Theorem xrmulc1cn 24269

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

Hypotheses

Ref	Expression
xrmulc1cn.k	|- J = (ordTop ` <_ )
xrmulc1cn.f	|- F = ( x e. RR* |-> ( x x e 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 14627	. . . 4 |- <_ e. TosetRel
2	1	a1i 11	. . 3 |- ( ph -> <_ e. TosetRel )
3		simpr 448	. . . . . . 7 |- ( ( ph /\ x e. RR* ) -> x e. RR* )
4		xrmulc1cn.c	. . . . . . . . 9 |- ( ph -> C e. RR+ )
5	4	adantr 452	. . . . . . . 8 |- ( ( ph /\ x e. RR* ) -> C e. RR+ )
6	5	rpxrd 10605	. . . . . . 7 |- ( ( ph /\ x e. RR* ) -> C e. RR* )
7	3, 6	xmulcld 10837	. . . . . 6 |- ( ( ph /\ x e. RR* ) -> ( x x e C ) e. RR* )
8	7	ralrimiva 2749	. . . . 5 |- ( ph -> A. x e. RR* ( x x e C ) e. RR* )
9		simpr 448	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> y e. RR* )
10	4	adantr 452	. . . . . . . . 9 |- ( ( ph /\ y e. RR* ) -> C e. RR+ )
11	10	rpred 10604	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> C e. RR )
12	10	rpne0d 10609	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> C =/= 0 )
13		xreceu 24121	. . . . . . . 8 |- ( ( y e. RR* /\ C e. RR /\ C =/= 0 ) -> E! x e. RR* ( C x e x ) = y )
14	9, 11, 12, 13	syl3anc 1184	. . . . . . 7 |- ( ( ph /\ y e. RR* ) -> E! x e. RR* ( C x e x ) = y )
15		eqcom 2406	. . . . . . . . 9 |- ( y = ( x x e C ) <-> ( x x e C ) = y )
16		simpr 448	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> x e. RR* )
17	6	adantlr 696	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> C e. RR* )
18		xmulcom 10801	. . . . . . . . . . 11 |- ( ( x e. RR* /\ C e. RR* ) -> ( x x e C ) = ( C x e x ) )
19	16, 17, 18	syl2anc 643	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( x x e C ) = ( C x e x ) )
20	19	eqeq1d 2412	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( ( x x e C ) = y <-> ( C x e x ) = y ) )
21	15, 20	syl5bb 249	. . . . . . . 8 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( y = ( x x e C ) <-> ( C x e x ) = y ) )
22	21	reubidva 2851	. . . . . . 7 |- ( ( ph /\ y e. RR* ) -> ( E! x e. RR* y = ( x x e C ) <-> E! x e. RR* ( C x e x ) = y ) )
23	14, 22	mpbird 224	. . . . . 6 |- ( ( ph /\ y e. RR* ) -> E! x e. RR* y = ( x x e C ) )
24	23	ralrimiva 2749	. . . . 5 |- ( ph -> A. y e. RR* E! x e. RR* y = ( x x e C ) )
25		xrmulc1cn.f	. . . . . 6 |- F = ( x e. RR* |-> ( x x e C ) )
26	25	f1ompt 5850	. . . . 5 |- ( F : RR* -1-1-onto-> RR* <-> ( A. x e. RR* ( x x e C ) e. RR* /\ A. y e. RR* E! x e. RR* y = ( x x e C ) ) )
27	8, 24, 26	sylanbrc 646	. . . 4 |- ( ph -> F : RR* -1-1-onto-> RR* )
28		simplr 732	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> x e. RR* )
29		simpr 448	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> y e. RR* )
30	4	ad2antrr 707	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> C e. RR+ )
31		xlemul1 10825	. . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* /\ C e. RR+ ) -> ( x <_ y <-> ( x x e C ) <_ ( y x e C ) ) )
32	28, 29, 30, 31	syl3anc 1184	. . . . . . 7 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( x <_ y <-> ( x x e C ) <_ ( y x e C ) ) )
33		ovex 6065	. . . . . . . . 9 |- ( x x e C ) e. _V
34	25	fvmpt2 5771	. . . . . . . . 9 |- ( ( x e. RR* /\ ( x x e C ) e. _V ) -> ( F ` x ) = ( x x e C ) )
35	28, 33, 34	sylancl 644	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( F ` x ) = ( x x e C ) )
36		oveq1 6047	. . . . . . . . . 10 |- ( x = y -> ( x x e C ) = ( y x e C ) )
37		ovex 6065	. . . . . . . . . 10 |- ( y x e C ) e. _V
38	36, 25, 37	fvmpt 5765	. . . . . . . . 9 |- ( y e. RR* -> ( F ` y ) = ( y x e C ) )
39	38	adantl 453	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( F ` y ) = ( y x e C ) )
40	35, 39	breq12d 4185	. . . . . . 7 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( ( F ` x ) <_ ( F ` y ) <-> ( x x e C ) <_ ( y x e C ) ) )
41	32, 40	bitr4d 248	. . . . . 6 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
42	41	ralrimiva 2749	. . . . 5 |- ( ( ph /\ x e. RR* ) -> A. y e. RR* ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
43	42	ralrimiva 2749	. . . 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 646	. . 3 |- ( ph -> F Isom <_ , <_ ( RR* , RR* ) )
46		ledm 14624	. . . 4 |- RR* = dom <_
47	46, 46	ordthmeolem 17786	. . 3 |- ( ( <_ e. TosetRel /\ <_ e. TosetRel /\ F Isom <_ , <_ ( RR* , RR* ) ) -> F e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
48	2, 2, 45, 47	syl3anc 1184	. 2 |- ( ph -> F e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
49		xrmulc1cn.k	. . 3 |- J = (ordTop ` <_ )
50	49, 49	oveq12i 6052	. 2 |- ( J Cn J ) = ( (ordTop ` <_ ) Cn (ordTop ` <_ ) )
51	48, 50	syl6eleqr 2495	1 |- ( ph -> F e. ( J Cn J ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E!wreu 2668   _Vcvv 2916   class class class wbr 4172   e. cmpt 4226   -1-1-onto->wf1o 5412   ` cfv 5413   Isom wiso 5414  (class class class)co 6040   RRcr 8945   0cc0 8946   RR*cxr 9075   <_ cle 9077   RR+crp 10568   x ecxmu 10665  ordTopcordt 13676   TosetRel ctsr 14580   Cn ccn 17242

This theorem is referenced by:  xrge0mulc1cn  24280

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-rp 10569  df-xneg 10666  df-xmul 10668  df-topgen 13622  df-ordt 13680  df-ps 14584  df-tsr 14585  df-top 16918  df-bases 16920  df-topon 16921  df-cn 17245