Theorem xrmulc1cn 28725

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 16451	. . . 4 |- <_ e. TosetRel
2	1	a1i 11	. . 3 |- ( ph -> <_ e. TosetRel )
3		simpr 462	. . . . . . 7 |- ( ( ph /\ x e. RR* ) -> x e. RR* )
4		xrmulc1cn.c	. . . . . . . . 9 |- ( ph -> C e. RR+ )
5	4	adantr 466	. . . . . . . 8 |- ( ( ph /\ x e. RR* ) -> C e. RR+ )
6	5	rpxrd 11338	. . . . . . 7 |- ( ( ph /\ x e. RR* ) -> C e. RR* )
7	3, 6	xmulcld 11584	. . . . . 6 |- ( ( ph /\ x e. RR* ) -> ( x xe C ) e. RR* )
8	7	ralrimiva 2837	. . . . 5 |- ( ph -> A. x e. RR* ( x xe C ) e. RR* )
9		simpr 462	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> y e. RR* )
10	4	adantr 466	. . . . . . . . 9 |- ( ( ph /\ y e. RR* ) -> C e. RR+ )
11	10	rpred 11337	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> C e. RR )
12	10	rpne0d 11342	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> C =/= 0 )
13		xreceu 28378	. . . . . . . 8 |- ( ( y e. RR* /\ C e. RR /\ C =/= 0 ) -> E! x e. RR* ( C xe x ) = y )
14	9, 11, 12, 13	syl3anc 1264	. . . . . . 7 |- ( ( ph /\ y e. RR* ) -> E! x e. RR* ( C xe x ) = y )
15		eqcom 2429	. . . . . . . . 9 |- ( y = ( x xe C ) <-> ( x xe C ) = y )
16		simpr 462	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> x e. RR* )
17	6	adantlr 719	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> C e. RR* )
18		xmulcom 11548	. . . . . . . . . . 11 |- ( ( x e. RR* /\ C e. RR* ) -> ( x xe C ) = ( C xe x ) )
19	16, 17, 18	syl2anc 665	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( x xe C ) = ( C xe x ) )
20	19	eqeq1d 2422	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( ( x xe C ) = y <-> ( C xe x ) = y ) )
21	15, 20	syl5bb 260	. . . . . . . 8 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( y = ( x xe C ) <-> ( C xe x ) = y ) )
22	21	reubidva 3010	. . . . . . 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 235	. . . . . 6 |- ( ( ph /\ y e. RR* ) -> E! x e. RR* y = ( x xe C ) )
24	23	ralrimiva 2837	. . . . 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 6051	. . . . 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 668	. . . 4 |- ( ph -> F : RR* -1-1-onto-> RR* )
28		simplr 760	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> x e. RR* )
29		simpr 462	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> y e. RR* )
30	4	ad2antrr 730	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> C e. RR+ )
31		xlemul1 11572	. . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* /\ C e. RR+ ) -> ( x <_ y <-> ( x xe C ) <_ ( y xe C ) ) )
32	28, 29, 30, 31	syl3anc 1264	. . . . . . 7 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( x <_ y <-> ( x xe C ) <_ ( y xe C ) ) )
33		ovex 6325	. . . . . . . . 9 |- ( x xe C ) e. _V
34	25	fvmpt2 5965	. . . . . . . . 9 |- ( ( x e. RR* /\ ( x xe C ) e. _V ) -> ( F ` x ) = ( x xe C ) )
35	28, 33, 34	sylancl 666	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( F ` x ) = ( x xe C ) )
36		oveq1 6304	. . . . . . . . . 10 |- ( x = y -> ( x xe C ) = ( y xe C ) )
37		ovex 6325	. . . . . . . . . 10 |- ( y xe C ) e. _V
38	36, 25, 37	fvmpt 5956	. . . . . . . . 9 |- ( y e. RR* -> ( F ` y ) = ( y xe C ) )
39	38	adantl 467	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( F ` y ) = ( y xe C ) )
40	35, 39	breq12d 4430	. . . . . . 7 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( ( F ` x ) <_ ( F ` y ) <-> ( x xe C ) <_ ( y xe C ) ) )
41	32, 40	bitr4d 259	. . . . . 6 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
42	41	ralrimiva 2837	. . . . 5 |- ( ( ph /\ x e. RR* ) -> A. y e. RR* ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
43	42	ralrimiva 2837	. . . 4 |- ( ph -> A. x e. RR* A. y e. RR* ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
44		df-isom 5602	. . . 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 668	. . 3 |- ( ph -> F Isom <_ , <_ ( RR* , RR* ) )
46		ledm 16448	. . . 4 |- RR* = dom <_
47	46, 46	ordthmeolem 20793	. . 3 |- ( ( <_ e. TosetRel /\ <_ e. TosetRel /\ F Isom <_ , <_ ( RR* , RR* ) ) -> F e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
48	2, 2, 45, 47	syl3anc 1264	. 2 |- ( ph -> F e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
49		xrmulc1cn.k	. . 3 |- J = (ordTop ` <_ )
50	49, 49	oveq12i 6309	. 2 |- ( J Cn J ) = ( (ordTop ` <_ ) Cn (ordTop ` <_ ) )
51	48, 50	syl6eleqr 2519	1 |- ( ph -> F e. ( J Cn J ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1867   =/= wne 2616   A.wral 2773   E!wreu 2775   _Vcvv 3078   class class class wbr 4417   |-> cmpt 4476   -1-1-onto->wf1o 5592   ` cfv 5593   Isom wiso 5594  (class class class)co 6297   RRcr 9534   0cc0 9535   RR*cxr 9670   <_ cle 9672   RR+crp 11298   xecxmu 11404  ordTopcordt 15375   TosetRel ctsr 16423   Cn ccn 20217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-cnex 9591  ax-resscn 9592  ax-1cn 9593  ax-icn 9594  ax-addcl 9595  ax-addrcl 9596  ax-mulcl 9597  ax-mulrcl 9598  ax-mulcom 9599  ax-addass 9600  ax-mulass 9601  ax-distr 9602  ax-i2m1 9603  ax-1ne0 9604  ax-1rid 9605  ax-rnegex 9606  ax-rrecex 9607  ax-cnre 9608  ax-pre-lttri 9609  ax-pre-lttrn 9610  ax-pre-ltadd 9611  ax-pre-mulgt0 9612

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4477  df-mpt 4478  df-tr 4513  df-eprel 4757  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-pred 5391  df-ord 5437  df-on 5438  df-lim 5439  df-suc 5440  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6259  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-om 6699  df-1st 6799  df-2nd 6800  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7923  df-pnf 9673  df-mnf 9674  df-xr 9675  df-ltxr 9676  df-le 9677  df-sub 9858  df-neg 9859  df-div 10266  df-rp 11299  df-xneg 11405  df-xmul 11407  df-topgen 15320  df-ordt 15377  df-ps 16424  df-tsr 16425  df-top 19898  df-bases 19899  df-topon 19900  df-cn 20220

This theorem is referenced by:  xrge0mulc1cn  28736