ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpncan Unicode version
Theorem xpncan 9654
Description: Extended real version of pncan 7968. (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 9613 . . . 4 |- ( B e. RR -> -e B = -u B )
21adantl 275 . . 3 |- ( ( A e. RR* /\ B e. RR ) -> -e B = -u B )
32oveq2d 5790 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = ( ( A +e B ) +e -u B ) )
4 renegcl 8023 . . . . . 6 |- ( B e. RR -> -u B e. RR )
54ad2antlr 480 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> -u B e. RR )
6 rexr 7811 . . . . . 6 |- ( -u B e. RR -> -u B e. RR* )
7 renepnf 7813 . . . . . 6 |- ( -u B e. RR -> -u B =/= +oo )
8 xaddmnf2 9632 . . . . . 6 |- ( ( -u B e. RR* /\ -u B =/= +oo ) -> ( -oo +e -u B ) = -oo )
96, 7, 8syl2anc 408 . . . . 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 5781 . . . . . 6 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
12 rexr 7811 . . . . . . . 8 |- ( B e. RR -> B e. RR* )
13 renepnf 7813 . . . . . . . 8 |- ( B e. RR -> B =/= +oo )
14 xaddmnf2 9632 . . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
1512, 13, 14syl2anc 408 . . . . . . 7 |- ( B e. RR -> ( -oo +e B ) = -oo )
1615adantl 275 . . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> ( -oo +e B ) = -oo )
1711, 16sylan9eqr 2194 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( A +e B ) = -oo )
1817oveq1d 5789 . . . 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 2182 . . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( ( A +e B ) +e -u B ) = A )
21 simpll 518 . . . . 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 480 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. RR* )
24 renemnf 7814 . . . . . 6 |- ( B e. RR -> B =/= -oo )
2524ad2antlr 480 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B =/= -oo )
264ad2antlr 480 . . . . . 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 7814 . . . . . 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 9652 . . . . 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 1232 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( ( A +e B ) +e -u B ) = ( A +e ( B +e -u B ) ) )
32 simplr 519 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. RR )
3332, 26rexaddd 9637 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = ( B + -u B ) )
3432recnd 7794 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. CC )
3534negidd 8063 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B + -u B ) = 0 )
3633, 35eqtrd 2172 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = 0 )
3736oveq2d 5790 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = ( A +e 0 ) )
38 xaddid1 9645 . . . . . 6 |- ( A e. RR* -> ( A +e 0 ) = A )
3938ad2antrr 479 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e 0 ) = A )
4037, 39eqtrd 2172 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = A )
4131, 40eqtrd 2172 . . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( ( A +e B ) +e -u B ) = A )
42 xrmnfdc 9626 . . . . . 6 |- ( A e. RR* -> DECID A = -oo )
43 exmiddc 821 . . . . . 6 |- (DECID A = -oo -> ( A = -oo \/ -. A = -oo ) )
4442, 43syl 14 . . . . 5 |- ( A e. RR* -> ( A = -oo \/ -. A = -oo ) )
45 df-ne 2309 . . . . . 6 |- ( A =/= -oo <-> -. A = -oo )
4645orbi2i 751 . . . . 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 787 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -u B ) = A )
503, 49eqtrd 2172 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 697  DECID wdc 819   = wceq 1331   e. wcel 1480   =/= wne 2308  (class class class)co 5774   RRcr 7619   0cc0 7620   + caddc 7623   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799   -ucneg 7934   -ecxne 9556   +ecxad 9557
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-sub 7935  df-neg 7936  df-xneg 9559  df-xadd 9560
This theorem is referenced by:  xnpcan  9655  xleadd1  9658  xrmaxaddlem  11029
  Copyright terms: Public domain W3C validator