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Theorem xrge0npcan 24046
Description: Extended non-negative real version of npcan 9247. (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 10926 . . . . . . . . 9  |-  ( 0 [,]  +oo )  C_  RR*
2 simpl1 960 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  A  e.  ( 0 [,]  +oo ) )
31, 2sseldi 3290 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  A  e.  RR* )
4 simpr 448 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  B  =  +oo )
5 simpl3 962 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  B  <_  A )
64, 5eqbrtrrd 4176 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  +oo  <_  A )
7 xgtpnf 24016 . . . . . . . . 9  |-  ( A  e.  RR*  ->  (  +oo  <_  A  <->  A  =  +oo ) )
87biimpa 471 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  +oo  <_  A )  ->  A  =  +oo )
93, 6, 8syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  A  =  +oo )
10 xnegeq 10726 . . . . . . . 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 6039 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( A + e  - e B )  =  (  +oo + e  - e  +oo ) )
13 pnfxr 10646 . . . . . . 7  |-  +oo  e.  RR*
14 xnegid 10755 . . . . . . 7  |-  (  +oo  e.  RR*  ->  (  +oo + e  - e  +oo )  =  0 )
1513, 14ax-mp 8 . . . . . 6  |-  (  +oo + e  - e  +oo )  =  0
1612, 15syl6eq 2436 . . . . 5  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( A + e  - e B )  =  0 )
1716oveq1d 6036 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( ( A + e  - e B ) + e B )  =  ( 0 + e B ) )
184oveq2d 6037 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( 0 + e B )  =  ( 0 + e  +oo ) )
19 xaddid2 10759 . . . . 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 2424 . . 3  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( ( A + e  - e B ) + e B )  =  +oo )
2221, 9eqtr4d 2423 . 2  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( ( A + e  - e B ) + e B )  =  A )
23 simpl1 960 . . . . 5  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  A  e.  ( 0 [,]  +oo )
)
241, 23sseldi 3290 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  A  e.  RR* )
25 xrge0neqmnf 24042 . . . . 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 961 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  B  e.  ( 0 [,]  +oo )
)
281, 27sseldi 3290 . . . . 5  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  B  e.  RR* )
2928xnegcld 10812 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  - e B  e. 
RR* )
30 simpr 448 . . . . 5  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  -.  B  =  +oo )
31 xnegneg 10733 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  - e  - e B  =  B )
32 xnegeq 10726 . . . . . . . . . 10  |-  (  - e B  =  -oo  -> 
- e  - e B  =  - e  -oo )
3331, 32sylan9req 2441 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  - e B  =  -oo )  ->  B  =  - e  -oo )
34 xnegmnf 10729 . . . . . . . . 9  |-  - e  -oo  =  +oo
3533, 34syl6eq 2436 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  - e B  =  -oo )  ->  B  =  +oo )
3635ex 424 . . . . . . 7  |-  ( B  e.  RR*  ->  (  - e B  =  -oo  ->  B  =  +oo )
)
3736con3and 429 . . . . . 6  |-  ( ( B  e.  RR*  /\  -.  B  =  +oo )  ->  -.  - e B  = 
-oo )
3837neneqad 2621 . . . . 5  |-  ( ( B  e.  RR*  /\  -.  B  =  +oo )  ->  - e B  =/=  -oo )
3928, 30, 38syl2anc 643 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  - e B  =/= 
-oo )
40 xrge0neqmnf 24042 . . . . 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 10761 . . . 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 1200 . . 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 10732 . . . . . . . 8  |-  ( B  e.  RR*  ->  - e B  e.  RR* )
45 xaddcom 10757 . . . . . . . 8  |-  ( ( 
- e B  e. 
RR*  /\  B  e.  RR* )  ->  (  - e B + e B )  =  ( B + e  - e B ) )
4644, 45mpancom 651 . . . . . . 7  |-  ( B  e.  RR*  ->  (  - e B + e B )  =  ( B + e  - e B ) )
47 xnegid 10755 . . . . . . 7  |-  ( B  e.  RR*  ->  ( B + e  - e B )  =  0 )
4846, 47eqtrd 2420 . . . . . 6  |-  ( B  e.  RR*  ->  (  - e B + e B )  =  0 )
4948oveq2d 6037 . . . . 5  |-  ( B  e.  RR*  ->  ( A + e (  - e B + e B ) )  =  ( A + e 0 ) )
50 xaddid1 10758 . . . . 5  |-  ( A  e.  RR*  ->  ( A + e 0 )  =  A )
5149, 50sylan9eqr 2442 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A + e (  - e B + e B ) )  =  A )
5224, 28, 51syl2anc 643 . . 3  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  ( A + e (  - e B + e B ) )  =  A )
5343, 52eqtrd 2420 . 2  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  ( ( A + e  - e B ) + e B )  =  A )
5422, 53pm2.61dan 767 1  |-  ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  B  <_  A
)  ->  ( ( A + e  - e B ) + e B )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   class class class wbr 4154  (class class class)co 6021   0cc0 8924    +oocpnf 9051    -oocmnf 9052   RR*cxr 9053    <_ cle 9055    - ecxne 10640   + ecxad 10641   [,]cicc 10852
This theorem is referenced by:  esumle  24246  esumlef  24251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-xneg 10643  df-xadd 10644  df-icc 10856
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