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Theorem xpncan 9896
Description: Extended real version of pncan 8188. (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 9855 . . . 4  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
21adantl 277 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  -e
B  =  -u B
)
32oveq2d 5908 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A +e
B ) +e  -e B )  =  ( ( A +e B ) +e -u B ) )
4 renegcl 8243 . . . . . 6  |-  ( B  e.  RR  ->  -u B  e.  RR )
54ad2antlr 489 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  -u B  e.  RR )
6 rexr 8028 . . . . . 6  |-  ( -u B  e.  RR  ->  -u B  e.  RR* )
7 renepnf 8030 . . . . . 6  |-  ( -u B  e.  RR  ->  -u B  =/= +oo )
8 xaddmnf2 9874 . . . . . 6  |-  ( (
-u B  e.  RR*  /\  -u B  =/= +oo )  ->  ( -oo +e -u B )  = -oo )
96, 7, 8syl2anc 411 . . . . 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 5899 . . . . . 6  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
12 rexr 8028 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
13 renepnf 8030 . . . . . . . 8  |-  ( B  e.  RR  ->  B  =/= +oo )
14 xaddmnf2 9874 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
1512, 13, 14syl2anc 411 . . . . . . 7  |-  ( B  e.  RR  ->  ( -oo +e B )  = -oo )
1615adantl 277 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( -oo +e B )  = -oo )
1711, 16sylan9eqr 2244 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( A +e B )  = -oo )
1817oveq1d 5907 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( ( A +e B ) +e -u B
)  =  ( -oo +e -u B
) )
19 simpr 110 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  A  = -oo )
2010, 18, 193eqtr4d 2232 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( ( A +e B ) +e -u B
)  =  A )
21 simpll 527 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  A  e.  RR* )
22 simpr 110 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  A  =/= -oo )
2312ad2antlr 489 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  RR* )
24 renemnf 8031 . . . . . 6  |-  ( B  e.  RR  ->  B  =/= -oo )
2524ad2antlr 489 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  =/= -oo )
264ad2antlr 489 . . . . . 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 8031 . . . . . 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 9894 . . . . 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 1265 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( ( A +e B ) +e -u B
)  =  ( A +e ( B +e -u B
) ) )
32 simplr 528 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  RR )
3332, 26rexaddd 9879 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B +e -u B )  =  ( B  +  -u B ) )
3432recnd 8011 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  CC )
3534negidd 8283 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B  +  -u B )  =  0 )
3633, 35eqtrd 2222 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B +e -u B )  =  0 )
3736oveq2d 5908 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e ( B +e -u B ) )  =  ( A +e 0 ) )
38 xaddid1 9887 . . . . . 6  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
3938ad2antrr 488 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e 0 )  =  A )
4037, 39eqtrd 2222 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e ( B +e -u B ) )  =  A )
4131, 40eqtrd 2222 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( ( A +e B ) +e -u B
)  =  A )
42 xrmnfdc 9868 . . . . . 6  |-  ( A  e.  RR*  -> DECID  A  = -oo )
43 exmiddc 837 . . . . . 6  |-  (DECID  A  = -oo  ->  ( A  = -oo  \/  -.  A  = -oo ) )
4442, 43syl 14 . . . . 5  |-  ( A  e.  RR*  ->  ( A  = -oo  \/  -.  A  = -oo )
)
45 df-ne 2361 . . . . . 6  |-  ( A  =/= -oo  <->  -.  A  = -oo )
4645orbi2i 763 . . . . 5  |-  ( ( A  = -oo  \/  A  =/= -oo )  <->  ( A  = -oo  \/  -.  A  = -oo ) )
4744, 46sylibr 134 . . . 4  |-  ( A  e.  RR*  ->  ( A  = -oo  \/  A  =/= -oo ) )
4847adantr 276 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  = -oo  \/  A  =/= -oo ) )
4920, 41, 48mpjaodan 799 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A +e
B ) +e -u B )  =  A )
503, 49eqtrd 2222 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 104    \/ wo 709  DECID wdc 835    = wceq 1364    e. wcel 2160    =/= wne 2360  (class class class)co 5892   RRcr 7835   0cc0 7836    + caddc 7839   +oocpnf 8014   -oocmnf 8015   RR*cxr 8016   -ucneg 8154    -ecxne 9794   +ecxad 9795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-addcom 7936  ax-addass 7938  ax-distr 7940  ax-i2m1 7941  ax-0id 7944  ax-rnegex 7945  ax-cnre 7947
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-pnf 8019  df-mnf 8020  df-xr 8021  df-sub 8155  df-neg 8156  df-xneg 9797  df-xadd 9798
This theorem is referenced by:  xnpcan  9897  xleadd1  9900  xrmaxaddlem  11295
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