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Theorem xpncan 9541
Description: Extended real version of pncan 7885. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xpncan  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A +e
B ) +e  -e B )  =  A )

Proof of Theorem xpncan
StepHypRef Expression
1 rexneg 9500 . . . 4  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
21adantl 273 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  -e
B  =  -u B
)
32oveq2d 5742 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A +e
B ) +e  -e B )  =  ( ( A +e B ) +e -u B ) )
4 renegcl 7940 . . . . . 6  |-  ( B  e.  RR  ->  -u B  e.  RR )
54ad2antlr 478 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  -u B  e.  RR )
6 rexr 7729 . . . . . 6  |-  ( -u B  e.  RR  ->  -u B  e.  RR* )
7 renepnf 7731 . . . . . 6  |-  ( -u B  e.  RR  ->  -u B  =/= +oo )
8 xaddmnf2 9519 . . . . . 6  |-  ( (
-u B  e.  RR*  /\  -u B  =/= +oo )  ->  ( -oo +e -u B )  = -oo )
96, 7, 8syl2anc 406 . . . . 5  |-  ( -u B  e.  RR  ->  ( -oo +e -u B )  = -oo )
105, 9syl 14 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( -oo +e -u B )  = -oo )
11 oveq1 5733 . . . . . 6  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
12 rexr 7729 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
13 renepnf 7731 . . . . . . . 8  |-  ( B  e.  RR  ->  B  =/= +oo )
14 xaddmnf2 9519 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
1512, 13, 14syl2anc 406 . . . . . . 7  |-  ( B  e.  RR  ->  ( -oo +e B )  = -oo )
1615adantl 273 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( -oo +e B )  = -oo )
1711, 16sylan9eqr 2167 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( A +e B )  = -oo )
1817oveq1d 5741 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( ( A +e B ) +e -u B
)  =  ( -oo +e -u B
) )
19 simpr 109 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  A  = -oo )
2010, 18, 193eqtr4d 2155 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( ( A +e B ) +e -u B
)  =  A )
21 simpll 501 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  A  e.  RR* )
22 simpr 109 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  A  =/= -oo )
2312ad2antlr 478 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  RR* )
24 renemnf 7732 . . . . . 6  |-  ( B  e.  RR  ->  B  =/= -oo )
2524ad2antlr 478 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  =/= -oo )
264ad2antlr 478 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  -u B  e.  RR )
2726, 6syl 14 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  -u B  e.  RR* )
28 renemnf 7732 . . . . . 6  |-  ( -u B  e.  RR  ->  -u B  =/= -oo )
2926, 28syl 14 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  -u B  =/= -oo )
30 xaddass 9539 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( -u B  e. 
RR*  /\  -u B  =/= -oo ) )  ->  (
( A +e
B ) +e -u B )  =  ( A +e ( B +e -u B ) ) )
3121, 22, 23, 25, 27, 29, 30syl222anc 1213 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( ( A +e B ) +e -u B
)  =  ( A +e ( B +e -u B
) ) )
32 simplr 502 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  RR )
3332, 26rexaddd 9524 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B +e -u B )  =  ( B  +  -u B ) )
3432recnd 7712 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  CC )
3534negidd 7980 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B  +  -u B )  =  0 )
3633, 35eqtrd 2145 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B +e -u B )  =  0 )
3736oveq2d 5742 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e ( B +e -u B ) )  =  ( A +e 0 ) )
38 xaddid1 9532 . . . . . 6  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
3938ad2antrr 477 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e 0 )  =  A )
4037, 39eqtrd 2145 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e ( B +e -u B ) )  =  A )
4131, 40eqtrd 2145 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( ( A +e B ) +e -u B
)  =  A )
42 xrmnfdc 9513 . . . . . 6  |-  ( A  e.  RR*  -> DECID  A  = -oo )
43 exmiddc 804 . . . . . 6  |-  (DECID  A  = -oo  ->  ( A  = -oo  \/  -.  A  = -oo ) )
4442, 43syl 14 . . . . 5  |-  ( A  e.  RR*  ->  ( A  = -oo  \/  -.  A  = -oo )
)
45 df-ne 2281 . . . . . 6  |-  ( A  =/= -oo  <->  -.  A  = -oo )
4645orbi2i 734 . . . . 5  |-  ( ( A  = -oo  \/  A  =/= -oo )  <->  ( A  = -oo  \/  -.  A  = -oo ) )
4744, 46sylibr 133 . . . 4  |-  ( A  e.  RR*  ->  ( A  = -oo  \/  A  =/= -oo ) )
4847adantr 272 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  = -oo  \/  A  =/= -oo ) )
4920, 41, 48mpjaodan 770 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A +e
B ) +e -u B )  =  A )
503, 49eqtrd 2145 1  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A +e
B ) +e  -e B )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 680  DECID wdc 802    = wceq 1312    e. wcel 1461    =/= wne 2280  (class class class)co 5726   RRcr 7540   0cc0 7541    + caddc 7544   +oocpnf 7715   -oocmnf 7716   RR*cxr 7717   -ucneg 7851    -ecxne 9443   +ecxad 9444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-addcom 7639  ax-addass 7641  ax-distr 7643  ax-i2m1 7644  ax-0id 7647  ax-rnegex 7648  ax-cnre 7650
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-pnf 7720  df-mnf 7721  df-xr 7722  df-sub 7852  df-neg 7853  df-xneg 9446  df-xadd 9447
This theorem is referenced by:  xnpcan  9542  xleadd1  9545  xrmaxaddlem  10915
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