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Theorem xaddcom 9673
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 9592 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 9592 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 recn 7776 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 7776 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
5 addcom 7922 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
63, 4, 5syl2an 287 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( B  +  A ) )
7 rexadd 9664 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
8 rexadd 9664 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B +e
A )  =  ( B  +  A ) )
98ancoms 266 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B +e
A )  =  ( B  +  A ) )
106, 7, 93eqtr4d 2183 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( B +e A ) )
11 oveq2 5789 . . . . . . 7  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
12 rexr 7834 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
13 renemnf 7837 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= -oo )
14 xaddpnf1 9658 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
1512, 13, 14syl2anc 409 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e +oo )  = +oo )
1611, 15sylan9eqr 2195 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
17 oveq1 5788 . . . . . . 7  |-  ( B  = +oo  ->  ( B +e A )  =  ( +oo +e A ) )
18 xaddpnf2 9659 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( +oo +e A )  = +oo )
1912, 13, 18syl2anc 409 . . . . . . 7  |-  ( A  e.  RR  ->  ( +oo +e A )  = +oo )
2017, 19sylan9eqr 2195 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B +e
A )  = +oo )
2116, 20eqtr4d 2176 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  =  ( B +e A ) )
22 oveq2 5789 . . . . . . 7  |-  ( B  = -oo  ->  ( A +e B )  =  ( A +e -oo ) )
23 renepnf 7836 . . . . . . . 8  |-  ( A  e.  RR  ->  A  =/= +oo )
24 xaddmnf1 9660 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
2512, 23, 24syl2anc 409 . . . . . . 7  |-  ( A  e.  RR  ->  ( A +e -oo )  = -oo )
2622, 25sylan9eqr 2195 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  = -oo )
27 oveq1 5788 . . . . . . 7  |-  ( B  = -oo  ->  ( B +e A )  =  ( -oo +e A ) )
28 xaddmnf2 9661 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( -oo +e A )  = -oo )
2912, 23, 28syl2anc 409 . . . . . . 7  |-  ( A  e.  RR  ->  ( -oo +e A )  = -oo )
3027, 29sylan9eqr 2195 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( B +e
A )  = -oo )
3126, 30eqtr4d 2176 . . . . 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 285 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
34 pnfaddmnf 9662 . . . . . . . 8  |-  ( +oo +e -oo )  =  0
35 mnfaddpnf 9663 . . . . . . . 8  |-  ( -oo +e +oo )  =  0
3634, 35eqtr4i 2164 . . . . . . 7  |-  ( +oo +e -oo )  =  ( -oo +e +oo )
37 simpr 109 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  B  = -oo )
3837oveq2d 5797 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( +oo +e -oo ) )
3937oveq1d 5796 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( B +e +oo )  =  ( -oo +e +oo ) )
4036, 38, 393eqtr4a 2199 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( B +e +oo ) )
41 xaddpnf2 9659 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
42 xaddpnf1 9658 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( B +e +oo )  = +oo )
4341, 42eqtr4d 2176 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  =  ( B +e +oo ) )
44 xrmnfdc 9655 . . . . . . . 8  |-  ( B  e.  RR*  -> DECID  B  = -oo )
45 exmiddc 822 . . . . . . . 8  |-  (DECID  B  = -oo  ->  ( B  = -oo  \/  -.  B  = -oo ) )
4644, 45syl 14 . . . . . . 7  |-  ( B  e.  RR*  ->  ( B  = -oo  \/  -.  B  = -oo )
)
47 df-ne 2310 . . . . . . . 8  |-  ( B  =/= -oo  <->  -.  B  = -oo )
4847orbi2i 752 . . . . . . 7  |-  ( ( B  = -oo  \/  B  =/= -oo )  <->  ( B  = -oo  \/  -.  B  = -oo ) )
4946, 48sylibr 133 . . . . . 6  |-  ( B  e.  RR*  ->  ( B  = -oo  \/  B  =/= -oo ) )
5040, 43, 49mpjaodan 788 . . . . 5  |-  ( B  e.  RR*  ->  ( +oo +e B )  =  ( B +e +oo ) )
5150adantl 275 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( +oo +e B )  =  ( B +e +oo )
)
52 simpl 108 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  A  = +oo )
5352oveq1d 5796 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( +oo +e B ) )
5452oveq2d 5797 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( B +e
A )  =  ( B +e +oo ) )
5551, 53, 543eqtr4d 2183 . . 3  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
5635, 34eqtr4i 2164 . . . . . . 7  |-  ( -oo +e +oo )  =  ( +oo +e -oo )
57 simpr 109 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  B  = +oo )
5857oveq2d 5797 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( -oo +e +oo ) )
5957oveq1d 5796 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( B +e -oo )  =  ( +oo +e -oo ) )
6056, 58, 593eqtr4a 2199 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( B +e -oo ) )
61 xaddmnf2 9661 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
62 xaddmnf1 9660 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( B +e -oo )  = -oo )
6361, 62eqtr4d 2176 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  =  ( B +e -oo ) )
64 xrpnfdc 9654 . . . . . . . 8  |-  ( B  e.  RR*  -> DECID  B  = +oo )
65 exmiddc 822 . . . . . . . 8  |-  (DECID  B  = +oo  ->  ( B  = +oo  \/  -.  B  = +oo ) )
6664, 65syl 14 . . . . . . 7  |-  ( B  e.  RR*  ->  ( B  = +oo  \/  -.  B  = +oo )
)
67 df-ne 2310 . . . . . . . 8  |-  ( B  =/= +oo  <->  -.  B  = +oo )
6867orbi2i 752 . . . . . . 7  |-  ( ( B  = +oo  \/  B  =/= +oo )  <->  ( B  = +oo  \/  -.  B  = +oo ) )
6966, 68sylibr 133 . . . . . 6  |-  ( B  e.  RR*  ->  ( B  = +oo  \/  B  =/= +oo ) )
7060, 63, 69mpjaodan 788 . . . . 5  |-  ( B  e.  RR*  ->  ( -oo +e B )  =  ( B +e -oo ) )
7170adantl 275 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( -oo +e B )  =  ( B +e -oo )
)
72 simpl 108 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  A  = -oo )
7372oveq1d 5796 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( -oo +e B ) )
7472oveq2d 5797 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( B +e
A )  =  ( B +e -oo ) )
7571, 73, 743eqtr4d 2183 . . 3  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( B +e A ) )
7633, 55, 753jaoian 1284 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  ( A +e B )  =  ( B +e
A ) )
771, 76sylanb 282 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  ( B +e A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 698  DECID wdc 820    \/ w3o 962    = wceq 1332    e. wcel 1481    =/= wne 2309  (class class class)co 5781   CCcc 7641   RRcr 7642   0cc0 7643    + caddc 7646   +oocpnf 7820   -oocmnf 7821   RR*cxr 7822   +ecxad 9586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1re 7737  ax-addrcl 7740  ax-addcom 7743  ax-rnegex 7752
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pnf 7825  df-mnf 7826  df-xr 7827  df-xadd 9589
This theorem is referenced by:  xaddid2  9675  xleadd2a  9686  xltadd2  9689  xadd4d  9697  xrmaxaddlem  11060
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