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Theorem xpncan 10273
Description: Extended real version of pncan 8532. (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 10232 . . . 4 |- ( B e. RR -> -e B = -u B )
21adantl 277 . . 3 |- ( ( A e. RR* /\ B e. RR ) -> -e B = -u B )
32oveq2d 6101 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = ( ( A +e B ) +e -u B ) )
4 renegcl 8587 . . . . . 6 |- ( B e. RR -> -u B e. RR )
54ad2antlr 493 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> -u B e. RR )
6 rexr 8371 . . . . . 6 |- ( -u B e. RR -> -u B e. RR* )
7 renepnf 8373 . . . . . 6 |- ( -u B e. RR -> -u B =/= +oo )
8 xaddmnf2 10251 . . . . . 6 |- ( ( -u B e. RR* /\ -u B =/= +oo ) -> ( -oo +e -u B ) = -oo )
96, 7, 8syl2anc 415 . . . . 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 6092 . . . . . 6 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
12 rexr 8371 . . . . . . . 8 |- ( B e. RR -> B e. RR* )
13 renepnf 8373 . . . . . . . 8 |- ( B e. RR -> B =/= +oo )
14 xaddmnf2 10251 . . . . . . . 8 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
1512, 13, 14syl2anc 415 . . . . . . 7 |- ( B e. RR -> ( -oo +e B ) = -oo )
1615adantl 277 . . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> ( -oo +e B ) = -oo )
1711, 16sylan9eqr 2293 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( A +e B ) = -oo )
1817oveq1d 6100 . . . 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 2281 . . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( ( A +e B ) +e -u B ) = A )
21 simpll 531 . . . . 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 493 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. RR* )
24 renemnf 8374 . . . . . 6 |- ( B e. RR -> B =/= -oo )
2524ad2antlr 493 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B =/= -oo )
264ad2antlr 493 . . . . . 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 8374 . . . . . 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 10271 . . . . 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 1294 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( ( A +e B ) +e -u B ) = ( A +e ( B +e -u B ) ) )
32 simplr 533 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. RR )
3332, 26rexaddd 10256 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = ( B + -u B ) )
3432recnd 8354 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. CC )
3534negidd 8627 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B + -u B ) = 0 )
3633, 35eqtrd 2271 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = 0 )
3736oveq2d 6101 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = ( A +e 0 ) )
38 xaddid1 10264 . . . . . 6 |- ( A e. RR* -> ( A +e 0 ) = A )
3938ad2antrr 492 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e 0 ) = A )
4037, 39eqtrd 2271 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = A )
4131, 40eqtrd 2271 . . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( ( A +e B ) +e -u B ) = A )
42 xrmnfdc 10245 . . . . . 6 |- ( A e. RR* -> DECID A = -oo )
43 exmiddc 848 . . . . . 6 |- (DECID A = -oo -> ( A = -oo \/ -. A = -oo ) )
4442, 43syl 14 . . . . 5 |- ( A e. RR* -> ( A = -oo \/ -. A = -oo ) )
45 df-ne 2421 . . . . . 6 |- ( A =/= -oo <-> -. A = -oo )
4645orbi2i 774 . . . . 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 810 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -u B ) = A )
503, 49eqtrd 2271 1 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = A )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420  (class class class)co 6085   RRcr 8178   0cc0 8179   + caddc 8182   +oocpnf 8357   -oocmnf 8358   RR*cxr 8359   -ucneg 8498   -ecxne 10171   +ecxad 10172
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-sub 8499  df-neg 8500  df-xneg 10174  df-xadd 10175
This theorem is used by:  xnpcan  10274  xleadd1  10277  xrmaxaddlem  12026
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