR )  ->  w  e.  RR )
32 fvres 5701 . . . . . . . . 9  |-  ( w  e.  RR  ->  (
( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w )  =  ( ( x  e.  CC  |->  ( x  x.  C ) ) `
 w ) )
33 recn 9368 . . . . . . . . . 10  |-  ( w  e.  RR  ->  w  e.  CC )
34 oveq1 6097 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  x.  C )  =  ( w  x.  C ) )
35 ovex 6115 . . . . . . . . . . 11  |-  ( w  x.  C )  e. 
_V
3634, 26, 35fvmpt 5771 . . . . . . . . . 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 2473 . . . . . . . 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 462 . . . . . . . 8  |-  ( (
ph  /\  w  e.  RR )  ->  C  e.  RR )
4131, 40remulcld 9410 . . . . . . 7  |-  ( (
ph  /\  w  e.  RR )  ->  ( w  x.  C )  e.  RR )
4239, 41eqeltrd 2515 . . . . . 6  |-  ( (
ph  /\  w  e.  RR )  ->  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  |`  RR ) `
 w )  e.  RR )
4342ralrimiva 2797 . . . . 5  |-  ( ph  ->  A. w  e.  RR  ( ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w
)  e.  RR )
44 fnfvrnss 5868 . . . . 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 656 . . . 4  |-  ( ph  ->  ran  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  C_  RR )
4618a1i 11 . . . 4  |-  ( ph  ->  RR  C_  CC )
47 cnrest2 18849 . . . 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 1213 . . 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 210 . 2  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  (
( ( TopOpen ` fld )t  RR )  Cn  (
( TopOpen ` fld )t  RR ) ) )
50 resmpt 5153 . . 3  |-  ( RR  C_  CC  ->  ( (
x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  =  ( x  e.  RR  |->  ( x  x.  C ) ) )
5118, 50ax-mp 5 . 2  |-  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  =  ( x  e.  RR  |->  ( x  x.  C ) )
52 rmulccn.1 . . . . 5  |-  J  =  ( topGen `  ran  (,) )
531tgioo2 20339 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
5452, 53eqtri 2461 . . . 4  |-  J  =  ( ( TopOpen ` fld )t  RR )
5554, 54oveq12i 6102 . . 3  |-  ( J  Cn  J )  =  ( ( ( TopOpen ` fld )t  RR )  Cn  ( ( TopOpen ` fld )t  RR ) )
5655eqcomi 2445 . 2  |-  ( ( ( TopOpen ` fld )t  RR )  Cn  (
( TopOpen ` fld )t  RR ) )  =  ( J  Cn  J
)
5749, 51, 563eltr3g 2523 1  |-  ( ph  ->  ( x  e.  RR  |->  ( x  x.  C
) )  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713    C_ wss 3325    e. cmpt 4347    X. cxp 4834   ran crn 4837    |` cres 4838    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   CCcc 9276   RRcr 9277    x. cmul 9283   (,)cioo 11296   ↾t crest 14355   TopOpenctopn 14356   topGenctg 14372  ℂfldccnfld 17777  TopOnctopon 18458    Cn ccn 18787    tX ctx 19092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cn 18790  df-cnp 18791  df-tx 19094  df-hmeo 19287  df-xms 19854  df-ms 19855  df-tms 19856
This theorem is referenced by:  rrvmulc  26766
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