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Theorem xrge0npcan 27662
Description: Extended nonnegative real version of npcan 9834. (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 11618 . . . . . . . . 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 3487 . . . . . . . 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 1002 . . . . . . . . 9  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  B  <_  A
)
64, 5eqbrtrrd 4459 . . . . . . . 8  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  -> +oo  <_  A )
7 xgepnf 27548 . . . . . . . . 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 11417 . . . . . . . 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 6299 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( A +e  -e B )  =  ( +oo +e  -e +oo )
)
13 pnfxr 11332 . . . . . . 7  |- +oo  e.  RR*
14 xnegid 11446 . . . . . . 7  |-  ( +oo  e.  RR*  ->  ( +oo +e  -e +oo )  =  0 )
1513, 14ax-mp 5 . . . . . 6  |-  ( +oo +e  -e +oo )  =  0
1612, 15syl6eq 2500 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( A +e  -e B )  =  0 )
1716oveq1d 6296 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  =  ( 0 +e B ) )
184oveq2d 6297 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( 0 +e B )  =  ( 0 +e +oo ) )
19 xaddid2 11450 . . . . 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 2488 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  B  = +oo )  ->  ( ( A +e  -e
B ) +e
B )  = +oo )
2221, 9eqtr4d 2487 . 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 3487 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  A  e.  RR* )
25 xrge0neqmnf 27657 . . . . 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 1001 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  e.  ( 0 [,] +oo ) )
281, 27sseldi 3487 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  e.  RR* )
2928xnegcld 11503 . . . 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 11424 . . . . . . . . 9  |-  ( B  e.  RR*  ->  -e  -e B  =  B )
32 xnegeq 11417 . . . . . . . . 9  |-  (  -e B  = -oo  -> 
-e  -e
B  =  -e -oo )
3331, 32sylan9req 2505 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  -e
B  = -oo )  ->  B  =  -e -oo )
34 xnegmnf 11420 . . . . . . . 8  |-  -e -oo  = +oo
3533, 34syl6eq 2500 . . . . . . 7  |-  ( ( B  e.  RR*  /\  -e
B  = -oo )  ->  B  = +oo )
3635stoic1a 1592 . . . . . 6  |-  ( ( B  e.  RR*  /\  -.  B  = +oo )  ->  -.  -e B  = -oo )
3736neqned 2646 . . . . 5  |-  ( ( B  e.  RR*  /\  -.  B  = +oo )  -> 
-e B  =/= -oo )
3828, 30, 37syl2anc 661 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  -e
B  =/= -oo )
39 xrge0neqmnf 27657 . . . . 5  |-  ( B  e.  ( 0 [,] +oo )  ->  B  =/= -oo )
4027, 39syl 16 . . . 4  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  B  =/= -oo )
41 xaddass 11452 . . . 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 1245 . . 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 11423 . . . . . . . 8  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
44 xaddcom 11448 . . . . . . . 8  |-  ( ( 
-e B  e. 
RR*  /\  B  e.  RR* )  ->  (  -e
B +e B )  =  ( B +e  -e
B ) )
4543, 44mpancom 669 . . . . . . 7  |-  ( B  e.  RR*  ->  (  -e B +e
B )  =  ( B +e  -e B ) )
46 xnegid 11446 . . . . . . 7  |-  ( B  e.  RR*  ->  ( B +e  -e
B )  =  0 )
4745, 46eqtrd 2484 . . . . . 6  |-  ( B  e.  RR*  ->  (  -e B +e
B )  =  0 )
4847oveq2d 6297 . . . . 5  |-  ( B  e.  RR*  ->  ( A +e (  -e B +e
B ) )  =  ( A +e 0 ) )
49 xaddid1 11449 . . . . 5  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
5048, 49sylan9eqr 2506 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e (  -e B +e
B ) )  =  A )
5124, 28, 50syl2anc 661 . . 3  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  ( A +e (  -e B +e
B ) )  =  A )
5242, 51eqtrd 2484 . 2  |-  ( ( ( A  e.  ( 0 [,] +oo )  /\  B  e.  (
0 [,] +oo )  /\  B  <_  A )  /\  -.  B  = +oo )  ->  (
( A +e  -e B ) +e B )  =  A )
5322, 52pm2.61dan 791 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   class class class wbr 4437  (class class class)co 6281   0cc0 9495   +oocpnf 9628   -oocmnf 9629   RR*cxr 9630    <_ cle 9632    -ecxne 11326   +ecxad 11327   [,]cicc 11543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-xneg 11329  df-xadd 11330  df-icc 11547
This theorem is referenced by:  esumle  28043  esumlef  28048
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