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Theorem xdivpnfrp 27297
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 11232 . . . . 5  |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  A  =/=  0 ) )
2 pnfxr 11317 . . . . 5  |- +oo  e.  RR*
31, 2jctil 537 . . . 4  |-  ( A  e.  RR+  ->  ( +oo  e.  RR*  /\  ( A  e.  RR  /\  A  =/=  0 ) ) )
4 3anass 977 . . . 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 27283 . . 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 11479 . . . . . . 7  |-  ( ( +oo  e.  RR*  /\  x  e.  RR*  /\  A  e.  RR+ )  ->  ( +oo  <_  x  <->  ( A xe +oo )  <_ 
( A xe x ) ) )
102, 9mp3an1 1311 . . . . . 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 11223 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e. 
RR* )
13 rpgt0 11227 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  < 
A )
14 xmulpnf1 11462 . . . . . . . 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 4457 . . . . 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 11461 . . . . . 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 27238 . . . . 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 27238 . . . . 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 6269 . 2  |-  ( A  e.  RR+  ->  ( iota_ x  e.  RR*  ( A xe x )  = +oo )  = +oo )
277, 26eqtrd 2508 1  |-  ( A  e.  RR+  ->  ( +oo /𝑒  A )  = +oo )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   iota_crio 6242  (class class class)co 6282   RRcr 9487   0cc0 9488   +oocpnf 9621   RR*cxr 9623    < clt 9624    <_ cle 9625   RR+crp 11216   xecxmu 11313   /𝑒 cxdiv 27281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-rp 11217  df-xneg 11314  df-xmul 11316  df-xdiv 27282
This theorem is referenced by:  xrpxdivcld  27299
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