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Theorem xaddcom 11539
Description: The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
Assertion
Ref Expression
xaddcom  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  ( B +e A ) )

Proof of Theorem xaddcom
StepHypRef Expression
1 elxr 11424 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11424 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 recn 9637 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 9637 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
5 addcom 9827 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
63, 4, 5syl2an 479 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( B  +  A ) )
7 rexadd 11533 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
8 rexadd 11533 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B +e
A )  =  ( B  +  A ) )
98ancoms 454 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B +e
A )  =  ( B  +  A ) )
106, 7, 93eqtr4d 2473 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( B +e A ) )
11 oveq2 6314 . . . . . . 7  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
12 rexr 9694 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
13 renemnf 9697 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= -oo )
14 xaddpnf1 11527 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
1512, 13, 14syl2anc 665 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e +oo )  = +oo )
1611, 15sylan9eqr 2485 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
17 oveq1 6313 . . . . . . 7  |-  ( B  = +oo  ->  ( B +e A )  =  ( +oo +e A ) )
18 xaddpnf2 11528 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( +oo +e A )  = +oo )
1912, 13, 18syl2anc 665 . . . . . . 7  |-  ( A  e.  RR  ->  ( +oo +e A )  = +oo )
2017, 19sylan9eqr 2485 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B +e
A )  = +oo )
2116, 20eqtr4d 2466 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  =  ( B +e A ) )
22 oveq2 6314 . . . . . . 7  |-  ( B  = -oo  ->  ( A +e B )  =  ( A +e -oo ) )
23 renepnf 9696 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= +oo )
24 xaddmnf1 11529 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
2512, 23, 24syl2anc 665 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e -oo )  = -oo )
2622, 25sylan9eqr 2485 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  = -oo )
27 oveq1 6313 . . . . . . 7  |-  ( B  = -oo  ->  ( B +e A )  =  ( -oo +e A ) )
28 xaddmnf2 11530 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( -oo +e A )  = -oo )
2912, 23, 28syl2anc 665 . . . . . . 7  |-  ( A  e.  RR  ->  ( -oo +e A )  = -oo )
3027, 29sylan9eqr 2485 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( B +e
A )  = -oo )
3126, 30eqtr4d 2466 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  =  ( B +e A ) )
3210, 21, 313jaodan 1330 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A +e B )  =  ( B +e A ) )
332, 32sylan2b 477 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
34 pnfaddmnf 11531 . . . . . . . 8  |-  ( +oo +e -oo )  =  0
35 mnfaddpnf 11532 . . . . . . . 8  |-  ( -oo +e +oo )  =  0
3634, 35eqtr4i 2454 . . . . . . 7  |-  ( +oo +e -oo )  =  ( -oo +e +oo )
37 simpr 462 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  B  = -oo )
3837oveq2d 6322 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( +oo +e -oo ) )
3937oveq1d 6321 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( B +e +oo )  =  ( -oo +e +oo ) )
4036, 38, 393eqtr4a 2489 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( B +e +oo ) )
41 xaddpnf2 11528 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
42 xaddpnf1 11527 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( B +e +oo )  = +oo )
4341, 42eqtr4d 2466 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  =  ( B +e +oo ) )
4440, 43pm2.61dane 2738 . . . . 5  |-  ( B  e.  RR*  ->  ( +oo +e B )  =  ( B +e +oo ) )
4544adantl 467 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( +oo +e B )  =  ( B +e +oo )
)
46 simpl 458 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  A  = +oo )
4746oveq1d 6321 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( +oo +e B ) )
4846oveq2d 6322 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( B +e
A )  =  ( B +e +oo ) )
4945, 47, 483eqtr4d 2473 . . 3  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
5035, 34eqtr4i 2454 . . . . . . 7  |-  ( -oo +e +oo )  =  ( +oo +e -oo )
51 simpr 462 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  B  = +oo )
5251oveq2d 6322 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( -oo +e +oo ) )
5351oveq1d 6321 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( B +e -oo )  =  ( +oo +e -oo ) )
5450, 52, 533eqtr4a 2489 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( B +e -oo ) )
55 xaddmnf2 11530 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
56 xaddmnf1 11529 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( B +e -oo )  = -oo )
5755, 56eqtr4d 2466 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  =  ( B +e -oo ) )
5854, 57pm2.61dane 2738 . . . . 5  |-  ( B  e.  RR*  ->  ( -oo +e B )  =  ( B +e -oo ) )
5958adantl 467 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( -oo +e B )  =  ( B +e -oo )
)
60 simpl 458 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  A  = -oo )
6160oveq1d 6321 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( -oo +e B ) )
6260oveq2d 6322 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( B +e
A )  =  ( B +e -oo ) )
6359, 61, 623eqtr4d 2473 . . 3  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
6433, 49, 633jaoian 1329 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  ( A +e B )  =  ( B +e
A ) )
651, 64sylanb 474 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  ( B +e A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1872    =/= wne 2614  (class class class)co 6306   CCcc 9545   RRcr 9546   0cc0 9547    + caddc 9550   +oocpnf 9680   -oocmnf 9681   RR*cxr 9682   +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-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-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-ov 6309  df-oprab 6310  df-mpt2 6311  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-xadd 11418
This theorem is referenced by:  xaddid2  11541  xleadd2a  11548  xltadd2  11551  xposdif  11556  xadd4d  11597  hashunx  12572  xrsnsgrp  19004  xrs1cmn  19008  blcld  21519  xrsxmet  21826  metdstri  21867  metdstriOLD  21882  vdgrf  25625  xaddeq0  28337  xlt2addrd  28345  xrge0npcan  28465  esumle  28888  esumlef  28892  measun  29042  difelcarsg  29151  xaddcomd  37502
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