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Theorem xrge0addgt0 26159
Description: The sum of nonnegative and positive numbers is positive. See addgtge0 9832 (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 9435 . . . 4  |-  0  e.  RR*
2 xaddid1 11214 . . . 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 11383 . . . . . 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 3359 . . . . 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 3359 . . . . 5  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  < 
B )  ->  B  e.  RR* )
12 xlt2add 11228 . . . . 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 1219 . . . 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 4331 . 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 6104 . . . . . 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 4309 . . . 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 3359 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )
)  /\  0  <  A )  /\  0  =  B )  ->  A  e.  RR* )
22 xaddid1 11214 . . . . . 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 4309 . . . 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 3359 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  B  e.  RR* )
30 pnfxr 11097 . . . . 5  |- +oo  e.  RR*
3130a1i 11 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  -> +oo  e.  RR* )
32 iccgelb 11357 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  B  e.  ( 0 [,] +oo ) )  ->  0  <_  B )
3327, 31, 28, 32syl3anc 1218 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )
)  /\  0  <  A )  ->  0  <_  B )
34 xrleloe 11126 . . . 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 1217 . 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 1369    e. wcel 1756   class class class wbr 4297  (class class class)co 6096   0cc0 9287   +oocpnf 9420   RR*cxr 9422    < clt 9423    <_ cle 9424   +ecxad 11092   [,]cicc 11308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-xneg 11094  df-xadd 11095  df-icc 11312
This theorem is referenced by:  xrge0adddir  26160
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