Theorem xrmulc1cn 28810

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 16551	. . . 4 |- <_ e. TosetRel
2	1	a1i 11	. . 3 |- ( ph -> <_ e. TosetRel )
3		simpr 468	. . . . . . 7 |- ( ( ph /\ x e. RR* ) -> x e. RR* )
4		xrmulc1cn.c	. . . . . . . . 9 |- ( ph -> C e. RR+ )
5	4	adantr 472	. . . . . . . 8 |- ( ( ph /\ x e. RR* ) -> C e. RR+ )
6	5	rpxrd 11365	. . . . . . 7 |- ( ( ph /\ x e. RR* ) -> C e. RR* )
7	3, 6	xmulcld 11613	. . . . . 6 |- ( ( ph /\ x e. RR* ) -> ( x xe C ) e. RR* )
8	7	ralrimiva 2809	. . . . 5 |- ( ph -> A. x e. RR* ( x xe C ) e. RR* )
9		simpr 468	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> y e. RR* )
10	4	adantr 472	. . . . . . . . 9 |- ( ( ph /\ y e. RR* ) -> C e. RR+ )
11	10	rpred 11364	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> C e. RR )
12	10	rpne0d 11369	. . . . . . . 8 |- ( ( ph /\ y e. RR* ) -> C =/= 0 )
13		xreceu 28466	. . . . . . . 8 |- ( ( y e. RR* /\ C e. RR /\ C =/= 0 ) -> E! x e. RR* ( C xe x ) = y )
14	9, 11, 12, 13	syl3anc 1292	. . . . . . 7 |- ( ( ph /\ y e. RR* ) -> E! x e. RR* ( C xe x ) = y )
15		eqcom 2478	. . . . . . . . 9 |- ( y = ( x xe C ) <-> ( x xe C ) = y )
16		simpr 468	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> x e. RR* )
17	6	adantlr 729	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> C e. RR* )
18		xmulcom 11577	. . . . . . . . . . 11 |- ( ( x e. RR* /\ C e. RR* ) -> ( x xe C ) = ( C xe x ) )
19	16, 17, 18	syl2anc 673	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( x xe C ) = ( C xe x ) )
20	19	eqeq1d 2473	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( ( x xe C ) = y <-> ( C xe x ) = y ) )
21	15, 20	syl5bb 265	. . . . . . . 8 |- ( ( ( ph /\ y e. RR* ) /\ x e. RR* ) -> ( y = ( x xe C ) <-> ( C xe x ) = y ) )
22	21	reubidva 2960	. . . . . . 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 240	. . . . . 6 |- ( ( ph /\ y e. RR* ) -> E! x e. RR* y = ( x xe C ) )
24	23	ralrimiva 2809	. . . . 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 6059	. . . . 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 677	. . . 4 |- ( ph -> F : RR* -1-1-onto-> RR* )
28		simplr 770	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> x e. RR* )
29		simpr 468	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> y e. RR* )
30	4	ad2antrr 740	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> C e. RR+ )
31		xlemul1 11601	. . . . . . . 8 |- ( ( x e. RR* /\ y e. RR* /\ C e. RR+ ) -> ( x <_ y <-> ( x xe C ) <_ ( y xe C ) ) )
32	28, 29, 30, 31	syl3anc 1292	. . . . . . 7 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( x <_ y <-> ( x xe C ) <_ ( y xe C ) ) )
33		ovex 6336	. . . . . . . . 9 |- ( x xe C ) e. _V
34	25	fvmpt2 5972	. . . . . . . . 9 |- ( ( x e. RR* /\ ( x xe C ) e. _V ) -> ( F ` x ) = ( x xe C ) )
35	28, 33, 34	sylancl 675	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( F ` x ) = ( x xe C ) )
36		oveq1 6315	. . . . . . . . . 10 |- ( x = y -> ( x xe C ) = ( y xe C ) )
37		ovex 6336	. . . . . . . . . 10 |- ( y xe C ) e. _V
38	36, 25, 37	fvmpt 5963	. . . . . . . . 9 |- ( y e. RR* -> ( F ` y ) = ( y xe C ) )
39	38	adantl 473	. . . . . . . 8 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( F ` y ) = ( y xe C ) )
40	35, 39	breq12d 4408	. . . . . . 7 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( ( F ` x ) <_ ( F ` y ) <-> ( x xe C ) <_ ( y xe C ) ) )
41	32, 40	bitr4d 264	. . . . . 6 |- ( ( ( ph /\ x e. RR* ) /\ y e. RR* ) -> ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
42	41	ralrimiva 2809	. . . . 5 |- ( ( ph /\ x e. RR* ) -> A. y e. RR* ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
43	42	ralrimiva 2809	. . . 4 |- ( ph -> A. x e. RR* A. y e. RR* ( x <_ y <-> ( F ` x ) <_ ( F ` y ) ) )
44		df-isom 5598	. . . 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 677	. . 3 |- ( ph -> F Isom <_ , <_ ( RR* , RR* ) )
46		ledm 16548	. . . 4 |- RR* = dom <_
47	46, 46	ordthmeolem 20893	. . 3 |- ( ( <_ e. TosetRel /\ <_ e. TosetRel /\ F Isom <_ , <_ ( RR* , RR* ) ) -> F e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
48	2, 2, 45, 47	syl3anc 1292	. 2 |- ( ph -> F e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
49		xrmulc1cn.k	. . 3 |- J = (ordTop ` <_ )
50	49, 49	oveq12i 6320	. 2 |- ( J Cn J ) = ( (ordTop ` <_ ) Cn (ordTop ` <_ ) )
51	48, 50	syl6eleqr 2560	1 |- ( ph -> F e. ( J Cn J ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   E!wreu 2758   _Vcvv 3031   class class class wbr 4395   |-> cmpt 4454   -1-1-onto->wf1o 5588   ` cfv 5589   Isom wiso 5590  (class class class)co 6308   RRcr 9556   0cc0 9557   RR*cxr 9692   <_ cle 9694   RR+crp 11325   xecxmu 11431  ordTopcordt 15475   TosetRel ctsr 16523   Cn ccn 20317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-rp 11326  df-xneg 11432  df-xmul 11434  df-topgen 15420  df-ordt 15477  df-ps 16524  df-tsr 16525  df-top 19998  df-bases 19999  df-topon 20000  df-cn 20320

This theorem is referenced by:  xrge0mulc1cn  28821