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Theorem xaddcom 12071
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 11950 . 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2 elxr 11950 . . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3 recn 10026 . . . . . . 7 |- ( A e. RR -> A e. CC )
4 recn 10026 . . . . . . 7 |- ( B e. RR -> B e. CC )
5 addcom 10222 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
63, 4, 5syl2an 494 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) )
7 rexadd 12063 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
8 rexadd 12063 . . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( B +e A ) = ( B + A ) )
98ancoms 469 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( B +e A ) = ( B + A ) )
106, 7, 93eqtr4d 2666 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( B +e A ) )
11 oveq2 6658 . . . . . . 7 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
12 rexr 10085 . . . . . . . 8 |- ( A e. RR -> A e. RR* )
13 renemnf 10088 . . . . . . . 8 |- ( A e. RR -> A =/= -oo )
14 xaddpnf1 12057 . . . . . . . 8 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
1512, 13, 14syl2anc 693 . . . . . . 7 |- ( A e. RR -> ( A +e +oo ) = +oo )
1611, 15sylan9eqr 2678 . . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
17 oveq1 6657 . . . . . . 7 |- ( B = +oo -> ( B +e A ) = ( +oo +e A ) )
18 xaddpnf2 12058 . . . . . . . 8 |- ( ( A e. RR* /\ A =/= -oo ) -> ( +oo +e A ) = +oo )
1912, 13, 18syl2anc 693 . . . . . . 7 |- ( A e. RR -> ( +oo +e A ) = +oo )
2017, 19sylan9eqr 2678 . . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( B +e A ) = +oo )
2116, 20eqtr4d 2659 . . . . 5 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = ( B +e A ) )
22 oveq2 6658 . . . . . . 7 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
23 renepnf 10087 . . . . . . . 8 |- ( A e. RR -> A =/= +oo )
24 xaddmnf1 12059 . . . . . . . 8 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
2512, 23, 24syl2anc 693 . . . . . . 7 |- ( A e. RR -> ( A +e -oo ) = -oo )
2622, 25sylan9eqr 2678 . . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
27 oveq1 6657 . . . . . . 7 |- ( B = -oo -> ( B +e A ) = ( -oo +e A ) )
28 xaddmnf2 12060 . . . . . . . 8 |- ( ( A e. RR* /\ A =/= +oo ) -> ( -oo +e A ) = -oo )
2912, 23, 28syl2anc 693 . . . . . . 7 |- ( A e. RR -> ( -oo +e A ) = -oo )
3027, 29sylan9eqr 2678 . . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( B +e A ) = -oo )
3126, 30eqtr4d 2659 . . . . 5 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = ( B +e A ) )
3210, 21, 313jaodan 1394 . . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A +e B ) = ( B +e A ) )
332, 32sylan2b 492 . . 3 |- ( ( A e. RR /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
34 pnfaddmnf 12061 . . . . . . . 8 |- ( +oo +e -oo ) = 0
35 mnfaddpnf 12062 . . . . . . . 8 |- ( -oo +e +oo ) = 0
3634, 35eqtr4i 2647 . . . . . . 7 |- ( +oo +e -oo ) = ( -oo +e +oo )
37 simpr 477 . . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> B = -oo )
3837oveq2d 6666 . . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( +oo +e -oo ) )
3937oveq1d 6665 . . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( B +e +oo ) = ( -oo +e +oo ) )
4036, 38, 393eqtr4a 2682 . . . . . 6 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( B +e +oo ) )
41 xaddpnf2 12058 . . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
42 xaddpnf1 12057 . . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B +e +oo ) = +oo )
4341, 42eqtr4d 2659 . . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = ( B +e +oo ) )
4440, 43pm2.61dane 2881 . . . . 5 |- ( B e. RR* -> ( +oo +e B ) = ( B +e +oo ) )
4544adantl 482 . . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( +oo +e B ) = ( B +e +oo ) )
46 simpl 473 . . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
4746oveq1d 6665 . . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( +oo +e B ) )
4846oveq2d 6666 . . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( B +e A ) = ( B +e +oo ) )
4945, 47, 483eqtr4d 2666 . . 3 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
5035, 34eqtr4i 2647 . . . . . . 7 |- ( -oo +e +oo ) = ( +oo +e -oo )
51 simpr 477 . . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> B = +oo )
5251oveq2d 6666 . . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( -oo +e +oo ) )
5351oveq1d 6665 . . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( B +e -oo ) = ( +oo +e -oo ) )
5450, 52, 533eqtr4a 2682 . . . . . 6 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( B +e -oo ) )
55 xaddmnf2 12060 . . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
56 xaddmnf1 12059 . . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( B +e -oo ) = -oo )
5755, 56eqtr4d 2659 . . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = ( B +e -oo ) )
5854, 57pm2.61dane 2881 . . . . 5 |- ( B e. RR* -> ( -oo +e B ) = ( B +e -oo ) )
5958adantl 482 . . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( -oo +e B ) = ( B +e -oo ) )
60 simpl 473 . . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> A = -oo )
6160oveq1d 6665 . . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( -oo +e B ) )
6260oveq2d 6666 . . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( B +e A ) = ( B +e -oo ) )
6359, 61, 623eqtr4d 2666 . . 3 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
6433, 49, 633jaoian 1393 . 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
651, 64sylanb 489 1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   +ecxad 11944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-xadd 11947
This theorem is referenced by:  xaddid2  12073  xleadd2a  12084  xltadd2  12087  xposdif  12092  xadd4d  12133  hashunx  13175  xrsnsgrp  19782  xrs1cmn  19786  blcld  22310  xrsxmet  22612  metdstri  22654  vtxdginducedm1  26439  xaddeq0  29518  xlt2addrd  29523  xrge0npcan  29694  esumle  30120  esumlef  30124  measun  30274  difelcarsg  30372  xaddcomd  39540
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