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Theorem xpncan 10223
Description: Extended real version of pncan 8495. (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 10182 . . . 4 |- ( B e. RR -> -e B = -u B )
21adantl 277 . . 3 |- ( ( A e. RR* /\ B e. RR ) -> -e B = -u B )
32oveq2d 6074 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -e B ) = ( ( A +e B ) +e -u B ) )
4 renegcl 8550 . . . . . 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 8335 . . . . . 6 |- ( -u B e. RR -> -u B e. RR* )
7 renepnf 8337 . . . . . 6 |- ( -u B e. RR -> -u B =/= +oo )
8 xaddmnf2 10201 . . . . . 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 6065 . . . . . 6 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
12 rexr 8335 . . . . . . . 8 |- ( B e. RR -> B e. RR* )
13 renepnf 8337 . . . . . . . 8 |- ( B e. RR -> B =/= +oo )
14 xaddmnf2 10201 . . . . . . . 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 2289 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A = -oo ) -> ( A +e B ) = -oo )
1817oveq1d 6073 . . . 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 2277 . . 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 8338 . . . . . 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 8338 . . . . . 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 10221 . . . . 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 1290 . . . 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 10206 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = ( B + -u B ) )
3432recnd 8318 . . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> B e. CC )
3534negidd 8590 . . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B + -u B ) = 0 )
3633, 35eqtrd 2267 . . . . . 6 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( B +e -u B ) = 0 )
3736oveq2d 6074 . . . . 5 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = ( A +e 0 ) )
38 xaddid1 10214 . . . . . 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 2267 . . . 4 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( A +e ( B +e -u B ) ) = A )
4131, 40eqtrd 2267 . . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ A =/= -oo ) -> ( ( A +e B ) +e -u B ) = A )
42 xrmnfdc 10195 . . . . . 6 |- ( A e. RR* -> DECID A = -oo )
43 exmiddc 844 . . . . . 6 |- (DECID A = -oo -> ( A = -oo \/ -. A = -oo ) )
4442, 43syl 14 . . . . 5 |- ( A e. RR* -> ( A = -oo \/ -. A = -oo ) )
45 df-ne 2415 . . . . . 6 |- ( A =/= -oo <-> -. A = -oo )
4645orbi2i 770 . . . . 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 806 . 2 |- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e B ) +e -u B ) = A )
503, 49eqtrd 2267 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 716  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414  (class class class)co 6058   RRcr 8142   0cc0 8143   + caddc 8146   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323   -ucneg 8461   -ecxne 10121   +ecxad 10122
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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-xr 8328  df-sub 8462  df-neg 8463  df-xneg 10124  df-xadd 10125
This theorem is referenced by:  xnpcan  10224  xleadd1  10227  xrmaxaddlem  11970
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