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Theorem xpncan 10105
Description: Extended real version of pncan 8384. (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 10064 . . . 4 |- ( B e. RR -> -e B = -u B )
21adantl 277 . . 3 |- ( ( A e. RR* /\ B e. RR ) -> -e B = -u B )
32oveq2d 6033 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = ( ( A +e B ) +e -u B ) )
4 renegcl 8439 . . . . . 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 8224 . . . . . 6 |- ( -u B e. RR -> -u B e. RR* )
7 renepnf 8226 . . . . . 6 |- ( -u B e. RR -> -u B =/= +oo )
8 xaddmnf2 10083 . . . . . 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 6024 . . . . . 6 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
12 rexr 8224 . . . . . . . 8 |- ( B e. RR -> B e. RR* )
13 renepnf 8226 . . . . . . . 8 |- ( B e. RR -> B =/= +oo )
14 xaddmnf2 10083 . . . . . . . 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 2286 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( A +e B ) = -oo )
1817oveq1d 6032 . . . 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 2274 . . 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 8227 . . . . . 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 8227 . . . . . 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 10103 . . . . 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 1289 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( ( A +e B ) +e -u B ) = ( A +e ( B +e -u B ) ) )
32 simplr 529 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. RR )
3332, 26rexaddd 10088 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = ( B + -u B ) )
3432recnd 8207 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. CC )
3534negidd 8479 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B + -u B ) = 0 )
3633, 35eqtrd 2264 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = 0 )
3736oveq2d 6033 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = ( A +e 0 ) )
38 xaddid1 10096 . . . . . 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 2264 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = A )
4131, 40eqtrd 2264 . . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( ( A +e B ) +e -u B ) = A )
42 xrmnfdc 10077 . . . . . 6 |- ( A e. RR* -> DECID A = -oo )
43 exmiddc 843 . . . . . 6 |- (DECID A = -oo -> ( A = -oo \/ -. A = -oo ) )
4442, 43syl 14 . . . . 5 |- ( A e. RR* -> ( A = -oo \/ -. A = -oo ) )
45 df-ne 2403 . . . . . 6 |- ( A =/= -oo <-> -. A = -oo )
4645orbi2i 769 . . . . 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 805 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -u B ) = A )
503, 49eqtrd 2264 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 715  DECID wdc 841   = wceq 1397   e. wcel 2202   =/= wne 2402  (class class class)co 6017   RRcr 8030   0cc0 8031   + caddc 8034   +oocpnf 8210   -oocmnf 8211   RR*cxr 8212   -ucneg 8350   -ecxne 10003   +ecxad 10004
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142
This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-xr 8217  df-sub 8351  df-neg 8352  df-xneg 10006  df-xadd 10007
This theorem is referenced by:  xnpcan  10106  xleadd1  10109  xrmaxaddlem  11820
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