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Theorem xrge0addgt0 24167
Description: The sum of nonnegative and positive numbers is positive. See addgtge0 9472 (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 9087 . . . 4  |-  0  e.  RR*
2 xaddid1 10781 . . . 4  |-  ( 0  e.  RR*  ->  ( 0 + e 0 )  =  0 )
31, 2ax-mp 8 . . 3  |-  ( 0 + e 0 )  =  0
4 simplr 732 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  0  <  A )
5 simpr 448 . . . 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 10949 . . . . . 6  |-  ( 0 [,]  +oo )  C_  RR*
8 simplll 735 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  A  e.  ( 0 [,]  +oo ) )
97, 8sseldi 3306 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  A  e.  RR* )
10 simpllr 736 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  B  e.  ( 0 [,]  +oo ) )
117, 10sseldi 3306 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  B  e.  RR* )
12 xlt2add 10795 . . . . 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 1185 . . . 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 661 . . 3  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  (
0 + e 0 )  <  ( A + e B ) )
153, 14syl5eqbrr 4206 . 2  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  0  <  ( A + e B ) )
16 simplr 732 . . 3  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  0  <  A )
17 oveq2 6048 . . . . . 6  |-  ( 0  =  B  ->  ( A + e 0 )  =  ( A + e B ) )
1817adantl 453 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  ( A + e 0 )  =  ( A + e B ) )
1918breq2d 4184 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  (
0  <  ( A + e 0 )  <->  0  <  ( A + e B ) ) )
20 simplll 735 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  A  e.  ( 0 [,]  +oo ) )
217, 20sseldi 3306 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  A  e.  RR* )
22 xaddid1 10781 . . . . . 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 4184 . . . 4  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  (
0  <  ( A + e 0 )  <->  0  <  A ) )
2519, 24bitr3d 247 . . 3  |-  ( ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  (
0  <  ( A + e B )  <->  0  <  A ) )
2616, 25mpbird 224 . 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 732 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo ) )  /\  0  <  A
)  ->  B  e.  ( 0 [,]  +oo ) )
297, 28sseldi 3306 . . 3  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo ) )  /\  0  <  A
)  ->  B  e.  RR* )
30 pnfxr 10669 . . . . 5  |-  +oo  e.  RR*
3130a1i 11 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo ) )  /\  0  <  A
)  ->  +oo  e.  RR* )
32 iccgelb 24089 . . . 4  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR*  /\  B  e.  ( 0 [,]  +oo ) )  ->  0  <_  B )
3327, 31, 28, 32syl3anc 1184 . . 3  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo ) )  /\  0  <  A
)  ->  0  <_  B )
34 xrleloe 10693 . . . 4  |-  ( ( 0  e.  RR*  /\  B  e.  RR* )  ->  (
0  <_  B  <->  ( 0  <  B  \/  0  =  B ) ) )
3534biimpa 471 . . 3  |-  ( ( ( 0  e.  RR*  /\  B  e.  RR* )  /\  0  <_  B )  ->  ( 0  < 
B  \/  0  =  B ) )
3627, 29, 33, 35syl21anc 1183 . 2  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo ) )  /\  0  <  A
)  ->  ( 0  <  B  \/  0  =  B ) )
3715, 26, 36mpjaodan 762 1  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo ) )  /\  0  <  A
)  ->  0  <  ( A + e B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4172  (class class class)co 6040   0cc0 8946    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077   + ecxad 10664   [,]cicc 10875
This theorem is referenced by:  xrge0adddir  24168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-xneg 10666  df-xadd 10667  df-icc 10879
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