Theorem xaddcom 10086

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

Step	Hyp	Ref	Expression
1		elxr 10001	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 10001	. . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		recn 8155	. . . . . . 7 |- ( A e. RR -> A e. CC )
4		recn 8155	. . . . . . 7 |- ( B e. RR -> B e. CC )
5		addcom 8306	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
6	3, 4, 5	syl2an 289	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) )
7		rexadd 10077	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
8		rexadd 10077	. . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( B +e A ) = ( B + A ) )
9	8	ancoms 268	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( B +e A ) = ( B + A ) )
10	6, 7, 9	3eqtr4d 2272	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( B +e A ) )
11		oveq2 6021	. . . . . . 7 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
12		rexr 8215	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
13		renemnf 8218	. . . . . . . 8 |- ( A e. RR -> A =/= -oo )
14		xaddpnf1 10071	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
15	12, 13, 14	syl2anc 411	. . . . . . 7 |- ( A e. RR -> ( A +e +oo ) = +oo )
16	11, 15	sylan9eqr 2284	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
17		oveq1 6020	. . . . . . 7 |- ( B = +oo -> ( B +e A ) = ( +oo +e A ) )
18		xaddpnf2 10072	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= -oo ) -> ( +oo +e A ) = +oo )
19	12, 13, 18	syl2anc 411	. . . . . . 7 |- ( A e. RR -> ( +oo +e A ) = +oo )
20	17, 19	sylan9eqr 2284	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( B +e A ) = +oo )
21	16, 20	eqtr4d 2265	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = ( B +e A ) )
22		oveq2 6021	. . . . . . 7 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
23		renepnf 8217	. . . . . . . 8 |- ( A e. RR -> A =/= +oo )
24		xaddmnf1 10073	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
25	12, 23, 24	syl2anc 411	. . . . . . 7 |- ( A e. RR -> ( A +e -oo ) = -oo )
26	22, 25	sylan9eqr 2284	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
27		oveq1 6020	. . . . . . 7 |- ( B = -oo -> ( B +e A ) = ( -oo +e A ) )
28		xaddmnf2 10074	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= +oo ) -> ( -oo +e A ) = -oo )
29	12, 23, 28	syl2anc 411	. . . . . . 7 |- ( A e. RR -> ( -oo +e A ) = -oo )
30	27, 29	sylan9eqr 2284	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( B +e A ) = -oo )
31	26, 30	eqtr4d 2265	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = ( B +e A ) )
32	10, 21, 31	3jaodan 1340	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A +e B ) = ( B +e A ) )
33	2, 32	sylan2b 287	. . 3 |- ( ( A e. RR /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
34		pnfaddmnf 10075	. . . . . . . 8 |- ( +oo +e -oo ) = 0
35		mnfaddpnf 10076	. . . . . . . 8 |- ( -oo +e +oo ) = 0
36	34, 35	eqtr4i 2253	. . . . . . 7 |- ( +oo +e -oo ) = ( -oo +e +oo )
37		simpr 110	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> B = -oo )
38	37	oveq2d 6029	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( +oo +e -oo ) )
39	37	oveq1d 6028	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( B +e +oo ) = ( -oo +e +oo ) )
40	36, 38, 39	3eqtr4a 2288	. . . . . 6 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( B +e +oo ) )
41		xaddpnf2 10072	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
42		xaddpnf1 10071	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B +e +oo ) = +oo )
43	41, 42	eqtr4d 2265	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = ( B +e +oo ) )
44		xrmnfdc 10068	. . . . . . . 8 |- ( B e. RR* -> DECID B = -oo )
45		exmiddc 841	. . . . . . . 8 |- (DECID B = -oo -> ( B = -oo \/ -. B = -oo ) )
46	44, 45	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = -oo \/ -. B = -oo ) )
47		df-ne 2401	. . . . . . . 8 |- ( B =/= -oo <-> -. B = -oo )
48	47	orbi2i 767	. . . . . . 7 |- ( ( B = -oo \/ B =/= -oo ) <-> ( B = -oo \/ -. B = -oo ) )
49	46, 48	sylibr 134	. . . . . 6 |- ( B e. RR* -> ( B = -oo \/ B =/= -oo ) )
50	40, 43, 49	mpjaodan 803	. . . . 5 |- ( B e. RR* -> ( +oo +e B ) = ( B +e +oo ) )
51	50	adantl 277	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( +oo +e B ) = ( B +e +oo ) )
52		simpl 109	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
53	52	oveq1d 6028	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( +oo +e B ) )
54	52	oveq2d 6029	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( B +e A ) = ( B +e +oo ) )
55	51, 53, 54	3eqtr4d 2272	. . 3 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
56	35, 34	eqtr4i 2253	. . . . . . 7 |- ( -oo +e +oo ) = ( +oo +e -oo )
57		simpr 110	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> B = +oo )
58	57	oveq2d 6029	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( -oo +e +oo ) )
59	57	oveq1d 6028	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( B +e -oo ) = ( +oo +e -oo ) )
60	56, 58, 59	3eqtr4a 2288	. . . . . 6 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( B +e -oo ) )
61		xaddmnf2 10074	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
62		xaddmnf1 10073	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( B +e -oo ) = -oo )
63	61, 62	eqtr4d 2265	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = ( B +e -oo ) )
64		xrpnfdc 10067	. . . . . . . 8 |- ( B e. RR* -> DECID B = +oo )
65		exmiddc 841	. . . . . . . 8 |- (DECID B = +oo -> ( B = +oo \/ -. B = +oo ) )
66	64, 65	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = +oo \/ -. B = +oo ) )
67		df-ne 2401	. . . . . . . 8 |- ( B =/= +oo <-> -. B = +oo )
68	67	orbi2i 767	. . . . . . 7 |- ( ( B = +oo \/ B =/= +oo ) <-> ( B = +oo \/ -. B = +oo ) )
69	66, 68	sylibr 134	. . . . . 6 |- ( B e. RR* -> ( B = +oo \/ B =/= +oo ) )
70	60, 63, 69	mpjaodan 803	. . . . 5 |- ( B e. RR* -> ( -oo +e B ) = ( B +e -oo ) )
71	70	adantl 277	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( -oo +e B ) = ( B +e -oo ) )
72		simpl 109	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> A = -oo )
73	72	oveq1d 6028	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( -oo +e B ) )
74	72	oveq2d 6029	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( B +e A ) = ( B +e -oo ) )
75	71, 73, 74	3eqtr4d 2272	. . 3 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
76	33, 55, 75	3jaoian 1339	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
77	1, 76	sylanb 284	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   \/ w3o 1001   = wceq 1395   e. wcel 2200   =/= wne 2400  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   + caddc 8025   +oocpnf 8201   -oocmnf 8202   RR*cxr 8203   +ecxad 9995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119  ax-addcom 8122  ax-rnegex 8131

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-xadd 9998

This theorem is referenced by:  xaddid2  10088  xleadd2a  10099  xltadd2  10102  xadd4d  10110  xrmaxaddlem  11811