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Theorem xrge0npcan 26155
Description: Extended nonnegative real version of npcan 9617. (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 11376 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
2 simpl1 991 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  A  e.  ( 0 [,] +oo )
)
31, 2sseldi 3352 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  A  e.  RR* )
4 simpr 461 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  B  = +oo )
5 simpl3 993 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  B  <_  A
)
64, 5eqbrtrrd 4312 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  -> +oo  <_  A )
7 xgepnf 26041 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( +oo  <_  A  <->  A  = +oo ) )
87biimpa 484 . . . . . . . 8  |-  ( ( A  e.  RR*  /\ +oo  <_  A )  ->  A  = +oo )
93, 6, 8syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  A  = +oo )
10 xnegeq 11175 . . . . . . . 8  |-  ( B  = +oo  ->  -e
B  =  -e +oo )
114, 10syl 16 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  -e B  = 
-e +oo )
129, 11oveq12d 6107 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( A +e  -e B )  =  ( +oo +e  -e +oo )
)
13 pnfxr 11090 . . . . . . 7  |- +oo  e.  RR*
14 xnegid 11204 . . . . . . 7  |-  ( +oo  e.  RR*  ->  ( +oo +e  -e +oo )  =  0 )
1513, 14ax-mp 5 . . . . . 6  |-  ( +oo +e  -e +oo )  =  0
1612, 15syl6eq 2489 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( A +e  -e B )  =  0 )
1716oveq1d 6104 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  =  ( 0 +e B ) )
184oveq2d 6105 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( 0 +e B )  =  ( 0 +e +oo ) )
19 xaddid2 11208 . . . . 5  |-  ( +oo  e.  RR*  ->  ( 0 +e +oo )  = +oo )
2013, 19mp1i 12 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( 0 +e +oo )  = +oo )
2117, 18, 203eqtrd 2477 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  = +oo )
2221, 9eqtr4d 2476 . 2  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  =  A )
23 simpl1 991 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  A  e.  ( 0 [,] +oo ) )
241, 23sseldi 3352 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  A  e.  RR* )
25 xrge0neqmnf 26150 . . . . 5  |-  ( A  e.  ( 0 [,] +oo )  ->  A  =/= -oo )
2623, 25syl 16 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  A  =/= -oo )
27 simpl2 992 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  e.  ( 0 [,] +oo ) )
281, 27sseldi 3352 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  e.  RR* )
2928xnegcld 11261 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  -e
B  e.  RR* )
30 simpr 461 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  -.  B  = +oo )
31 xnegneg 11182 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  -e  -e B  =  B )
32 xnegeq 11175 . . . . . . . . . 10  |-  (  -e B  = -oo  -> 
-e  -e
B  =  -e -oo )
3331, 32sylan9req 2494 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  -e
B  = -oo )  ->  B  =  -e -oo )
34 xnegmnf 11178 . . . . . . . . 9  |-  -e -oo  = +oo
3533, 34syl6eq 2489 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  -e
B  = -oo )  ->  B  = +oo )
3635ex 434 . . . . . . 7  |-  ( B  e.  RR*  ->  (  -e B  = -oo  ->  B  = +oo )
)
3736con3dimp 441 . . . . . 6  |-  ( ( B  e.  RR*  /\  -.  B  = +oo )  ->  -.  -e B  = -oo )
3837neneqad 2679 . . . . 5  |-  ( ( B  e.  RR*  /\  -.  B  = +oo )  -> 
-e B  =/= -oo )
3928, 30, 38syl2anc 661 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  -e
B  =/= -oo )
40 xrge0neqmnf 26150 . . . . 5  |-  ( B  e.  ( 0 [,] +oo )  ->  B  =/= -oo )
4127, 40syl 16 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  =/= -oo )
42 xaddass 11210 . . . 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 ) ) )
4324, 26, 29, 39, 28, 41, 42syl222anc 1234 . . 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 ) ) )
44 xnegcl 11181 . . . . . . . 8  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
45 xaddcom 11206 . . . . . . . 8  |-  ( ( 
-e B  e. 
RR*  /\  B  e.  RR* )  ->  (  -e
B +e B )  =  ( B +e  -e
B ) )
4644, 45mpancom 669 . . . . . . 7  |-  ( B  e.  RR*  ->  (  -e B +e
B )  =  ( B +e  -e B ) )
47 xnegid 11204 . . . . . . 7  |-  ( B  e.  RR*  ->  ( B +e  -e
B )  =  0 )
4846, 47eqtrd 2473 . . . . . 6  |-  ( B  e.  RR*  ->  (  -e B +e
B )  =  0 )
4948oveq2d 6105 . . . . 5  |-  ( B  e.  RR*  ->  ( A +e (  -e B +e
B ) )  =  ( A +e 0 ) )
50 xaddid1 11207 . . . . 5  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
5149, 50sylan9eqr 2495 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e (  -e B +e
B ) )  =  A )
5224, 28, 51syl2anc 661 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  ( A +e (  -e B +e
B ) )  =  A )
5343, 52eqtrd 2473 . 2  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  (
( A +e  -e B ) +e B )  =  A )
5422, 53pm2.61dan 789 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   class class class wbr 4290  (class class class)co 6089   0cc0 9280   +oocpnf 9413   -oocmnf 9414   RR*cxr 9415    <_ cle 9417    -ecxne 11084   +ecxad 11085   [,]cicc 11301
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-xneg 11087  df-xadd 11088  df-icc 11305
This theorem is referenced by:  esumle  26506  esumlef  26511
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