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Theorem xaddcom 11433
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 11321 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11321 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 recn 9578 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 9578 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
5 addcom 9761 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
63, 4, 5syl2an 477 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( B  +  A ) )
7 rexadd 11427 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
8 rexadd 11427 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B +e
A )  =  ( B  +  A ) )
98ancoms 453 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B +e
A )  =  ( B  +  A ) )
106, 7, 93eqtr4d 2518 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( B +e A ) )
11 oveq2 6290 . . . . . . 7  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
12 rexr 9635 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
13 renemnf 9638 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= -oo )
14 xaddpnf1 11421 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
1512, 13, 14syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e +oo )  = +oo )
1611, 15sylan9eqr 2530 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
17 oveq1 6289 . . . . . . 7  |-  ( B  = +oo  ->  ( B +e A )  =  ( +oo +e A ) )
18 xaddpnf2 11422 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( +oo +e A )  = +oo )
1912, 13, 18syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( +oo +e A )  = +oo )
2017, 19sylan9eqr 2530 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B +e
A )  = +oo )
2116, 20eqtr4d 2511 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  =  ( B +e A ) )
22 oveq2 6290 . . . . . . 7  |-  ( B  = -oo  ->  ( A +e B )  =  ( A +e -oo ) )
23 renepnf 9637 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= +oo )
24 xaddmnf1 11423 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
2512, 23, 24syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e -oo )  = -oo )
2622, 25sylan9eqr 2530 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  = -oo )
27 oveq1 6289 . . . . . . 7  |-  ( B  = -oo  ->  ( B +e A )  =  ( -oo +e A ) )
28 xaddmnf2 11424 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( -oo +e A )  = -oo )
2912, 23, 28syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( -oo +e A )  = -oo )
3027, 29sylan9eqr 2530 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( B +e
A )  = -oo )
3126, 30eqtr4d 2511 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  =  ( B +e A ) )
3210, 21, 313jaodan 1294 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A +e B )  =  ( B +e A ) )
332, 32sylan2b 475 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
34 pnfaddmnf 11425 . . . . . . . 8  |-  ( +oo +e -oo )  =  0
35 mnfaddpnf 11426 . . . . . . . 8  |-  ( -oo +e +oo )  =  0
3634, 35eqtr4i 2499 . . . . . . 7  |-  ( +oo +e -oo )  =  ( -oo +e +oo )
37 simpr 461 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  B  = -oo )
3837oveq2d 6298 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( +oo +e -oo ) )
3937oveq1d 6297 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( B +e +oo )  =  ( -oo +e +oo ) )
4036, 38, 393eqtr4a 2534 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( B +e +oo ) )
41 xaddpnf2 11422 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
42 xaddpnf1 11421 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( B +e +oo )  = +oo )
4341, 42eqtr4d 2511 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  =  ( B +e +oo ) )
4440, 43pm2.61dane 2785 . . . . 5  |-  ( B  e.  RR*  ->  ( +oo +e B )  =  ( B +e +oo ) )
4544adantl 466 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( +oo +e B )  =  ( B +e +oo )
)
46 simpl 457 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  A  = +oo )
4746oveq1d 6297 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( +oo +e B ) )
4846oveq2d 6298 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( B +e
A )  =  ( B +e +oo ) )
4945, 47, 483eqtr4d 2518 . . 3  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
5035, 34eqtr4i 2499 . . . . . . 7  |-  ( -oo +e +oo )  =  ( +oo +e -oo )
51 simpr 461 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  B  = +oo )
5251oveq2d 6298 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( -oo +e +oo ) )
5351oveq1d 6297 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( B +e -oo )  =  ( +oo +e -oo ) )
5450, 52, 533eqtr4a 2534 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( B +e -oo ) )
55 xaddmnf2 11424 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
56 xaddmnf1 11423 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( B +e -oo )  = -oo )
5755, 56eqtr4d 2511 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  =  ( B +e -oo ) )
5854, 57pm2.61dane 2785 . . . . 5  |-  ( B  e.  RR*  ->  ( -oo +e B )  =  ( B +e -oo ) )
5958adantl 466 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( -oo +e B )  =  ( B +e -oo )
)
60 simpl 457 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  A  = -oo )
6160oveq1d 6297 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( -oo +e B ) )
6260oveq2d 6298 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( B +e
A )  =  ( B +e -oo ) )
6359, 61, 623eqtr4d 2518 . . 3  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
6433, 49, 633jaoian 1293 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  ( A +e B )  =  ( B +e
A ) )
651, 64sylanb 472 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  ( B +e A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488    + caddc 9491   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623   +ecxad 11312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-xadd 11315
This theorem is referenced by:  xaddid2  11435  xleadd2a  11442  xltadd2  11445  xposdif  11450  xadd4d  11491  hashunx  12418  xrs1cmn  18226  blcld  20743  xrsxmet  21049  metdstri  21090  vdgrf  24574  xaddeq0  27241  xlt2addrd  27246  xrge0npcan  27346  esumle  27705  esumlef  27710  measun  27822
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