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Theorem rmulccn 23383
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 2316 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldtopon 18344 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
32a1i 10 . . . . 5  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
43cnmptid 17411 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  x )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
5 rmulccn.2 . . . . . . 7  |-  ( ph  ->  C  e.  RR )
65recnd 8906 . . . . . 6  |-  ( ph  ->  C  e.  CC )
73, 3, 6cnmptc 17412 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  C )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
8 ax-mulf 8862 . . . . . . . . 9  |-  x.  :
( CC  X.  CC )
--> CC
9 ffn 5427 . . . . . . . . 9  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
108, 9ax-mp 8 . . . . . . . 8  |-  x.  Fn  ( CC  X.  CC )
11 fnov 5994 . . . . . . . 8  |-  (  x.  Fn  ( CC  X.  CC )  <->  x.  =  (
y  e.  CC , 
z  e.  CC  |->  ( y  x.  z ) ) )
1210, 11mpbi 199 . . . . . . 7  |-  x.  =  ( y  e.  CC ,  z  e.  CC  |->  ( y  x.  z
) )
131mulcn 18423 . . . . . . 7  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
1412, 13eqeltrri 2387 . . . . . 6  |-  ( y  e.  CC ,  z  e.  CC  |->  ( y  x.  z ) )  e.  ( ( (
TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
)
1514a1i 10 . . . . 5  |-  ( ph  ->  ( y  e.  CC ,  z  e.  CC  |->  ( y  x.  z
) )  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
16 oveq12 5909 . . . . 5  |-  ( ( y  =  x  /\  z  =  C )  ->  ( y  x.  z
)  =  ( x  x.  C ) )
173, 4, 7, 3, 3, 15, 16cnmpt12 17417 . . . 4  |-  ( ph  ->  ( x  e.  CC  |->  ( x  x.  C
) )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
18 ax-resscn 8839 . . . 4  |-  RR  C_  CC
192toponunii 16726 . . . . 5  |-  CC  =  U. ( TopOpen ` fld )
2019cnrest 17069 . . . 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 643 . . 3  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  (
( ( TopOpen ` fld )t  RR )  Cn  ( TopOpen
` fld
) ) )
22 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
236adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  C  e.  CC )
2422, 23mulcld 8900 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  C )  e.  CC )
2524ralrimiva 2660 . . . . . . 7  |-  ( ph  ->  A. x  e.  CC  ( x  x.  C
)  e.  CC )
26 eqid 2316 . . . . . . . 8  |-  ( x  e.  CC  |->  ( x  x.  C ) )  =  ( x  e.  CC  |->  ( x  x.  C ) )
2726fnmpt 5407 . . . . . . 7  |-  ( A. x  e.  CC  (
x  x.  C )  e.  CC  ->  (
x  e.  CC  |->  ( x  x.  C ) )  Fn  CC )
2825, 27syl 15 . . . . . 6  |-  ( ph  ->  ( x  e.  CC  |->  ( x  x.  C
) )  Fn  CC )
29 fnssres 5394 . . . . . 6  |-  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  Fn  CC  /\  RR  C_  CC )  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  Fn  RR )
3028, 18, 29sylancl 643 . . . . 5  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  Fn  RR )
31 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  w  e.  RR )  ->  w  e.  RR )
32 fvres 5580 . . . . . . . . 9  |-  ( w  e.  RR  ->  (
( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w )  =  ( ( x  e.  CC  |->  ( x  x.  C ) ) `
 w ) )
3318sseli 3210 . . . . . . . . . 10  |-  ( w  e.  RR  ->  w  e.  CC )
34 oveq1 5907 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  x.  C )  =  ( w  x.  C ) )
35 ovex 5925 . . . . . . . . . . 11  |-  ( w  x.  C )  e. 
_V
3634, 26, 35fvmpt 5640 . . . . . . . . . 10  |-  ( w  e.  CC  ->  (
( x  e.  CC  |->  ( x  x.  C
) ) `  w
)  =  ( w  x.  C ) )
3733, 36syl 15 . . . . . . . . 9  |-  ( w  e.  RR  ->  (
( x  e.  CC  |->  ( x  x.  C
) ) `  w
)  =  ( w  x.  C ) )
3832, 37eqtrd 2348 . . . . . . . 8  |-  ( w  e.  RR  ->  (
( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w )  =  ( w  x.  C ) )
3931, 38syl 15 . . . . . . 7  |-  ( (
ph  /\  w  e.  RR )  ->  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  |`  RR ) `
 w )  =  ( w  x.  C
) )
405adantr 451 . . . . . . . 8  |-  ( (
ph  /\  w  e.  RR )  ->  C  e.  RR )
4131, 40remulcld 8908 . . . . . . 7  |-  ( (
ph  /\  w  e.  RR )  ->  ( w  x.  C )  e.  RR )
4239, 41eqeltrd 2390 . . . . . 6  |-  ( (
ph  /\  w  e.  RR )  ->  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  |`  RR ) `
 w )  e.  RR )
4342ralrimiva 2660 . . . . 5  |-  ( ph  ->  A. w  e.  RR  ( ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w
)  e.  RR )
44 fnfvrnss 5725 . . . . 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 642 . . . 4  |-  ( ph  ->  ran  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  C_  RR )
4618a1i 10 . . . 4  |-  ( ph  ->  RR  C_  CC )
47 cnrest2 17070 . . . 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 1182 . . 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 201 . 2  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  (
( ( TopOpen ` fld )t  RR )  Cn  (
( TopOpen ` fld )t  RR ) ) )
50 resmpt 5037 . . 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 18361 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
5452, 53eqtri 2336 . . . 4  |-  J  =  ( ( TopOpen ` fld )t  RR )
5554, 54oveq12i 5912 . . 3  |-  ( J  Cn  J )  =  ( ( ( TopOpen ` fld )t  RR )  Cn  ( ( TopOpen ` fld )t  RR ) )
5655eqcomi 2320 . 2  |-  ( ( ( TopOpen ` fld )t  RR )  Cn  (
( TopOpen ` fld )t  RR ) )  =  ( J  Cn  J
)
5749, 51, 563eltr3g 2398 1  |-  ( ph  ->  ( x  e.  RR  |->  ( x  x.  C
) )  e.  ( J  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577    C_ wss 3186    e. cmpt 4114    X. cxp 4724   ran crn 4727    |` cres 4728    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   CCcc 8780   RRcr 8781    x. cmul 8787   (,)cioo 10703   ↾t crest 13374   TopOpenctopn 13375   topGenctg 13391  ℂfldccnfld 16432  TopOnctopon 16688    Cn ccn 17010    tX ctx 17311
This theorem is referenced by:  rrvmulc  23885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-mulf 8862
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-oi 7270  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ioo 10707  df-icc 10710  df-fz 10830  df-fzo 10918  df-seq 11094  df-exp 11152  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-hom 13279  df-cco 13280  df-rest 13376  df-topn 13377  df-topgen 13393  df-pt 13394  df-prds 13397  df-xrs 13452  df-0g 13453  df-gsum 13454  df-qtop 13459  df-imas 13460  df-xps 13462  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-submnd 14465  df-mulg 14541  df-cntz 14842  df-cmn 15140  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-cnfld 16433  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cn 17013  df-cnp 17014  df-tx 17313  df-hmeo 17502  df-xms 17937  df-ms 17938  df-tms 17939
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