Theorem xrmulc1cn 26329

Description: The operation multiplying an extended real number by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)

Hypotheses

Ref	Expression
xrmulc1cn.k	|- J = (ordTop ` <_ )
xrmulc1cn.f	|- F = ( x e. RR* |-> ( x xe C ) )
xrmulc1cn.c	|- ( ph -> C e. RR+ )

Assertion

Ref	Expression
xrmulc1cn	|- ( ph -> F e. ( J Cn J ) )

Distinct variable groups:   x, C   x, F   ph, x

Allowed substitution hint:   J( x)

Proof of Theorem xrmulc1cn

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		letsr 15389	. . . 4 |- <_ e. TosetRel
2	1	a1i 11	. . 3 |- ( ph -> <_ e. TosetRel )
3		simpr 461	. . . . . . 7 |- ( ( ph /\ x e. RR* ) -> x e. RR* )
4		xrmulc1cn.c	. . . . . . . . 9 |- ( ph -> C e. RR+ )
5	4	adantr 465	. . . . . . . 8 |- ( ( ph /\ x e. RR* ) -> C e. RR+ )
6	5	rpxrd 11020	. . . . . . 7 |- ( ( ph /\ x e. RR* ) -> C e. RR* )
7	3, 6	xmulcld 11257	. . . . . 6 |- ( ( ph /\ x e. RR* ) -> ( x xe C ) e. RR* )
8	7	ralrimiva 2794	. . . . 5 |- ( ph -> A. x e. RR* ( x xe C ) e. RR* )
9		simpr 461	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> y e. RR* )
10	4	adantr 465	. . . . . . . . 9 |- ( ( ph /\ y e. RR* ) -> C e. RR+ )
11	10	rpred 11019	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> C e. RR )
12	10	rpne0d 11024	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> C =/= 0 )
13		xreceu 26065	. . . . . . . 8 |- ( ( y e. RR* /\ C e. RR /\ C =/= 0 ) -> E! x e. RR* ( C xe x ) = y )
14	9, 11, 12, 13	syl3anc 1218	. . . . . . 7 |- ( ( ph /\ y e. RR* ) -> E! x e. RR* ( C xe x ) = y )
15		eqcom 2440	. . . . . . . . 9 |- ( y = ( x xe C ) <-> ( x xe C ) = y )
16		simpr 461	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> x e. RR* )
17	6	adantlr 714	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> C e. RR* )
18		xmulcom 11221	. . . . . . . . . . 11 |- ( ( x e. RR* /\ C e. RR* ) -> ( x xe C ) = ( C xe x ) )
19	16, 17, 18	syl2anc 661	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( x xe C ) = ( C xe x ) )
20	19	eqeq1d 2446	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( ( x xe C ) = y <-> ( C xe x ) = y ) )
21	15, 20	syl5bb 257	. . . . . . . 8 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( y = ( x xe C ) <-> ( C xe x ) = y ) )
22	21	reubidva 2899	. . . . . . 7 |- ( ( ph /\ y e. RR* ) -> ( E! x e. RR* y = ( x xe C ) <-> E! x e. RR* ( C xe x ) = y ) )
23	14, 22	mpbird 232	. . . . . 6 |- ( ( ph /\ y e. RR* ) -> E! x e. RR* y = ( x xe C ) )
24	23	ralrimiva 2794	. . . . 5 |- ( ph -> A. y e. RR* E! x e. RR* y = ( x xe C ) )
25		xrmulc1cn.f	. . . . . 6 |- F = ( x e. RR* |-> ( x xe C ) )
26	25	f1ompt 5860	. . . . 5 |- ( F : RR* -1-1-onto-> RR* <-> ( A. x e. RR* ( x xe C ) e. RR* /\ A. y e. RR* E! x e. RR* y = ( x xe C ) ) )
27	8, 24, 26	sylanbrc 664	. . . 4 |- ( ph -> F : RR* -1-1-onto-> RR* )
28		simplr 754	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> x e. RR* )
29		simpr 461	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> y e. RR* )
30	4	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> C e. RR+ )
31		xlemul1 11245	. . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* /\ C e. RR+ ) -> ( x <_ y <-> ( x xe C ) <_ ( y xe C ) ) )
32	28, 29, 30, 31	syl3anc 1218	. . . . . . 7 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( x <_ y <-> ( x xe C ) <_ ( y xe C ) ) )
33		ovex 6111	. . . . . . . . 9 |- ( x xe C ) e. _V
34	25	fvmpt2 5776	. . . . . . . . 9 |- ( ( x e. RR* /\ ( x xe C ) e. _V ) -> ( F ` x ) = ( x xe C ) )
35	28, 33, 34	sylancl 662	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( F ` x ) = ( x xe C ) )
36		oveq1 6093	. . . . . . . . . 10 |- ( x = y -> ( x xe C ) = ( y xe C ) )
37		ovex 6111	. . . . . . . . . 10 |- ( y xe C ) e. _V
38	36, 25, 37	fvmpt 5769	. . . . . . . . 9 |- ( y e. RR* -> ( F ` y ) = ( y xe C ) )
39	38	adantl 466	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( F ` y ) = ( y xe C ) )
40	35, 39	breq12d 4300	. . . . . . 7 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( ( F ` x ) <_ ( F ` y ) <-> ( x xe C ) <_ ( y xe C ) ) )
41	32, 40	bitr4d 256	. . . . . 6 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
42	41	ralrimiva 2794	. . . . 5 |- ( ( ph /\ x e. RR* ) -> A. y e. RR* ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
43	42	ralrimiva 2794	. . . 4 |- ( ph -> A. x e. RR* A. y e. RR* ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
44		df-isom 5422	. . . 4 |- ( F Isom <_ , <_ ( RR* , RR* ) <-> ( F : RR* -1-1-onto-> RR* /\ A. x e. RR* A. y e. RR* ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) ) )
45	27, 43, 44	sylanbrc 664	. . 3 |- ( ph -> F Isom <_ , <_ ( RR* , RR* ) )
46		ledm 15386	. . . 4 |- RR* = dom <_
47	46, 46	ordthmeolem 19354	. . 3 |- ( ( <_ e. TosetRel /\ <_ e. TosetRel /\ F Isom <_ , <_ ( RR* , RR* ) ) -> F e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
48	2, 2, 45, 47	syl3anc 1218	. 2 |- ( ph -> F e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
49		xrmulc1cn.k	. . 3 |- J = (ordTop ` <_ )
50	49, 49	oveq12i 6098	. 2 |- ( J Cn J ) = ( (ordTop ` <_ ) Cn (ordTop ` <_ ) )
51	48, 50	syl6eleqr 2529	1 |- ( ph -> F e. ( J Cn J ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2601   A.wral 2710   E!wreu 2712   _Vcvv 2967   class class class wbr 4287   e. cmpt 4345   -1-1-onto->wf1o 5412   ` cfv 5413   Isom wiso 5414  (class class class)co 6086   RRcr 9273   0cc0 9274   RR*cxr 9409   <_ cle 9411   RR+crp 10983   xecxmu 11080  ordTopcordt 14429   TosetRel ctsr 15361   Cn ccn 18808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fi 7653  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-rp 10984  df-xneg 11081  df-xmul 11083  df-topgen 14374  df-ordt 14431  df-ps 15362  df-tsr 15363  df-top 18483  df-bases 18485  df-topon 18486  df-cn 18811

This theorem is referenced by:  xrge0mulc1cn  26340