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Theorem xrge0npcan 28122
Description: Extended nonnegative real version of npcan 9864. (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 11659 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
2 simpl1 1000 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  A  e.  ( 0 [,] +oo )
)
31, 2sseldi 3439 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  A  e.  RR* )
4 simpr 459 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  B  = +oo )
5 simpl3 1002 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  B  <_  A
)
64, 5eqbrtrrd 4416 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  -> +oo  <_  A )
7 xgepnf 27997 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( +oo  <_  A  <->  A  = +oo ) )
87biimpa 482 . . . . . . . 8  |-  ( ( A  e.  RR*  /\ +oo  <_  A )  ->  A  = +oo )
93, 6, 8syl2anc 659 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  A  = +oo )
10 xnegeq 11458 . . . . . . . 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 6295 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( A +e  -e B )  =  ( +oo +e  -e +oo )
)
13 pnfxr 11373 . . . . . . 7  |- +oo  e.  RR*
14 xnegid 11487 . . . . . . 7  |-  ( +oo  e.  RR*  ->  ( +oo +e  -e +oo )  =  0 )
1513, 14ax-mp 5 . . . . . 6  |-  ( +oo +e  -e +oo )  =  0
1612, 15syl6eq 2459 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( A +e  -e B )  =  0 )
1716oveq1d 6292 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  =  ( 0 +e B ) )
184oveq2d 6293 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( 0 +e B )  =  ( 0 +e +oo ) )
19 xaddid2 11491 . . . . 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 2447 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  = +oo )
2221, 9eqtr4d 2446 . 2  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  =  A )
23 simpl1 1000 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  A  e.  ( 0 [,] +oo ) )
241, 23sseldi 3439 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  A  e.  RR* )
25 xrge0neqmnf 28117 . . . . 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 1001 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  e.  ( 0 [,] +oo ) )
281, 27sseldi 3439 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  e.  RR* )
2928xnegcld 11544 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  -e
B  e.  RR* )
30 simpr 459 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  -.  B  = +oo )
31 xnegneg 11465 . . . . . . . . 9  |-  ( B  e.  RR*  ->  -e  -e B  =  B )
32 xnegeq 11458 . . . . . . . . 9  |-  (  -e B  = -oo  -> 
-e  -e
B  =  -e -oo )
3331, 32sylan9req 2464 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  -e
B  = -oo )  ->  B  =  -e -oo )
34 xnegmnf 11461 . . . . . . . 8  |-  -e -oo  = +oo
3533, 34syl6eq 2459 . . . . . . 7  |-  ( ( B  e.  RR*  /\  -e
B  = -oo )  ->  B  = +oo )
3635stoic1a 1625 . . . . . 6  |-  ( ( B  e.  RR*  /\  -.  B  = +oo )  ->  -.  -e B  = -oo )
3736neqned 2606 . . . . 5  |-  ( ( B  e.  RR*  /\  -.  B  = +oo )  -> 
-e B  =/= -oo )
3828, 30, 37syl2anc 659 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  -e
B  =/= -oo )
39 xrge0neqmnf 28117 . . . . 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 11493 . . . 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 1246 . . 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 11464 . . . . . . . 8  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
44 xaddcom 11489 . . . . . . . 8  |-  ( ( 
-e B  e. 
RR*  /\  B  e.  RR* )  ->  (  -e
B +e B )  =  ( B +e  -e
B ) )
4543, 44mpancom 667 . . . . . . 7  |-  ( B  e.  RR*  ->  (  -e B +e
B )  =  ( B +e  -e B ) )
46 xnegid 11487 . . . . . . 7  |-  ( B  e.  RR*  ->  ( B +e  -e
B )  =  0 )
4745, 46eqtrd 2443 . . . . . 6  |-  ( B  e.  RR*  ->  (  -e B +e
B )  =  0 )
4847oveq2d 6293 . . . . 5  |-  ( B  e.  RR*  ->  ( A +e (  -e B +e
B ) )  =  ( A +e 0 ) )
49 xaddid1 11490 . . . . 5  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
5048, 49sylan9eqr 2465 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e (  -e B +e
B ) )  =  A )
5124, 28, 50syl2anc 659 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  ( A +e (  -e B +e
B ) )  =  A )
5242, 51eqtrd 2443 . 2  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  (
( A +e  -e B ) +e B )  =  A )
5322, 52pm2.61dan 792 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394  (class class class)co 6277   0cc0 9521   +oocpnf 9654   -oocmnf 9655   RR*cxr 9656    <_ cle 9658    -ecxne 11367   +ecxad 11368   [,]cicc 11584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-xneg 11370  df-xadd 11371  df-icc 11588
This theorem is referenced by:  esumle  28491  esumlef  28495  carsgclctunlem2  28753
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