Theorem xaddass 9935

Description: Associativity of extended real addition. The correct condition here is "it is not the case that both +oo and -oo appear as one of A , B , C, i.e. -. { +oo , -oo } C_ { A , B , C }", but this condition is difficult to work with, so we break the theorem into two parts: this one, where -oo is not present in A , B , C, and xaddass2 9936, where +oo is not present. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xaddass	|- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )

Proof of Theorem xaddass

Step	Hyp	Ref	Expression
1		recn 8005	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
2		recn 8005	. . . . . . . . . 10 |- ( B e. RR -> B e. CC )
3		recn 8005	. . . . . . . . . 10 |- ( C e. RR -> C e. CC )
4		addass 8002	. . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5	1, 2, 3, 4	syl3an 1291	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
6	5	3expa 1205	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
7		readdcl 7998	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
8		rexadd 9918	. . . . . . . . 9 |- ( ( ( A + B ) e. RR /\ C e. RR ) -> ( ( A + B ) +e C ) = ( ( A + B ) + C ) )
9	7, 8	sylan 283	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A + B ) +e C ) = ( ( A + B ) + C ) )
10		readdcl 7998	. . . . . . . . . 10 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
11		rexadd 9918	. . . . . . . . . 10 |- ( ( A e. RR /\ ( B + C ) e. RR ) -> ( A +e ( B + C ) ) = ( A + ( B + C ) ) )
12	10, 11	sylan2 286	. . . . . . . . 9 |- ( ( A e. RR /\ ( B e. RR /\ C e. RR ) ) -> ( A +e ( B + C ) ) = ( A + ( B + C ) ) )
13	12	anassrs 400	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A +e ( B + C ) ) = ( A + ( B + C ) ) )
14	6, 9, 13	3eqtr4d 2236	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A + B ) +e C ) = ( A +e ( B + C ) ) )
15		rexadd 9918	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
16	15	adantr 276	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A +e B ) = ( A + B ) )
17	16	oveq1d 5933	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A +e B ) +e C ) = ( ( A + B ) +e C ) )
18		rexadd 9918	. . . . . . . . 9 |- ( ( B e. RR /\ C e. RR ) -> ( B +e C ) = ( B + C ) )
19	18	adantll 476	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( B +e C ) = ( B + C ) )
20	19	oveq2d 5934	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A +e ( B +e C ) ) = ( A +e ( B + C ) ) )
21	14, 17, 20	3eqtr4d 2236	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
22	21	adantll 476	. . . . 5 |- ( ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ ( A e. RR /\ B e. RR ) ) /\ C e. RR ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
23		oveq2 5926	. . . . . . . . 9 |- ( C = +oo -> ( ( A +e B ) +e C ) = ( ( A +e B ) +e +oo ) )
24		simp1l 1023	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> A e. RR* )
25		simp2l 1025	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> B e. RR* )
26		xaddcl 9926	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* )
27	24, 25, 26	syl2anc 411	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A +e B ) e. RR* )
28		xaddnemnf 9923	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) ) -> ( A +e B ) =/= -oo )
29	28	3adant3 1019	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A +e B ) =/= -oo )
30		xaddpnf1 9912	. . . . . . . . . 10 |- ( ( ( A +e B ) e. RR* /\ ( A +e B ) =/= -oo ) -> ( ( A +e B ) +e +oo ) = +oo )
31	27, 29, 30	syl2anc 411	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( ( A +e B ) +e +oo ) = +oo )
32	23, 31	sylan9eqr 2248	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( ( A +e B ) +e C ) = +oo )
33		xaddpnf1 9912	. . . . . . . . . 10 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
34	33	3ad2ant1 1020	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A +e +oo ) = +oo )
35	34	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( A +e +oo ) = +oo )
36	32, 35	eqtr4d 2229	. . . . . . 7 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( ( A +e B ) +e C ) = ( A +e +oo ) )
37		oveq2 5926	. . . . . . . . 9 |- ( C = +oo -> ( B +e C ) = ( B +e +oo ) )
38		xaddpnf1 9912	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= -oo ) -> ( B +e +oo ) = +oo )
39	38	3ad2ant2 1021	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( B +e +oo ) = +oo )
40	37, 39	sylan9eqr 2248	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( B +e C ) = +oo )
41	40	oveq2d 5934	. . . . . . 7 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( A +e ( B +e C ) ) = ( A +e +oo ) )
42	36, 41	eqtr4d 2229	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ C = +oo ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
43	42	adantlr 477	. . . . 5 |- ( ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ ( A e. RR /\ B e. RR ) ) /\ C = +oo ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
44		simp3 1001	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( C e. RR* /\ C =/= -oo ) )
45		xrnemnf 9843	. . . . . . 7 |- ( ( C e. RR* /\ C =/= -oo ) <-> ( C e. RR \/ C = +oo ) )
46	44, 45	sylib 122	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( C e. RR \/ C = +oo ) )
47	46	adantr 276	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ ( A e. RR /\ B e. RR ) ) -> ( C e. RR \/ C = +oo ) )
48	22, 43, 47	mpjaodan 799	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ ( A e. RR /\ B e. RR ) ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
49	48	anassrs 400	. . 3 |- ( ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A e. RR ) /\ B e. RR ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
50		xaddpnf2 9913	. . . . . . . 8 |- ( ( C e. RR* /\ C =/= -oo ) -> ( +oo +e C ) = +oo )
51	50	3ad2ant3 1022	. . . . . . 7 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( +oo +e C ) = +oo )
52	51, 34	eqtr4d 2229	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( +oo +e C ) = ( A +e +oo ) )
53	52	adantr 276	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( +oo +e C ) = ( A +e +oo ) )
54		oveq2 5926	. . . . . . 7 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
55	54, 34	sylan9eqr 2248	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( A +e B ) = +oo )
56	55	oveq1d 5933	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( ( A +e B ) +e C ) = ( +oo +e C ) )
57		oveq1 5925	. . . . . . 7 |- ( B = +oo -> ( B +e C ) = ( +oo +e C ) )
58	57, 51	sylan9eqr 2248	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( B +e C ) = +oo )
59	58	oveq2d 5934	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( A +e ( B +e C ) ) = ( A +e +oo ) )
60	53, 56, 59	3eqtr4d 2236	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ B = +oo ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
61	60	adantlr 477	. . 3 |- ( ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A e. RR ) /\ B = +oo ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
62		simpl2 1003	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A e. RR ) -> ( B e. RR* /\ B =/= -oo ) )
63		xrnemnf 9843	. . . 4 |- ( ( B e. RR* /\ B =/= -oo ) <-> ( B e. RR \/ B = +oo ) )
64	62, 63	sylib 122	. . 3 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A e. RR ) -> ( B e. RR \/ B = +oo ) )
65	49, 61, 64	mpjaodan 799	. 2 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A e. RR ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
66		simpl3 1004	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( C e. RR* /\ C =/= -oo ) )
67	66, 50	syl 14	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( +oo +e C ) = +oo )
68		simpl2l 1052	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> B e. RR* )
69		simpl3l 1054	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> C e. RR* )
70		xaddcl 9926	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> ( B +e C ) e. RR* )
71	68, 69, 70	syl2anc 411	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( B +e C ) e. RR* )
72		simpl2 1003	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( B e. RR* /\ B =/= -oo ) )
73		xaddnemnf 9923	. . . . . 6 |- ( ( ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( B +e C ) =/= -oo )
74	72, 66, 73	syl2anc 411	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( B +e C ) =/= -oo )
75		xaddpnf2 9913	. . . . 5 |- ( ( ( B +e C ) e. RR* /\ ( B +e C ) =/= -oo ) -> ( +oo +e ( B +e C ) ) = +oo )
76	71, 74, 75	syl2anc 411	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( +oo +e ( B +e C ) ) = +oo )
77	67, 76	eqtr4d 2229	. . 3 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( +oo +e C ) = ( +oo +e ( B +e C ) ) )
78		simpr 110	. . . . . 6 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> A = +oo )
79	78	oveq1d 5933	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( A +e B ) = ( +oo +e B ) )
80		xaddpnf2 9913	. . . . . 6 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
81	72, 80	syl 14	. . . . 5 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( +oo +e B ) = +oo )
82	79, 81	eqtrd 2226	. . . 4 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( A +e B ) = +oo )
83	82	oveq1d 5933	. . 3 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( ( A +e B ) +e C ) = ( +oo +e C ) )
84	78	oveq1d 5933	. . 3 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( A +e ( B +e C ) ) = ( +oo +e ( B +e C ) ) )
85	77, 83, 84	3eqtr4d 2236	. 2 |- ( ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) /\ A = +oo ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )
86		simp1 999	. . 3 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A e. RR* /\ A =/= -oo ) )
87		xrnemnf 9843	. . 3 |- ( ( A e. RR* /\ A =/= -oo ) <-> ( A e. RR \/ A = +oo ) )
88	86, 87	sylib 122	. 2 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( A e. RR \/ A = +oo ) )
89	65, 85, 88	mpjaodan 799	1 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) ) -> ( ( A +e B ) +e C ) = ( A +e ( B +e C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   =/= wne 2364  (class class class)co 5918   CCcc 7870   RRcr 7871   + caddc 7875   +oocpnf 8051   -oocmnf 8052   RR*cxr 8053   +ecxad 9836

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969  ax-addass 7974  ax-rnegex 7981

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-xr 8058  df-xadd 9839

This theorem is referenced by:  xaddass2  9936  xpncan  9937  xadd4d  9951