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Theorem xrge0npcan 27332
Description: Extended nonnegative real version of npcan 9818. (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 11596 . . . . . . . . 9  |-  ( 0 [,] +oo )  C_  RR*
2 simpl1 994 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  A  e.  ( 0 [,] +oo )
)
31, 2sseldi 3495 . . . . . . . 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 996 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  B  <_  A
)
64, 5eqbrtrrd 4462 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  -> +oo  <_  A )
7 xgepnf 27224 . . . . . . . . 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 11395 . . . . . . . 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 6293 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( A +e  -e B )  =  ( +oo +e  -e +oo )
)
13 pnfxr 11310 . . . . . . 7  |- +oo  e.  RR*
14 xnegid 11424 . . . . . . 7  |-  ( +oo  e.  RR*  ->  ( +oo +e  -e +oo )  =  0 )
1513, 14ax-mp 5 . . . . . 6  |-  ( +oo +e  -e +oo )  =  0
1612, 15syl6eq 2517 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( A +e  -e B )  =  0 )
1716oveq1d 6290 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  =  ( 0 +e B ) )
184oveq2d 6291 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( 0 +e B )  =  ( 0 +e +oo ) )
19 xaddid2 11428 . . . . 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 2505 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  = +oo )
2221, 9eqtr4d 2504 . 2  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  =  A )
23 simpl1 994 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  A  e.  ( 0 [,] +oo ) )
241, 23sseldi 3495 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  A  e.  RR* )
25 xrge0neqmnf 27327 . . . . 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 995 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  e.  ( 0 [,] +oo ) )
281, 27sseldi 3495 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  e.  RR* )
2928xnegcld 11481 . . . 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 11402 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  -e  -e B  =  B )
32 xnegeq 11395 . . . . . . . . . 10  |-  (  -e B  = -oo  -> 
-e  -e
B  =  -e -oo )
3331, 32sylan9req 2522 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  -e
B  = -oo )  ->  B  =  -e -oo )
34 xnegmnf 11398 . . . . . . . . 9  |-  -e -oo  = +oo
3533, 34syl6eq 2517 . . . . . . . 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 )
3837neqned 2663 . . . . 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 27327 . . . . 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 11430 . . . 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 1239 . . 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 11401 . . . . . . . 8  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
45 xaddcom 11426 . . . . . . . 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 11424 . . . . . . 7  |-  ( B  e.  RR*  ->  ( B +e  -e
B )  =  0 )
4846, 47eqtrd 2501 . . . . . 6  |-  ( B  e.  RR*  ->  (  -e B +e
B )  =  0 )
4948oveq2d 6291 . . . . 5  |-  ( B  e.  RR*  ->  ( A +e (  -e B +e
B ) )  =  ( A +e 0 ) )
50 xaddid1 11427 . . . . 5  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
5149, 50sylan9eqr 2523 . . . 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 2501 . 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440  (class class class)co 6275   0cc0 9481   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616    <_ cle 9618    -ecxne 11304   +ecxad 11305   [,]cicc 11521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-xneg 11307  df-xadd 11308  df-icc 11525
This theorem is referenced by:  esumle  27691  esumlef  27696
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