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Theorem xrge0npcan 28531
Description: Extended nonnegative real version of npcan 9904. (Contributed by Thierry Arnoux, 9-Jun-2017.)
Assertion
Ref Expression
xrge0npcan  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  B  <_  A )  -> 
( ( A +e  -e B ) +e B )  =  A )

Proof of Theorem xrge0npcan
StepHypRef Expression
1 iccssxr 11742 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
2 simpl1 1033 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  A  e.  ( 0 [,] +oo )
)
31, 2sseldi 3416 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  A  e.  RR* )
4 simpr 468 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  B  = +oo )
5 simpl3 1035 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  B  <_  A
)
64, 5eqbrtrrd 4418 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  -> +oo  <_  A )
7 xgepnf 28402 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( +oo  <_  A  <->  A  = +oo ) )
87biimpa 492 . . . . . . . 8  |-  ( ( A  e.  RR*  /\ +oo  <_  A )  ->  A  = +oo )
93, 6, 8syl2anc 673 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  A  = +oo )
10 xnegeq 11523 . . . . . . . 8  |-  ( B  = +oo  ->  -e
B  =  -e +oo )
114, 10syl 17 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  -e B  = 
-e +oo )
129, 11oveq12d 6326 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( A +e  -e B )  =  ( +oo +e  -e +oo )
)
13 pnfxr 11435 . . . . . . 7  |- +oo  e.  RR*
14 xnegid 11553 . . . . . . 7  |-  ( +oo  e.  RR*  ->  ( +oo +e  -e +oo )  =  0 )
1513, 14ax-mp 5 . . . . . 6  |-  ( +oo +e  -e +oo )  =  0
1612, 15syl6eq 2521 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( A +e  -e B )  =  0 )
1716oveq1d 6323 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  =  ( 0 +e B ) )
184oveq2d 6324 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( 0 +e B )  =  ( 0 +e +oo ) )
19 xaddid2 11557 . . . . 5  |-  ( +oo  e.  RR*  ->  ( 0 +e +oo )  = +oo )
2013, 19mp1i 13 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( 0 +e +oo )  = +oo )
2117, 18, 203eqtrd 2509 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  = +oo )
2221, 9eqtr4d 2508 . 2  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  =  A )
23 simpl1 1033 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  A  e.  ( 0 [,] +oo ) )
241, 23sseldi 3416 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  A  e.  RR* )
25 xrge0neqmnf 11762 . . . . 5  |-  ( A  e.  ( 0 [,] +oo )  ->  A  =/= -oo )
2623, 25syl 17 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  A  =/= -oo )
27 simpl2 1034 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  e.  ( 0 [,] +oo ) )
281, 27sseldi 3416 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  e.  RR* )
2928xnegcld 11611 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  -e
B  e.  RR* )
30 simpr 468 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  -.  B  = +oo )
31 xnegneg 11530 . . . . . . . . 9  |-  ( B  e.  RR*  ->  -e  -e B  =  B )
32 xnegeq 11523 . . . . . . . . 9  |-  (  -e B  = -oo  -> 
-e  -e
B  =  -e -oo )
3331, 32sylan9req 2526 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  -e
B  = -oo )  ->  B  =  -e -oo )
34 xnegmnf 11526 . . . . . . . 8  |-  -e -oo  = +oo
3533, 34syl6eq 2521 . . . . . . 7  |-  ( ( B  e.  RR*  /\  -e
B  = -oo )  ->  B  = +oo )
3635stoic1a 1663 . . . . . 6  |-  ( ( B  e.  RR*  /\  -.  B  = +oo )  ->  -.  -e B  = -oo )
3736neqned 2650 . . . . 5  |-  ( ( B  e.  RR*  /\  -.  B  = +oo )  -> 
-e B  =/= -oo )
3828, 30, 37syl2anc 673 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  -e
B  =/= -oo )
39 xrge0neqmnf 11762 . . . . 5  |-  ( B  e.  ( 0 [,] +oo )  ->  B  =/= -oo )
4027, 39syl 17 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  =/= -oo )
41 xaddass 11560 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  (  -e B  e.  RR*  /\  -e
B  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )
)  ->  ( ( A +e  -e
B ) +e
B )  =  ( A +e ( 
-e B +e B ) ) )
4224, 26, 29, 38, 28, 40, 41syl222anc 1308 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  (
( A +e  -e B ) +e B )  =  ( A +e
(  -e B +e B ) ) )
43 xnegcl 11529 . . . . . . . 8  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
44 xaddcom 11555 . . . . . . . 8  |-  ( ( 
-e B  e. 
RR*  /\  B  e.  RR* )  ->  (  -e
B +e B )  =  ( B +e  -e
B ) )
4543, 44mpancom 682 . . . . . . 7  |-  ( B  e.  RR*  ->  (  -e B +e
B )  =  ( B +e  -e B ) )
46 xnegid 11553 . . . . . . 7  |-  ( B  e.  RR*  ->  ( B +e  -e
B )  =  0 )
4745, 46eqtrd 2505 . . . . . 6  |-  ( B  e.  RR*  ->  (  -e B +e
B )  =  0 )
4847oveq2d 6324 . . . . 5  |-  ( B  e.  RR*  ->  ( A +e (  -e B +e
B ) )  =  ( A +e 0 ) )
49 xaddid1 11556 . . . . 5  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
5048, 49sylan9eqr 2527 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e (  -e B +e
B ) )  =  A )
5124, 28, 50syl2anc 673 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  ( A +e (  -e B +e
B ) )  =  A )
5242, 51eqtrd 2505 . 2  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  (
( A +e  -e B ) +e B )  =  A )
5322, 52pm2.61dan 808 1  |-  ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  ( 0 [,] +oo )  /\  B  <_  A )  -> 
( ( A +e  -e B ) +e B )  =  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395  (class class class)co 6308   0cc0 9557   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    <_ cle 9694    -ecxne 11429   +ecxad 11430   [,]cicc 11663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-xneg 11432  df-xadd 11433  df-icc 11667
This theorem is referenced by:  esumle  28953  esumlef  28957  carsgclctunlem2  29224
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