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Theorem xpncan 9807
Description: Extended real version of pncan 8104. (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 9766 . . . 4 |- ( B e. RR -> -e B = -u B )
21adantl 275 . . 3 |- ( ( A e. RR* /\ B e. RR ) -> -e B = -u B )
32oveq2d 5858 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = ( ( A +e B ) +e -u B ) )
4 renegcl 8159 . . . . . 6 |- ( B e. RR -> -u B e. RR )
54ad2antlr 481 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> -u B e. RR )
6 rexr 7944 . . . . . 6 |- ( -u B e. RR -> -u B e. RR* )
7 renepnf 7946 . . . . . 6 |- ( -u B e. RR -> -u B =/= +oo )
8 xaddmnf2 9785 . . . . . 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 5849 . . . . . 6 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
12 rexr 7944 . . . . . . . 8 |- ( B e. RR -> B e. RR* )
13 renepnf 7946 . . . . . . . 8 |- ( B e. RR -> B =/= +oo )
14 xaddmnf2 9785 . . . . . . . 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 2221 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( A +e B ) = -oo )
1817oveq1d 5857 . . . 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 2208 . . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( ( A +e B ) +e -u B ) = A )
21 simpll 519 . . . . 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 481 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. RR* )
24 renemnf 7947 . . . . . 6 |- ( B e. RR -> B =/= -oo )
2524ad2antlr 481 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B =/= -oo )
264ad2antlr 481 . . . . . 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 7947 . . . . . 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 9805 . . . . 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 1244 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( ( A +e B ) +e -u B ) = ( A +e ( B +e -u B ) ) )
32 simplr 520 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. RR )
3332, 26rexaddd 9790 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = ( B + -u B ) )
3432recnd 7927 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. CC )
3534negidd 8199 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B + -u B ) = 0 )
3633, 35eqtrd 2198 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = 0 )
3736oveq2d 5858 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = ( A +e 0 ) )
38 xaddid1 9798 . . . . . 6 |- ( A e. RR* -> ( A +e 0 ) = A )
3938ad2antrr 480 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e 0 ) = A )
4037, 39eqtrd 2198 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = A )
4131, 40eqtrd 2198 . . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( ( A +e B ) +e -u B ) = A )
42 xrmnfdc 9779 . . . . . 6 |- ( A e. RR* -> DECID A = -oo )
43 exmiddc 826 . . . . . 6 |- (DECID A = -oo -> ( A = -oo \/ -. A = -oo ) )
4442, 43syl 14 . . . . 5 |- ( A e. RR* -> ( A = -oo \/ -. A = -oo ) )
45 df-ne 2337 . . . . . 6 |- ( A =/= -oo <-> -. A = -oo )
4645orbi2i 752 . . . . 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 788 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -u B ) = A )
503, 49eqtrd 2198 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 698  DECID wdc 824   = wceq 1343   e. wcel 2136   =/= wne 2336  (class class class)co 5842   RRcr 7752   0cc0 7753   + caddc 7756   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   -ucneg 8070   -ecxne 9705   +ecxad 9706
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-sub 8071  df-neg 8072  df-xneg 9708  df-xadd 9709
This theorem is referenced by:  xnpcan  9808  xleadd1  9811  xrmaxaddlem  11201
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