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Theorem xpncan 11545
Description: Extended real version of pncan 9889. (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 11512 . . . 4  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
21adantl 467 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  -e
B  =  -u B
)
32oveq2d 6322 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A +e
B ) +e  -e B )  =  ( ( A +e B ) +e -u B ) )
4 renegcl 9945 . . . . . 6  |-  ( B  e.  RR  ->  -u B  e.  RR )
54ad2antlr 731 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  -u B  e.  RR )
6 rexr 9694 . . . . . 6  |-  ( -u B  e.  RR  ->  -u B  e.  RR* )
7 renepnf 9696 . . . . . 6  |-  ( -u B  e.  RR  ->  -u B  =/= +oo )
8 xaddmnf2 11530 . . . . . 6  |-  ( (
-u B  e.  RR*  /\  -u B  =/= +oo )  ->  ( -oo +e -u B )  = -oo )
96, 7, 8syl2anc 665 . . . . 5  |-  ( -u B  e.  RR  ->  ( -oo +e -u B )  = -oo )
105, 9syl 17 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( -oo +e -u B )  = -oo )
11 oveq1 6313 . . . . . 6  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
12 rexr 9694 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
13 renepnf 9696 . . . . . . . 8  |-  ( B  e.  RR  ->  B  =/= +oo )
14 xaddmnf2 11530 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
1512, 13, 14syl2anc 665 . . . . . . 7  |-  ( B  e.  RR  ->  ( -oo +e B )  = -oo )
1615adantl 467 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( -oo +e B )  = -oo )
1711, 16sylan9eqr 2485 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( A +e B )  = -oo )
1817oveq1d 6321 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( ( A +e B ) +e -u B
)  =  ( -oo +e -u B
) )
19 simpr 462 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  A  = -oo )
2010, 18, 193eqtr4d 2473 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  = -oo )  ->  ( ( A +e B ) +e -u B
)  =  A )
21 simpll 758 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  A  e.  RR* )
22 simpr 462 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  A  =/= -oo )
2312ad2antlr 731 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  RR* )
24 renemnf 9697 . . . . . 6  |-  ( B  e.  RR  ->  B  =/= -oo )
2524ad2antlr 731 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  =/= -oo )
264ad2antlr 731 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  -u B  e.  RR )
2726, 6syl 17 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  -u B  e.  RR* )
28 renemnf 9697 . . . . . 6  |-  ( -u B  e.  RR  ->  -u B  =/= -oo )
2926, 28syl 17 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  -u B  =/= -oo )
30 xaddass 11543 . . . . 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 1280 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( ( A +e B ) +e -u B
)  =  ( A +e ( B +e -u B
) ) )
32 simplr 760 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  RR )
33 rexadd 11533 . . . . . . . 8  |-  ( ( B  e.  RR  /\  -u B  e.  RR )  ->  ( B +e -u B )  =  ( B  +  -u B ) )
3432, 26, 33syl2anc 665 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B +e -u B )  =  ( B  +  -u B ) )
3532recnd 9677 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  B  e.  CC )
3635negidd 9984 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B  +  -u B )  =  0 )
3734, 36eqtrd 2463 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( B +e -u B )  =  0 )
3837oveq2d 6322 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e ( B +e -u B ) )  =  ( A +e 0 ) )
39 xaddid1 11540 . . . . . 6  |-  ( A  e.  RR*  ->  ( A +e 0 )  =  A )
4039ad2antrr 730 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e 0 )  =  A )
4138, 40eqtrd 2463 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( A +e ( B +e -u B ) )  =  A )
4231, 41eqtrd 2463 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR )  /\  A  =/= -oo )  ->  ( ( A +e B ) +e -u B
)  =  A )
4320, 42pm2.61dane 2738 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A +e
B ) +e -u B )  =  A )
443, 43eqtrd 2463 1  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( A +e
B ) +e  -e B )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614  (class class class)co 6306   RRcr 9546   0cc0 9547    + caddc 9550   +oocpnf 9680   -oocmnf 9681   RR*cxr 9682   -ucneg 9869    -ecxne 11414   +ecxad 11415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-1st 6808  df-2nd 6809  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-sub 9870  df-neg 9871  df-xneg 11417  df-xadd 11418
This theorem is referenced by:  xnpcan  11546  xleadd1  11549  xaddeq0  28337
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