Theorem xrmulc1cn 26214

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 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 15380	. . . 4 |- <_ e. TosetRel
2	1	a1i 11	. . 3 |- ( ph -> <_ e. TosetRel )
3		simpr 458	. . . . . . 7 |- ( ( ph /\ x e. RR* ) -> x e. RR* )
4		xrmulc1cn.c	. . . . . . . . 9 |- ( ph -> C e. RR+ )
5	4	adantr 462	. . . . . . . 8 |- ( ( ph /\ x e. RR* ) -> C e. RR+ )
6	5	rpxrd 11016	. . . . . . 7 |- ( ( ph /\ x e. RR* ) -> C e. RR* )
7	3, 6	xmulcld 11253	. . . . . 6 |- ( ( ph /\ x e. RR* ) -> ( x xe C ) e. RR* )
8	7	ralrimiva 2789	. . . . 5 |- ( ph -> A. x e. RR* ( x xe C ) e. RR* )
9		simpr 458	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> y e. RR* )
10	4	adantr 462	. . . . . . . . 9 |- ( ( ph /\ y e. RR* ) -> C e. RR+ )
11	10	rpred 11015	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> C e. RR )
12	10	rpne0d 11020	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> C =/= 0 )
13		xreceu 25920	. . . . . . . 8 |- ( ( y e. RR* /\ C e. RR /\ C =/= 0 ) -> E! x e. RR* ( C xe x ) = y )
14	9, 11, 12, 13	syl3anc 1211	. . . . . . 7 |- ( ( ph /\ y e. RR* ) -> E! x e. RR* ( C xe x ) = y )
15		eqcom 2435	. . . . . . . . 9 |- ( y = ( x xe C ) <-> ( x xe C ) = y )
16		simpr 458	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> x e. RR* )
17	6	adantlr 707	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> C e. RR* )
18		xmulcom 11217	. . . . . . . . . . 11 |- ( ( x e. RR* /\ C e. RR* ) -> ( x xe C ) = ( C xe x ) )
19	16, 17, 18	syl2anc 654	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( x xe C ) = ( C xe x ) )
20	19	eqeq1d 2441	. . . . . . . . 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 2894	. . . . . . 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 2789	. . . . 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 5853	. . . . 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 657	. . . 4 |- ( ph -> F : RR* -1-1-onto-> RR* )
28		simplr 747	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> x e. RR* )
29		simpr 458	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> y e. RR* )
30	4	ad2antrr 718	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> C e. RR+ )
31		xlemul1 11241	. . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* /\ C e. RR+ ) -> ( x <_ y <-> ( x xe C ) <_ ( y xe C ) ) )
32	28, 29, 30, 31	syl3anc 1211	. . . . . . 7 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( x <_ y <-> ( x xe C ) <_ ( y xe C ) ) )
33		ovex 6105	. . . . . . . . 9 |- ( x xe C ) e. _V
34	25	fvmpt2 5769	. . . . . . . . 9 |- ( ( x e. RR* /\ ( x xe C ) e. _V ) -> ( F ` x ) = ( x xe C ) )
35	28, 33, 34	sylancl 655	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( F ` x ) = ( x xe C ) )
36		oveq1 6087	. . . . . . . . . 10 |- ( x = y -> ( x xe C ) = ( y xe C ) )
37		ovex 6105	. . . . . . . . . 10 |- ( y xe C ) e. _V
38	36, 25, 37	fvmpt 5762	. . . . . . . . 9 |- ( y e. RR* -> ( F ` y ) = ( y xe C ) )
39	38	adantl 463	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( F ` y ) = ( y xe C ) )
40	35, 39	breq12d 4293	. . . . . . 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 2789	. . . . 5 |- ( ( ph /\ x e. RR* ) -> A. y e. RR* ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
43	42	ralrimiva 2789	. . . 4 |- ( ph -> A. x e. RR* A. y e. RR* ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
44		df-isom 5415	. . . 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 657	. . 3 |- ( ph -> F Isom <_ , <_ ( RR* , RR* ) )
46		ledm 15377	. . . 4 |- RR* = dom <_
47	46, 46	ordthmeolem 19216	. . 3 |- ( ( <_ e. TosetRel /\ <_ e. TosetRel /\ F Isom <_ , <_ ( RR* , RR* ) ) -> F e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
48	2, 2, 45, 47	syl3anc 1211	. 2 |- ( ph -> F e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
49		xrmulc1cn.k	. . 3 |- J = (ordTop ` <_ )
50	49, 49	oveq12i 6092	. 2 |- ( J Cn J ) = ( (ordTop ` <_ ) Cn (ordTop ` <_ ) )
51	48, 50	syl6eleqr 2524	1 |- ( ph -> F e. ( J Cn J ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1362   e. wcel 1755   =/= wne 2596   A.wral 2705   E!wreu 2707   _Vcvv 2962   class class class wbr 4280   e. cmpt 4338   -1-1-onto->wf1o 5405   ` cfv 5406   Isom wiso 5407  (class class class)co 6080   RRcr 9269   0cc0 9270   RR*cxr 9405   <_ cle 9407   RR+crp 10979   xecxmu 11076  ordTopcordt 14420   TosetRel ctsr 15352   Cn ccn 18670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fi 7649  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-rp 10980  df-xneg 11077  df-xmul 11079  df-topgen 14365  df-ordt 14422  df-ps 15353  df-tsr 15354  df-top 18345  df-bases 18347  df-topon 18348  df-cn 18673

This theorem is referenced by:  xrge0mulc1cn  26225