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Theorem xaddcom 11462
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 11350 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11350 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 recn 9599 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 9599 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
5 addcom 9783 . . . . . . 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 11456 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
8 rexadd 11456 . . . . . . 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 2508 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( B +e A ) )
11 oveq2 6304 . . . . . . 7  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
12 rexr 9656 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
13 renemnf 9659 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= -oo )
14 xaddpnf1 11450 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
1512, 13, 14syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e +oo )  = +oo )
1611, 15sylan9eqr 2520 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
17 oveq1 6303 . . . . . . 7  |-  ( B  = +oo  ->  ( B +e A )  =  ( +oo +e A ) )
18 xaddpnf2 11451 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( +oo +e A )  = +oo )
1912, 13, 18syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( +oo +e A )  = +oo )
2017, 19sylan9eqr 2520 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B +e
A )  = +oo )
2116, 20eqtr4d 2501 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  =  ( B +e A ) )
22 oveq2 6304 . . . . . . 7  |-  ( B  = -oo  ->  ( A +e B )  =  ( A +e -oo ) )
23 renepnf 9658 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= +oo )
24 xaddmnf1 11452 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
2512, 23, 24syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e -oo )  = -oo )
2622, 25sylan9eqr 2520 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  = -oo )
27 oveq1 6303 . . . . . . 7  |-  ( B  = -oo  ->  ( B +e A )  =  ( -oo +e A ) )
28 xaddmnf2 11453 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( -oo +e A )  = -oo )
2912, 23, 28syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( -oo +e A )  = -oo )
3027, 29sylan9eqr 2520 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( B +e
A )  = -oo )
3126, 30eqtr4d 2501 . . . . 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 11454 . . . . . . . 8  |-  ( +oo +e -oo )  =  0
35 mnfaddpnf 11455 . . . . . . . 8  |-  ( -oo +e +oo )  =  0
3634, 35eqtr4i 2489 . . . . . . 7  |-  ( +oo +e -oo )  =  ( -oo +e +oo )
37 simpr 461 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  B  = -oo )
3837oveq2d 6312 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( +oo +e -oo ) )
3937oveq1d 6311 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( B +e +oo )  =  ( -oo +e +oo ) )
4036, 38, 393eqtr4a 2524 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( B +e +oo ) )
41 xaddpnf2 11451 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
42 xaddpnf1 11450 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( B +e +oo )  = +oo )
4341, 42eqtr4d 2501 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  =  ( B +e +oo ) )
4440, 43pm2.61dane 2775 . . . . 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 6311 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( +oo +e B ) )
4846oveq2d 6312 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( B +e
A )  =  ( B +e +oo ) )
4945, 47, 483eqtr4d 2508 . . 3  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
5035, 34eqtr4i 2489 . . . . . . 7  |-  ( -oo +e +oo )  =  ( +oo +e -oo )
51 simpr 461 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  B  = +oo )
5251oveq2d 6312 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( -oo +e +oo ) )
5351oveq1d 6311 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( B +e -oo )  =  ( +oo +e -oo ) )
5450, 52, 533eqtr4a 2524 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( B +e -oo ) )
55 xaddmnf2 11453 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
56 xaddmnf1 11452 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( B +e -oo )  = -oo )
5755, 56eqtr4d 2501 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  =  ( B +e -oo ) )
5854, 57pm2.61dane 2775 . . . . 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 6311 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( -oo +e B ) )
6260oveq2d 6312 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( B +e
A )  =  ( B +e -oo ) )
6359, 61, 623eqtr4d 2508 . . 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 1395    e. wcel 1819    =/= wne 2652  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509    + caddc 9512   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644   +ecxad 11341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-xadd 11344
This theorem is referenced by:  xaddid2  11464  xleadd2a  11471  xltadd2  11474  xposdif  11479  xadd4d  11520  hashunx  12456  xrsnsgrp  18580  xrs1cmn  18584  blcld  21133  xrsxmet  21439  metdstri  21480  vdgrf  25024  xaddeq0  27723  xlt2addrd  27728  xrge0npcan  27836  esumle  28220  esumlef  28225  measun  28343  difelcarsg  28440
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