Theorem xaddcom 9953

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 9868	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9868	. . . 4 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		recn 8029	. . . . . . 7 |- ( A e. RR -> A e. CC )
4		recn 8029	. . . . . . 7 |- ( B e. RR -> B e. CC )
5		addcom 8180	. . . . . . 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 9944	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
8		rexadd 9944	. . . . . . 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 2239	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( B +e A ) )
11		oveq2 5933	. . . . . . 7 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
12		rexr 8089	. . . . . . . 8 |- ( A e. RR -> A e. RR* )
13		renemnf 8092	. . . . . . . 8 |- ( A e. RR -> A =/= -oo )
14		xaddpnf1 9938	. . . . . . . 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 2251	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = +oo )
17		oveq1 5932	. . . . . . 7 |- ( B = +oo -> ( B +e A ) = ( +oo +e A ) )
18		xaddpnf2 9939	. . . . . . . 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 2251	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( B +e A ) = +oo )
21	16, 20	eqtr4d 2232	. . . . 5 |- ( ( A e. RR /\ B = +oo ) -> ( A +e B ) = ( B +e A ) )
22		oveq2 5933	. . . . . . 7 |- ( B = -oo -> ( A +e B ) = ( A +e -oo ) )
23		renepnf 8091	. . . . . . . 8 |- ( A e. RR -> A =/= +oo )
24		xaddmnf1 9940	. . . . . . . 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 2251	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = -oo )
27		oveq1 5932	. . . . . . 7 |- ( B = -oo -> ( B +e A ) = ( -oo +e A ) )
28		xaddmnf2 9941	. . . . . . . 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 2251	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( B +e A ) = -oo )
31	26, 30	eqtr4d 2232	. . . . 5 |- ( ( A e. RR /\ B = -oo ) -> ( A +e B ) = ( B +e A ) )
32	10, 21, 31	3jaodan 1317	. . . 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 9942	. . . . . . . 8 |- ( +oo +e -oo ) = 0
35		mnfaddpnf 9943	. . . . . . . 8 |- ( -oo +e +oo ) = 0
36	34, 35	eqtr4i 2220	. . . . . . 7 |- ( +oo +e -oo ) = ( -oo +e +oo )
37		simpr 110	. . . . . . . 8 |- ( ( B e. RR* /\ B = -oo ) -> B = -oo )
38	37	oveq2d 5941	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( +oo +e -oo ) )
39	37	oveq1d 5940	. . . . . . 7 |- ( ( B e. RR* /\ B = -oo ) -> ( B +e +oo ) = ( -oo +e +oo ) )
40	36, 38, 39	3eqtr4a 2255	. . . . . 6 |- ( ( B e. RR* /\ B = -oo ) -> ( +oo +e B ) = ( B +e +oo ) )
41		xaddpnf2 9939	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
42		xaddpnf1 9938	. . . . . . 7 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B +e +oo ) = +oo )
43	41, 42	eqtr4d 2232	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = ( B +e +oo ) )
44		xrmnfdc 9935	. . . . . . . 8 |- ( B e. RR* -> DECID B = -oo )
45		exmiddc 837	. . . . . . . 8 |- (DECID B = -oo -> ( B = -oo \/ -. B = -oo ) )
46	44, 45	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = -oo \/ -. B = -oo ) )
47		df-ne 2368	. . . . . . . 8 |- ( B =/= -oo <-> -. B = -oo )
48	47	orbi2i 763	. . . . . . 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 799	. . . . 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 5940	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( +oo +e B ) )
54	52	oveq2d 5941	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( B +e A ) = ( B +e +oo ) )
55	51, 53, 54	3eqtr4d 2239	. . 3 |- ( ( A = +oo /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
56	35, 34	eqtr4i 2220	. . . . . . 7 |- ( -oo +e +oo ) = ( +oo +e -oo )
57		simpr 110	. . . . . . . 8 |- ( ( B e. RR* /\ B = +oo ) -> B = +oo )
58	57	oveq2d 5941	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( -oo +e +oo ) )
59	57	oveq1d 5940	. . . . . . 7 |- ( ( B e. RR* /\ B = +oo ) -> ( B +e -oo ) = ( +oo +e -oo ) )
60	56, 58, 59	3eqtr4a 2255	. . . . . 6 |- ( ( B e. RR* /\ B = +oo ) -> ( -oo +e B ) = ( B +e -oo ) )
61		xaddmnf2 9941	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
62		xaddmnf1 9940	. . . . . . 7 |- ( ( B e. RR* /\ B =/= +oo ) -> ( B +e -oo ) = -oo )
63	61, 62	eqtr4d 2232	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = ( B +e -oo ) )
64		xrpnfdc 9934	. . . . . . . 8 |- ( B e. RR* -> DECID B = +oo )
65		exmiddc 837	. . . . . . . 8 |- (DECID B = +oo -> ( B = +oo \/ -. B = +oo ) )
66	64, 65	syl 14	. . . . . . 7 |- ( B e. RR* -> ( B = +oo \/ -. B = +oo ) )
67		df-ne 2368	. . . . . . . 8 |- ( B =/= +oo <-> -. B = +oo )
68	67	orbi2i 763	. . . . . . 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 799	. . . . 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 5940	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( -oo +e B ) )
74	72	oveq2d 5941	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( B +e A ) = ( B +e -oo ) )
75	71, 73, 74	3eqtr4d 2239	. . 3 |- ( ( A = -oo /\ B e. RR* ) -> ( A +e B ) = ( B +e A ) )
76	33, 55, 75	3jaoian 1316	. 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 709  DECID wdc 835   \/ w3o 979   = wceq 1364   e. wcel 2167   =/= wne 2367  (class class class)co 5925   CCcc 7894   RRcr 7895   0cc0 7896   + caddc 7899   +oocpnf 8075   -oocmnf 8076   RR*cxr 8077   +ecxad 9862

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993  ax-addcom 7996  ax-rnegex 8005

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-xadd 9865

This theorem is referenced by:  xaddid2  9955  xleadd2a  9966  xltadd2  9969  xadd4d  9977  xrmaxaddlem  11442