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Theorem xpncan 9821
Description: Extended real version of pncan 8118. (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 9780 . . . 4  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
21adantl 275 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  -e
B  =  -u B
)
32oveq2d 5867 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A +e
B ) +e  -e B )  =  ( ( A +e B ) +e -u B ) )
4 renegcl 8173 . . . . . 6  |-  ( B  e.  RR  ->  -u B  e.  RR )
54ad2antlr 486 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  -u B  e.  RR )
6 rexr 7958 . . . . . 6  |-  ( -u B  e.  RR  ->  -u B  e.  RR* )
7 renepnf 7960 . . . . . 6  |-  ( -u B  e.  RR  ->  -u B  =/= +oo )
8 xaddmnf2 9799 . . . . . 6  |-  ( (
-u B  e.  RR*  /\  -u B  =/= +oo )  ->  ( -oo +e -u B )  = -oo )
96, 7, 8syl2anc 409 . . . . 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 5858 . . . . . 6  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
12 rexr 7958 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
13 renepnf 7960 . . . . . . . 8  |-  ( B  e.  RR  ->  B  =/= +oo )
14 xaddmnf2 9799 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
1512, 13, 14syl2anc 409 . . . . . . 7  |-  ( B  e.  RR  ->  ( -oo +e B )  = -oo )
1615adantl 275 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( -oo +e B )  = -oo )
1711, 16sylan9eqr 2225 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( A +e B )  = -oo )
1817oveq1d 5866 . . . 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 2213 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( ( A +e B ) +e -u B
)  =  A )
21 simpll 524 . . . . 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 486 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  RR* )
24 renemnf 7961 . . . . . 6  |-  ( B  e.  RR  ->  B  =/= -oo )
2524ad2antlr 486 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  =/= -oo )
264ad2antlr 486 . . . . . 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 7961 . . . . . 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 9819 . . . . 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 1249 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( ( A +e B ) +e -u B
)  =  ( A +e ( B +e -u B
) ) )
32 simplr 525 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  RR )
3332, 26rexaddd 9804 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B +e -u B )  =  ( B  +  -u B ) )
3432recnd 7941 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  CC )
3534negidd 8213 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B  +  -u B )  =  0 )
3633, 35eqtrd 2203 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B +e -u B )  =  0 )
3736oveq2d 5867 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e ( B +e -u B ) )  =  ( A +e 0 ) )
38 xaddid1 9812 . . . . . 6  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
3938ad2antrr 485 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e 0 )  =  A )
4037, 39eqtrd 2203 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e ( B +e -u B ) )  =  A )
4131, 40eqtrd 2203 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( ( A +e B ) +e -u B
)  =  A )
42 xrmnfdc 9793 . . . . . 6  |-  ( A  e.  RR*  -> DECID  A  = -oo )
43 exmiddc 831 . . . . . 6  |-  (DECID  A  = -oo  ->  ( A  = -oo  \/  -.  A  = -oo ) )
4442, 43syl 14 . . . . 5  |-  ( A  e.  RR*  ->  ( A  = -oo  \/  -.  A  = -oo )
)
45 df-ne 2341 . . . . . 6  |-  ( A  =/= -oo  <->  -.  A  = -oo )
4645orbi2i 757 . . . . 5  |-  ( ( A  = -oo  \/  A  =/= -oo )  <->  ( A  = -oo  \/  -.  A  = -oo ) )
4744, 46sylibr 133 . . . 4  |-  ( A  e.  RR*  ->  ( A  = -oo  \/  A  =/= -oo ) )
4847adantr 274 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A  = -oo  \/  A  =/= -oo ) )
4920, 41, 48mpjaodan 793 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A +e
B ) +e -u B )  =  A )
503, 49eqtrd 2203 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 703  DECID wdc 829    = wceq 1348    e. wcel 2141    =/= wne 2340  (class class class)co 5851   RRcr 7766   0cc0 7767    + caddc 7770   +oocpnf 7944   -oocmnf 7945   RR*cxr 7946   -ucneg 8084    -ecxne 9719   +ecxad 9720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-addcom 7867  ax-addass 7869  ax-distr 7871  ax-i2m1 7872  ax-0id 7875  ax-rnegex 7876  ax-cnre 7878
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-pnf 7949  df-mnf 7950  df-xr 7951  df-sub 8085  df-neg 8086  df-xneg 9722  df-xadd 9723
This theorem is referenced by:  xnpcan  9822  xleadd1  9825  xrmaxaddlem  11216
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