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Theorem xaddcom 11309
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 11197 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11197 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 recn 9473 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 9473 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
5 addcom 9656 . . . . . . 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 11303 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
8 rexadd 11303 . . . . . . 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 2502 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( B +e A ) )
11 oveq2 6198 . . . . . . 7  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
12 rexr 9530 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
13 renemnf 9533 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= -oo )
14 xaddpnf1 11297 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
1512, 13, 14syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e +oo )  = +oo )
1611, 15sylan9eqr 2514 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
17 oveq1 6197 . . . . . . 7  |-  ( B  = +oo  ->  ( B +e A )  =  ( +oo +e A ) )
18 xaddpnf2 11298 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( +oo +e A )  = +oo )
1912, 13, 18syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( +oo +e A )  = +oo )
2017, 19sylan9eqr 2514 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B +e
A )  = +oo )
2116, 20eqtr4d 2495 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  =  ( B +e A ) )
22 oveq2 6198 . . . . . . 7  |-  ( B  = -oo  ->  ( A +e B )  =  ( A +e -oo ) )
23 renepnf 9532 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= +oo )
24 xaddmnf1 11299 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
2512, 23, 24syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e -oo )  = -oo )
2622, 25sylan9eqr 2514 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  = -oo )
27 oveq1 6197 . . . . . . 7  |-  ( B  = -oo  ->  ( B +e A )  =  ( -oo +e A ) )
28 xaddmnf2 11300 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( -oo +e A )  = -oo )
2912, 23, 28syl2anc 661 . . . . . . 7  |-  ( A  e.  RR  ->  ( -oo +e A )  = -oo )
3027, 29sylan9eqr 2514 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( B +e
A )  = -oo )
3126, 30eqtr4d 2495 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  =  ( B +e A ) )
3210, 21, 313jaodan 1285 . . . 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 11301 . . . . . . . 8  |-  ( +oo +e -oo )  =  0
35 mnfaddpnf 11302 . . . . . . . 8  |-  ( -oo +e +oo )  =  0
3634, 35eqtr4i 2483 . . . . . . 7  |-  ( +oo +e -oo )  =  ( -oo +e +oo )
37 simpr 461 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  B  = -oo )
3837oveq2d 6206 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( +oo +e -oo ) )
3937oveq1d 6205 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( B +e +oo )  =  ( -oo +e +oo ) )
4036, 38, 393eqtr4a 2518 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( B +e +oo ) )
41 xaddpnf2 11298 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
42 xaddpnf1 11297 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( B +e +oo )  = +oo )
4341, 42eqtr4d 2495 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  =  ( B +e +oo ) )
4440, 43pm2.61dane 2766 . . . . 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 6205 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( +oo +e B ) )
4846oveq2d 6206 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( B +e
A )  =  ( B +e +oo ) )
4945, 47, 483eqtr4d 2502 . . 3  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
5035, 34eqtr4i 2483 . . . . . . 7  |-  ( -oo +e +oo )  =  ( +oo +e -oo )
51 simpr 461 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  B  = +oo )
5251oveq2d 6206 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( -oo +e +oo ) )
5351oveq1d 6205 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( B +e -oo )  =  ( +oo +e -oo ) )
5450, 52, 533eqtr4a 2518 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( B +e -oo ) )
55 xaddmnf2 11300 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
56 xaddmnf1 11299 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( B +e -oo )  = -oo )
5755, 56eqtr4d 2495 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  =  ( B +e -oo ) )
5854, 57pm2.61dane 2766 . . . . 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 6205 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( -oo +e B ) )
6260oveq2d 6206 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( B +e
A )  =  ( B +e -oo ) )
6359, 61, 623eqtr4d 2502 . . 3  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
6433, 49, 633jaoian 1284 . 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 964    = wceq 1370    e. wcel 1758    =/= wne 2644  (class class class)co 6190   CCcc 9381   RRcr 9382   0cc0 9383    + caddc 9386   +oocpnf 9516   -oocmnf 9517   RR*cxr 9518   +ecxad 11188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-po 4739  df-so 4740  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-xadd 11191
This theorem is referenced by:  xaddid2  11311  xleadd2a  11318  xltadd2  11321  xposdif  11326  xadd4d  11367  hashunx  12251  xrs1cmn  17962  blcld  20196  xrsxmet  20502  metdstri  20543  vdgrf  23703  xaddeq0  26180  xlt2addrd  26185  xrge0npcan  26291  esumle  26642  esumlef  26647  measun  26759
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