Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrge0npcan Structured version   Unicode version

Theorem xrge0npcan 24208
Description: Extended non-negative real version of npcan 9306. (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 10985 . . . . . . . . 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 3338 . . . . . . . 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 4226 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  ->  +oo  <_  A )
7 xgepnf 24108 . . . . . . . . 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 10785 . . . . . . . 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 6091 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( A + e  - e B )  =  (  +oo + e  - e  +oo ) )
13 pnfxr 10705 . . . . . . 7  |-  +oo  e.  RR*
14 xnegid 10814 . . . . . . 7  |-  (  +oo  e.  RR*  ->  (  +oo + e  - e  +oo )  =  0 )
1513, 14ax-mp 8 . . . . . 6  |-  (  +oo + e  - e  +oo )  =  0
1612, 15syl6eq 2483 . . . . 5  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( A + e  - e B )  =  0 )
1716oveq1d 6088 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( ( A + e  - e B ) + e B )  =  ( 0 + e B ) )
184oveq2d 6089 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( 0 + e B )  =  ( 0 + e  +oo ) )
19 xaddid2 10818 . . . . 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 2471 . . 3  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  B  =  +oo )  -> 
( ( A + e  - e B ) + e B )  =  +oo )
2221, 9eqtr4d 2470 . 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 3338 . . . 4  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  A  e.  RR* )
25 xrge0neqmnf 24204 . . . . 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 3338 . . . . 5  |-  ( ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
0 [,]  +oo )  /\  B  <_  A )  /\  -.  B  =  +oo )  ->  B  e.  RR* )
2928xnegcld 10871 . . . 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 10792 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  - e  - e B  =  B )
32 xnegeq 10785 . . . . . . . . . 10  |-  (  - e B  =  -oo  -> 
- e  - e B  =  - e  -oo )
3331, 32sylan9req 2488 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  - e B  =  -oo )  ->  B  =  - e  -oo )
34 xnegmnf 10788 . . . . . . . . 9  |-  - e  -oo  =  +oo
3533, 34syl6eq 2483 . . . . . . . 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 2668 . . . . 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 24204 . . . . 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 10820 . . . 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 10791 . . . . . . . 8  |-  ( B  e.  RR*  ->  - e B  e.  RR* )
45 xaddcom 10816 . . . . . . . 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 10814 . . . . . . 7  |-  ( B  e.  RR*  ->  ( B + e  - e B )  =  0 )
4846, 47eqtrd 2467 . . . . . 6  |-  ( B  e.  RR*  ->  (  - e B + e B )  =  0 )
4948oveq2d 6089 . . . . 5  |-  ( B  e.  RR*  ->  ( A + e (  - e B + e B ) )  =  ( A + e 0 ) )
50 xaddid1 10817 . . . . 5  |-  ( A  e.  RR*  ->  ( A + e 0 )  =  A )
5149, 50sylan9eqr 2489 . . . 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 2467 . 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 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204  (class class class)co 6073   0cc0 8982    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    <_ cle 9113    - ecxne 10699   + ecxad 10700   [,]cicc 10911
This theorem is referenced by:  esumle  24441  esumlef  24446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-xneg 10702  df-xadd 10703  df-icc 10915
  Copyright terms: Public domain W3C validator