Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xdivpnfrp Structured version   Unicode version

Theorem xdivpnfrp 26246
Description: Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
Assertion
Ref Expression
xdivpnfrp  |-  ( A  e.  RR+  ->  ( +oo /𝑒  A )  = +oo )

Proof of Theorem xdivpnfrp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rprene0 11111 . . . . 5  |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  A  =/=  0 ) )
2 pnfxr 11196 . . . . 5  |- +oo  e.  RR*
31, 2jctil 537 . . . 4  |-  ( A  e.  RR+  ->  ( +oo  e.  RR*  /\  ( A  e.  RR  /\  A  =/=  0 ) ) )
4 3anass 969 . . . 4  |-  ( ( +oo  e.  RR*  /\  A  e.  RR  /\  A  =/=  0 )  <->  ( +oo  e.  RR*  /\  ( A  e.  RR  /\  A  =/=  0 ) ) )
53, 4sylibr 212 . . 3  |-  ( A  e.  RR+  ->  ( +oo  e.  RR*  /\  A  e.  RR  /\  A  =/=  0 ) )
6 xdivval 26232 . . 3  |-  ( ( +oo  e.  RR*  /\  A  e.  RR  /\  A  =/=  0 )  ->  ( +oo /𝑒  A )  =  ( iota_ x  e.  RR*  ( A xe x )  = +oo ) )
75, 6syl 16 . 2  |-  ( A  e.  RR+  ->  ( +oo /𝑒  A )  =  ( iota_ x  e. 
RR*  ( A xe x )  = +oo ) )
82a1i 11 . . 3  |-  ( A  e.  RR+  -> +oo  e.  RR* )
9 xlemul2 11358 . . . . . . 7  |-  ( ( +oo  e.  RR*  /\  x  e.  RR*  /\  A  e.  RR+ )  ->  ( +oo  <_  x  <->  ( A xe +oo )  <_ 
( A xe x ) ) )
102, 9mp3an1 1302 . . . . . 6  |-  ( ( x  e.  RR*  /\  A  e.  RR+ )  ->  ( +oo  <_  x  <->  ( A xe +oo )  <_  ( A xe x ) ) )
1110ancoms 453 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR* )  ->  ( +oo  <_  x  <->  ( A xe +oo )  <_  ( A xe x ) ) )
12 rpxr 11102 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e. 
RR* )
13 rpgt0 11106 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  < 
A )
14 xmulpnf1 11341 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
1512, 13, 14syl2anc 661 . . . . . . 7  |-  ( A  e.  RR+  ->  ( A xe +oo )  = +oo )
1615adantr 465 . . . . . 6  |-  ( ( A  e.  RR+  /\  x  e.  RR* )  ->  ( A xe +oo )  = +oo )
1716breq1d 4403 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR* )  ->  (
( A xe +oo )  <_  ( A xe x )  <-> +oo  <_  ( A xe x ) ) )
1811, 17bitr2d 254 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR* )  ->  ( +oo  <_  ( A xe x )  <-> +oo  <_  x
) )
19 xmulcl 11340 . . . . . 6  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A xe x )  e.  RR* )
2012, 19sylan 471 . . . . 5  |-  ( ( A  e.  RR+  /\  x  e.  RR* )  ->  ( A xe x )  e.  RR* )
21 xgepnf 26187 . . . . 5  |-  ( ( A xe x )  e.  RR*  ->  ( +oo  <_  ( A xe x )  <-> 
( A xe x )  = +oo ) )
2220, 21syl 16 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR* )  ->  ( +oo  <_  ( A xe x )  <->  ( A xe x )  = +oo ) )
23 xgepnf 26187 . . . . 5  |-  ( x  e.  RR*  ->  ( +oo  <_  x  <->  x  = +oo ) )
2423adantl 466 . . . 4  |-  ( ( A  e.  RR+  /\  x  e.  RR* )  ->  ( +oo  <_  x  <->  x  = +oo ) )
2518, 22, 243bitr3d 283 . . 3  |-  ( ( A  e.  RR+  /\  x  e.  RR* )  ->  (
( A xe x )  = +oo  <->  x  = +oo ) )
268, 25riota5 6180 . 2  |-  ( A  e.  RR+  ->  ( iota_ x  e.  RR*  ( A xe x )  = +oo )  = +oo )
277, 26eqtrd 2492 1  |-  ( A  e.  RR+  ->  ( +oo /𝑒  A )  = +oo )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4393   iota_crio 6153  (class class class)co 6193   RRcr 9385   0cc0 9386   +oocpnf 9519   RR*cxr 9521    < clt 9522    <_ cle 9523   RR+crp 11095   xecxmu 11192   /𝑒 cxdiv 26230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-rp 11096  df-xneg 11193  df-xmul 11195  df-xdiv 26231
This theorem is referenced by:  xrpxdivcld  26248
  Copyright terms: Public domain W3C validator