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Theorem rmulccn 27776
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 21156 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
32a1i 11 . . . . 5  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
43cnmptid 20028 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  x )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
5 rmulccn.2 . . . . . . 7  |-  ( ph  ->  C  e.  RR )
65recnd 9620 . . . . . 6  |-  ( ph  ->  C  e.  CC )
73, 3, 6cnmptc 20029 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  C )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
8 ax-mulf 9570 . . . . . . . . 9  |-  x.  :
( CC  X.  CC )
--> CC
9 ffn 5717 . . . . . . . . 9  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
108, 9ax-mp 5 . . . . . . . 8  |-  x.  Fn  ( CC  X.  CC )
11 fnov 6391 . . . . . . . 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 21237 . . . . . . 7  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
1412, 13eqeltrri 2526 . . . . . 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 6286 . . . . 5  |-  ( ( y  =  x  /\  z  =  C )  ->  ( y  x.  z
)  =  ( x  x.  C ) )
173, 4, 7, 3, 3, 15, 16cnmpt12 20034 . . . 4  |-  ( ph  ->  ( x  e.  CC  |->  ( x  x.  C
) )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
18 ax-resscn 9547 . . . 4  |-  RR  C_  CC
192toponunii 19300 . . . . 5  |-  CC  =  U. ( TopOpen ` fld )
2019cnrest 19652 . . . 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 662 . . 3  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  (
( ( TopOpen ` fld )t  RR )  Cn  ( TopOpen
` fld
) ) )
22 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
236adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  C  e.  CC )
2422, 23mulcld 9614 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  C )  e.  CC )
2524ralrimiva 2855 . . . . . . 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 5693 . . . . . . 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 5680 . . . . . 6  |-  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  Fn  CC  /\  RR  C_  CC )  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  Fn  RR )
3028, 18, 29sylancl 662 . . . . 5  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  Fn  RR )
31 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  w  e.  RR )  ->  w  e.  RR )
32 fvres 5866 . . . . . . . . 9  |-  ( w  e.  RR  ->  (
( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w )  =  ( ( x  e.  CC  |->  ( x  x.  C ) ) `
 w ) )
33 recn 9580 . . . . . . . . . 10  |-  ( w  e.  RR  ->  w  e.  CC )
34 oveq1 6284 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  x.  C )  =  ( w  x.  C ) )
35 ovex 6305 . . . . . . . . . . 11  |-  ( w  x.  C )  e. 
_V
3634, 26, 35fvmpt 5937 . . . . . . . . . 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 2482 . . . . . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  w  e.  RR )  ->  C  e.  RR )
4131, 40remulcld 9622 . . . . . . 7  |-  ( (
ph  /\  w  e.  RR )  ->  ( w  x.  C )  e.  RR )
4239, 41eqeltrd 2529 . . . . . 6  |-  ( (
ph  /\  w  e.  RR )  ->  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  |`  RR ) `
 w )  e.  RR )
4342ralrimiva 2855 . . . . 5  |-  ( ph  ->  A. w  e.  RR  ( ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w
)  e.  RR )
44 fnfvrnss 6040 . . . . 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 661 . . . 4  |-  ( ph  ->  ran  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  C_  RR )
4618a1i 11 . . . 4  |-  ( ph  ->  RR  C_  CC )
47 cnrest2 19653 . . . 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 1227 . . 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 5309 . . 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 21174 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
5452, 53eqtri 2470 . . . 4  |-  J  =  ( ( TopOpen ` fld )t  RR )
5554, 54oveq12i 6289 . . 3  |-  ( J  Cn  J )  =  ( ( ( TopOpen ` fld )t  RR )  Cn  ( ( TopOpen ` fld )t  RR ) )
5655eqcomi 2454 . 2  |-  ( ( ( TopOpen ` fld )t  RR )  Cn  (
( TopOpen ` fld )t  RR ) )  =  ( J  Cn  J
)
5749, 51, 563eltr3g 2545 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 1381    e. wcel 1802   A.wral 2791    C_ wss 3458    |-> cmpt 4491    X. cxp 4983   ran crn 4986    |` cres 4987    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   CCcc 9488   RRcr 9489    x. cmul 9495   (,)cioo 11533   ↾t crest 14690   TopOpenctopn 14691   topGenctg 14707  ℂfldccnfld 18288  TopOnctopon 19262    Cn ccn 19591    tX ctx 19927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-mulf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-fi 7869  df-sup 7899  df-oi 7933  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-ioo 11537  df-icc 11540  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-hom 14593  df-cco 14594  df-rest 14692  df-topn 14693  df-0g 14711  df-gsum 14712  df-topgen 14713  df-pt 14714  df-prds 14717  df-xrs 14771  df-qtop 14776  df-imas 14777  df-xps 14779  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-mulg 15929  df-cntz 16224  df-cmn 16669  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cn 19594  df-cnp 19595  df-tx 19929  df-hmeo 20122  df-xms 20689  df-ms 20690  df-tms 20691
This theorem is referenced by:  rrvmulc  28258
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