Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrge0addgt0 Structured version   Unicode version

Theorem xrge0addgt0 27505
Description: The sum of nonnegative and positive numbers is positive. See addgtge0 10052 (Contributed by Thierry Arnoux, 6-Jul-2017.)
Assertion
Ref Expression
xrge0addgt0  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  0  <  ( A +e B ) )

Proof of Theorem xrge0addgt0
StepHypRef Expression
1 0xr 9652 . . . 4  |-  0  e.  RR*
2 xaddid1 11450 . . . 4  |-  ( 0  e.  RR*  ->  ( 0 +e 0 )  =  0 )
31, 2ax-mp 5 . . 3  |-  ( 0 +e 0 )  =  0
4 simplr 754 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  0  <  A )
5 simpr 461 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  0  <  B )
61a1i 11 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  0  e.  RR* )
7 iccssxr 11619 . . . . . 6  |-  ( 0 [,] +oo )  C_  RR*
8 simplll 757 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  A  e.  ( 0 [,] +oo ) )
97, 8sseldi 3507 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  A  e.  RR* )
10 simpllr 758 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  B  e.  ( 0 [,] +oo ) )
117, 10sseldi 3507 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  B  e.  RR* )
12 xlt2add 11464 . . . . 5  |-  ( ( ( 0  e.  RR*  /\  0  e.  RR* )  /\  ( A  e.  RR*  /\  B  e.  RR* )
)  ->  ( (
0  <  A  /\  0  <  B )  -> 
( 0 +e 0 )  <  ( A +e B ) ) )
136, 6, 9, 11, 12syl22anc 1229 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  (
( 0  <  A  /\  0  <  B )  ->  ( 0 +e 0 )  < 
( A +e
B ) ) )
144, 5, 13mp2and 679 . . 3  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  (
0 +e 0 )  <  ( A +e B ) )
153, 14syl5eqbrr 4487 . 2  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  0  <  ( A +e
B ) )
16 simplr 754 . . 3  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  0  <  A )
17 oveq2 6303 . . . . . 6  |-  ( 0  =  B  ->  ( A +e 0 )  =  ( A +e B ) )
1817adantl 466 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  ( A +e 0 )  =  ( A +e B ) )
1918breq2d 4465 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  (
0  <  ( A +e 0 )  <->  0  <  ( A +e B ) ) )
20 simplll 757 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  A  e.  ( 0 [,] +oo ) )
217, 20sseldi 3507 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  A  e.  RR* )
22 xaddid1 11450 . . . . . 6  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
2321, 22syl 16 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  ( A +e 0 )  =  A )
2423breq2d 4465 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  (
0  <  ( A +e 0 )  <->  0  <  A ) )
2519, 24bitr3d 255 . . 3  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  (
0  <  ( A +e B )  <->  0  <  A ) )
2616, 25mpbird 232 . 2  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  0  <  ( A +e
B ) )
271a1i 11 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  0  e.  RR* )
28 simplr 754 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  B  e.  ( 0 [,] +oo ) )
297, 28sseldi 3507 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  B  e.  RR* )
30 pnfxr 11333 . . . . 5  |- +oo  e.  RR*
3130a1i 11 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  -> +oo  e.  RR* )
32 iccgelb 11593 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  B  e.  ( 0 [,] +oo ) )  ->  0  <_  B )
3327, 31, 28, 32syl3anc 1228 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  0  <_  B )
34 xrleloe 11362 . . . 4  |-  ( ( 0  e.  RR*  /\  B  e.  RR* )  ->  (
0  <_  B  <->  ( 0  <  B  \/  0  =  B ) ) )
3534biimpa 484 . . 3  |-  ( ( ( 0  e.  RR*  /\  B  e.  RR* )  /\  0  <_  B )  ->  ( 0  < 
B  \/  0  =  B ) )
3627, 29, 33, 35syl21anc 1227 . 2  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  ( 0  <  B  \/  0  =  B ) )
3715, 26, 36mpjaodan 784 1  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  0  <  ( A +e B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4453  (class class class)co 6295   0cc0 9504   +oocpnf 9637   RR*cxr 9639    < clt 9640    <_ cle 9641   +ecxad 11328   [,]cicc 11544
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-xneg 11330  df-xadd 11331  df-icc 11548
This theorem is referenced by:  xrge0adddir  27506
  Copyright terms: Public domain W3C validator