Theorem xaddcom 9644

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 9563	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9563	. . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		recn 7753	. . . . . . 7 |- ( A e. RR -> A e. CC )
4		recn 7753	. . . . . . 7 |- ( B e. RR -> B e. CC )
5		addcom 7899	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
6	3, 4, 5	syl2an 287	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) )
7		rexadd 9635	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
8		rexadd 9635	. . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( B +e A ) = ( B + A ) )
9	8	ancoms 266	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( B +e A ) = ( B + A ) )
10	6, 7, 9	3eqtr4d 2182	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( B +e A ) )
11		oveq2 5782	. . . . . . 7 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
12		rexr 7811	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
13		renemnf 7814	. . . . . . . 8 |- ( A e. RR -> A =/= -oo )
14		xaddpnf1 9629	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
15	12, 13, 14	syl2anc 408	. . . . . . 7 |- ( A e. RR -> ( A +e +oo ) = +oo )
16	11, 15	sylan9eqr 2194	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
17		oveq1 5781	. . . . . . 7 |- ( B = +oo -> ( B +e A ) = ( +oo +e A ) )
18		xaddpnf2 9630	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= -oo ) -> ( +oo +e A ) = +oo )
19	12, 13, 18	syl2anc 408	. . . . . . 7 |- ( A e. RR -> ( +oo +e A ) = +oo )
20	17, 19	sylan9eqr 2194	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( B +e A ) = +oo )
21	16, 20	eqtr4d 2175	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = ( B +e A ) )
22		oveq2 5782	. . . . . . 7 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
23		renepnf 7813	. . . . . . . 8 |- ( A e. RR -> A =/= +oo )
24		xaddmnf1 9631	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
25	12, 23, 24	syl2anc 408	. . . . . . 7 |- ( A e. RR -> ( A +e -oo ) = -oo )
26	22, 25	sylan9eqr 2194	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
27		oveq1 5781	. . . . . . 7 |- ( B = -oo -> ( B +e A ) = ( -oo +e A ) )
28		xaddmnf2 9632	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= +oo ) -> ( -oo +e A ) = -oo )
29	12, 23, 28	syl2anc 408	. . . . . . 7 |- ( A e. RR -> ( -oo +e A ) = -oo )
30	27, 29	sylan9eqr 2194	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( B +e A ) = -oo )
31	26, 30	eqtr4d 2175	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = ( B +e A ) )
32	10, 21, 31	3jaodan 1284	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A +e B ) = ( B +e A ) )
33	2, 32	sylan2b 285	. . 3 |- ( ( A e. RR /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
34		pnfaddmnf 9633	. . . . . . . 8 |- ( +oo +e -oo ) = 0
35		mnfaddpnf 9634	. . . . . . . 8 |- ( -oo +e +oo ) = 0
36	34, 35	eqtr4i 2163	. . . . . . 7 |- ( +oo +e -oo ) = ( -oo +e +oo )
37		simpr 109	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> B = -oo )
38	37	oveq2d 5790	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( +oo +e -oo ) )
39	37	oveq1d 5789	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( B +e +oo ) = ( -oo +e +oo ) )
40	36, 38, 39	3eqtr4a 2198	. . . . . 6 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( B +e +oo ) )
41		xaddpnf2 9630	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
42		xaddpnf1 9629	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B +e +oo ) = +oo )
43	41, 42	eqtr4d 2175	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = ( B +e +oo ) )
44		xrmnfdc 9626	. . . . . . . 8 |- ( B e. RR* -> DECID B = -oo )
45		exmiddc 821	. . . . . . . 8 |- (DECID B = -oo -> ( B = -oo \/ -. B = -oo ) )
46	44, 45	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = -oo \/ -. B = -oo ) )
47		df-ne 2309	. . . . . . . 8 |- ( B =/= -oo <-> -. B = -oo )
48	47	orbi2i 751	. . . . . . 7 |- ( ( B = -oo \/ B =/= -oo ) <-> ( B = -oo \/ -. B = -oo ) )
49	46, 48	sylibr 133	. . . . . 6 |- ( B e. RR* -> ( B = -oo \/ B =/= -oo ) )
50	40, 43, 49	mpjaodan 787	. . . . 5 |- ( B e. RR* -> ( +oo +e B ) = ( B +e +oo ) )
51	50	adantl 275	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( +oo +e B ) = ( B +e +oo ) )
52		simpl 108	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> A = +oo )
53	52	oveq1d 5789	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( +oo +e B ) )
54	52	oveq2d 5790	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( B +e A ) = ( B +e +oo ) )
55	51, 53, 54	3eqtr4d 2182	. . 3 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
56	35, 34	eqtr4i 2163	. . . . . . 7 |- ( -oo +e +oo ) = ( +oo +e -oo )
57		simpr 109	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> B = +oo )
58	57	oveq2d 5790	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( -oo +e +oo ) )
59	57	oveq1d 5789	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( B +e -oo ) = ( +oo +e -oo ) )
60	56, 58, 59	3eqtr4a 2198	. . . . . 6 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( B +e -oo ) )
61		xaddmnf2 9632	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
62		xaddmnf1 9631	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( B +e -oo ) = -oo )
63	61, 62	eqtr4d 2175	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = ( B +e -oo ) )
64		xrpnfdc 9625	. . . . . . . 8 |- ( B e. RR* -> DECID B = +oo )
65		exmiddc 821	. . . . . . . 8 |- (DECID B = +oo -> ( B = +oo \/ -. B = +oo ) )
66	64, 65	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = +oo \/ -. B = +oo ) )
67		df-ne 2309	. . . . . . . 8 |- ( B =/= +oo <-> -. B = +oo )
68	67	orbi2i 751	. . . . . . 7 |- ( ( B = +oo \/ B =/= +oo ) <-> ( B = +oo \/ -. B = +oo ) )
69	66, 68	sylibr 133	. . . . . 6 |- ( B e. RR* -> ( B = +oo \/ B =/= +oo ) )
70	60, 63, 69	mpjaodan 787	. . . . 5 |- ( B e. RR* -> ( -oo +e B ) = ( B +e -oo ) )
71	70	adantl 275	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( -oo +e B ) = ( B +e -oo ) )
72		simpl 108	. . . . 5 |- ( ( A = -oo /\ B e. RR* ) -> A = -oo )
73	72	oveq1d 5789	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( -oo +e B ) )
74	72	oveq2d 5790	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( B +e A ) = ( B +e -oo ) )
75	71, 73, 74	3eqtr4d 2182	. . 3 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
76	33, 55, 75	3jaoian 1283	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
77	1, 76	sylanb 282	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697  DECID wdc 819   \/ w3o 961   = wceq 1331   e. wcel 1480   =/= wne 2308  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   + caddc 7623   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799   +ecxad 9557

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1re 7714  ax-addrcl 7717  ax-addcom 7720  ax-rnegex 7729

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-xadd 9560

This theorem is referenced by:  xaddid2  9646  xleadd2a  9657  xltadd2  9660  xadd4d  9668  xrmaxaddlem  11029