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Theorem xdivpnfrp 27781
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 11261 . . . . 5  |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  A  =/=  0 ) )
2 pnfxr 11346 . . . . 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 27767 . . 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 11508 . . . . . . 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 11252 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e. 
RR* )
13 rpgt0 11256 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  < 
A )
14 xmulpnf1 11491 . . . . . . . 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 4466 . . . . 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 11490 . . . . . 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 27720 . . . . 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 27720 . . . . 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 6283 . 2  |-  ( A  e.  RR+  ->  ( iota_ x  e.  RR*  ( A xe x )  = +oo )  = +oo )
277, 26eqtrd 2498 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 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   iota_crio 6257  (class class class)co 6296   RRcr 9508   0cc0 9509   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646   RR+crp 11245   xecxmu 11342   /𝑒 cxdiv 27765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-rp 11246  df-xneg 11343  df-xmul 11345  df-xdiv 27766
This theorem is referenced by:  xrpxdivcld  27783
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