Theorem xaddcom 9485

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 9404	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9404	. . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		recn 7625	. . . . . . 7 |- ( A e. RR -> A e. CC )
4		recn 7625	. . . . . . 7 |- ( B e. RR -> B e. CC )
5		addcom 7770	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
6	3, 4, 5	syl2an 285	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) )
7		rexadd 9476	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
8		rexadd 9476	. . . . . . 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 2142	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( B +e A ) )
11		oveq2 5714	. . . . . . 7 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
12		rexr 7683	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
13		renemnf 7686	. . . . . . . 8 |- ( A e. RR -> A =/= -oo )
14		xaddpnf1 9470	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
15	12, 13, 14	syl2anc 406	. . . . . . 7 |- ( A e. RR -> ( A +e +oo ) = +oo )
16	11, 15	sylan9eqr 2154	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
17		oveq1 5713	. . . . . . 7 |- ( B = +oo -> ( B +e A ) = ( +oo +e A ) )
18		xaddpnf2 9471	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= -oo ) -> ( +oo +e A ) = +oo )
19	12, 13, 18	syl2anc 406	. . . . . . 7 |- ( A e. RR -> ( +oo +e A ) = +oo )
20	17, 19	sylan9eqr 2154	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( B +e A ) = +oo )
21	16, 20	eqtr4d 2135	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = ( B +e A ) )
22		oveq2 5714	. . . . . . 7 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
23		renepnf 7685	. . . . . . . 8 |- ( A e. RR -> A =/= +oo )
24		xaddmnf1 9472	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= +oo ) -> ( A +e -oo ) = -oo )
25	12, 23, 24	syl2anc 406	. . . . . . 7 |- ( A e. RR -> ( A +e -oo ) = -oo )
26	22, 25	sylan9eqr 2154	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
27		oveq1 5713	. . . . . . 7 |- ( B = -oo -> ( B +e A ) = ( -oo +e A ) )
28		xaddmnf2 9473	. . . . . . . 8 |- ( ( A e. RR* /\ A =/= +oo ) -> ( -oo +e A ) = -oo )
29	12, 23, 28	syl2anc 406	. . . . . . 7 |- ( A e. RR -> ( -oo +e A ) = -oo )
30	27, 29	sylan9eqr 2154	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( B +e A ) = -oo )
31	26, 30	eqtr4d 2135	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = ( B +e A ) )
32	10, 21, 31	3jaodan 1252	. . . 4 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A +e B ) = ( B +e A ) )
33	2, 32	sylan2b 283	. . 3 |- ( ( A e. RR /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
34		pnfaddmnf 9474	. . . . . . . 8 |- ( +oo +e -oo ) = 0
35		mnfaddpnf 9475	. . . . . . . 8 |- ( -oo +e +oo ) = 0
36	34, 35	eqtr4i 2123	. . . . . . 7 |- ( +oo +e -oo ) = ( -oo +e +oo )
37		simpr 109	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> B = -oo )
38	37	oveq2d 5722	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( +oo +e -oo ) )
39	37	oveq1d 5721	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( B +e +oo ) = ( -oo +e +oo ) )
40	36, 38, 39	3eqtr4a 2158	. . . . . 6 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( B +e +oo ) )
41		xaddpnf2 9471	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
42		xaddpnf1 9470	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B +e +oo ) = +oo )
43	41, 42	eqtr4d 2135	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = ( B +e +oo ) )
44		xrmnfdc 9467	. . . . . . . 8 |- ( B e. RR* -> DECID B = -oo )
45		exmiddc 788	. . . . . . . 8 |- (DECID B = -oo -> ( B = -oo \/ -. B = -oo ) )
46	44, 45	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = -oo \/ -. B = -oo ) )
47		df-ne 2268	. . . . . . . 8 |- ( B =/= -oo <-> -. B = -oo )
48	47	orbi2i 720	. . . . . . 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 753	. . . . 5 |- ( B e. RR* -> ( +oo +e B ) = ( B +e +oo ) )
51	50	adantl 273	. . . 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 5721	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( +oo +e B ) )
54	52	oveq2d 5722	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( B +e A ) = ( B +e +oo ) )
55	51, 53, 54	3eqtr4d 2142	. . 3 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
56	35, 34	eqtr4i 2123	. . . . . . 7 |- ( -oo +e +oo ) = ( +oo +e -oo )
57		simpr 109	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> B = +oo )
58	57	oveq2d 5722	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( -oo +e +oo ) )
59	57	oveq1d 5721	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( B +e -oo ) = ( +oo +e -oo ) )
60	56, 58, 59	3eqtr4a 2158	. . . . . 6 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( B +e -oo ) )
61		xaddmnf2 9473	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
62		xaddmnf1 9472	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( B +e -oo ) = -oo )
63	61, 62	eqtr4d 2135	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = ( B +e -oo ) )
64		xrpnfdc 9466	. . . . . . . 8 |- ( B e. RR* -> DECID B = +oo )
65		exmiddc 788	. . . . . . . 8 |- (DECID B = +oo -> ( B = +oo \/ -. B = +oo ) )
66	64, 65	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = +oo \/ -. B = +oo ) )
67		df-ne 2268	. . . . . . . 8 |- ( B =/= +oo <-> -. B = +oo )
68	67	orbi2i 720	. . . . . . 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 753	. . . . 5 |- ( B e. RR* -> ( -oo +e B ) = ( B +e -oo ) )
71	70	adantl 273	. . . 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 5721	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( -oo +e B ) )
74	72	oveq2d 5722	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( B +e A ) = ( B +e -oo ) )
75	71, 73, 74	3eqtr4d 2142	. . 3 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
76	33, 55, 75	3jaoian 1251	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
77	1, 76	sylanb 280	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 670  DECID wdc 786   \/ w3o 929   = wceq 1299   e. wcel 1448   =/= wne 2267  (class class class)co 5706   CCcc 7498   RRcr 7499   0cc0 7500   + caddc 7503   +oocpnf 7669   -oocmnf 7670   RR*cxr 7671   +ecxad 9398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1re 7589  ax-addrcl 7592  ax-addcom 7595  ax-rnegex 7604

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-xadd 9401

This theorem is referenced by:  xaddid2  9487  xleadd2a  9498  xltadd2  9501  xadd4d  9509  xrmaxaddlem  10868