Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rmulccn Structured version   Unicode version

Theorem rmulccn 28148
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 2454 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldtopon 21459 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
32a1i 11 . . . . 5  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
43cnmptid 20331 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  x )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
5 rmulccn.2 . . . . . . 7  |-  ( ph  ->  C  e.  RR )
65recnd 9611 . . . . . 6  |-  ( ph  ->  C  e.  CC )
73, 3, 6cnmptc 20332 . . . . 5  |-  ( ph  ->  ( x  e.  CC  |->  C )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
8 ax-mulf 9561 . . . . . . . . 9  |-  x.  :
( CC  X.  CC )
--> CC
9 ffn 5713 . . . . . . . . 9  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
108, 9ax-mp 5 . . . . . . . 8  |-  x.  Fn  ( CC  X.  CC )
11 fnov 6383 . . . . . . . 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 21540 . . . . . . 7  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
1412, 13eqeltrri 2539 . . . . . 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 6279 . . . . 5  |-  ( ( y  =  x  /\  z  =  C )  ->  ( y  x.  z
)  =  ( x  x.  C ) )
173, 4, 7, 3, 3, 15, 16cnmpt12 20337 . . . 4  |-  ( ph  ->  ( x  e.  CC  |->  ( x  x.  C
) )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) ) )
18 ax-resscn 9538 . . . 4  |-  RR  C_  CC
192toponunii 19603 . . . . 5  |-  CC  =  U. ( TopOpen ` fld )
2019cnrest 19956 . . . 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 660 . . 3  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  e.  (
( ( TopOpen ` fld )t  RR )  Cn  ( TopOpen
` fld
) ) )
22 simpr 459 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  x  e.  CC )
236adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  CC )  ->  C  e.  CC )
2422, 23mulcld 9605 . . . . . . . 8  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  C )  e.  CC )
2524ralrimiva 2868 . . . . . . 7  |-  ( ph  ->  A. x  e.  CC  ( x  x.  C
)  e.  CC )
26 eqid 2454 . . . . . . . 8  |-  ( x  e.  CC  |->  ( x  x.  C ) )  =  ( x  e.  CC  |->  ( x  x.  C ) )
2726fnmpt 5689 . . . . . . 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 5676 . . . . . 6  |-  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  Fn  CC  /\  RR  C_  CC )  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  Fn  RR )
3028, 18, 29sylancl 660 . . . . 5  |-  ( ph  ->  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  Fn  RR )
31 simpr 459 . . . . . . . 8  |-  ( (
ph  /\  w  e.  RR )  ->  w  e.  RR )
32 fvres 5862 . . . . . . . . 9  |-  ( w  e.  RR  ->  (
( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w )  =  ( ( x  e.  CC  |->  ( x  x.  C ) ) `
 w ) )
33 recn 9571 . . . . . . . . . 10  |-  ( w  e.  RR  ->  w  e.  CC )
34 oveq1 6277 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  x.  C )  =  ( w  x.  C ) )
35 ovex 6298 . . . . . . . . . . 11  |-  ( w  x.  C )  e. 
_V
3634, 26, 35fvmpt 5931 . . . . . . . . . 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 2495 . . . . . . . 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 463 . . . . . . . 8  |-  ( (
ph  /\  w  e.  RR )  ->  C  e.  RR )
4131, 40remulcld 9613 . . . . . . 7  |-  ( (
ph  /\  w  e.  RR )  ->  ( w  x.  C )  e.  RR )
4239, 41eqeltrd 2542 . . . . . 6  |-  ( (
ph  /\  w  e.  RR )  ->  ( ( ( x  e.  CC  |->  ( x  x.  C
) )  |`  RR ) `
 w )  e.  RR )
4342ralrimiva 2868 . . . . 5  |-  ( ph  ->  A. w  e.  RR  ( ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR ) `  w
)  e.  RR )
44 fnfvrnss 6035 . . . . 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 659 . . . 4  |-  ( ph  ->  ran  ( ( x  e.  CC  |->  ( x  x.  C ) )  |`  RR )  C_  RR )
4618a1i 11 . . . 4  |-  ( ph  ->  RR  C_  CC )
47 cnrest2 19957 . . . 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 1226 . . 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 5311 . . 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 21477 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
5452, 53eqtri 2483 . . . 4  |-  J  =  ( ( TopOpen ` fld )t  RR )
5554, 54oveq12i 6282 . . 3  |-  ( J  Cn  J )  =  ( ( ( TopOpen ` fld )t  RR )  Cn  ( ( TopOpen ` fld )t  RR ) )
5655eqcomi 2467 . 2  |-  ( ( ( TopOpen ` fld )t  RR )  Cn  (
( TopOpen ` fld )t  RR ) )  =  ( J  Cn  J
)
5749, 51, 563eltr3g 2558 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 367    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461    |-> cmpt 4497    X. cxp 4986   ran crn 4989    |` cres 4990    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   CCcc 9479   RRcr 9480    x. cmul 9486   (,)cioo 11532   ↾t crest 14913   TopOpenctopn 14914   topGenctg 14930  ℂfldccnfld 18618  TopOnctopon 19565    Cn ccn 19895    tX ctx 20230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-rest 14915  df-topn 14916  df-0g 14934  df-gsum 14935  df-topgen 14936  df-pt 14937  df-prds 14940  df-xrs 14994  df-qtop 14999  df-imas 15000  df-xps 15002  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cn 19898  df-cnp 19899  df-tx 20232  df-hmeo 20425  df-xms 20992  df-ms 20993  df-tms 20994
This theorem is referenced by:  rrvmulc  28659
  Copyright terms: Public domain W3C validator