Theorem infleinflem2 39587

Description: Lemma for infleinf 39588, when inf ( B , RR* , < ) = -oo. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
infleinflem2.a	|- ( ph -> A C_ RR* )
infleinflem2.b	|- ( ph -> B C_ RR* )
infleinflem2.r	|- ( ph -> R e. RR )
infleinflem2.x	|- ( ph -> X e. B )
infleinflem2.t	|- ( ph -> X < ( R - 2 ) )
infleinflem2.z	|- ( ph -> Z e. A )
infleinflem2.l	|- ( ph -> Z <_ ( X +e 1 ) )

Assertion

Ref	Expression
infleinflem2	|- ( ph -> Z < R )

Proof of Theorem infleinflem2

Step	Hyp	Ref	Expression
1		infleinflem2.r	. . . 4 |- ( ph -> R e. RR )
2	1	adantr 481	. . 3 |- ( ( ph /\ Z = -oo ) -> R e. RR )
3		simpr 477	. . 3 |- ( ( ph /\ Z = -oo ) -> Z = -oo )
4		simpr 477	. . . 4 |- ( ( R e. RR /\ Z = -oo ) -> Z = -oo )
5		mnflt 11957	. . . . 5 |- ( R e. RR -> -oo < R )
6	5	adantr 481	. . . 4 |- ( ( R e. RR /\ Z = -oo ) -> -oo < R )
7	4, 6	eqbrtrd 4675	. . 3 |- ( ( R e. RR /\ Z = -oo ) -> Z < R )
8	2, 3, 7	syl2anc 693	. 2 |- ( ( ph /\ Z = -oo ) -> Z < R )
9		simpl 473	. . 3 |- ( ( ph /\ -. Z = -oo ) -> ph )
10		neqne 2802	. . . 4 |- ( -. Z = -oo -> Z =/= -oo )
11	10	adantl 482	. . 3 |- ( ( ph /\ -. Z = -oo ) -> Z =/= -oo )
12	1	adantr 481	. . . . 5 |- ( ( ph /\ Z =/= -oo ) -> R e. RR )
13		id 22	. . . . . . . 8 |- ( ph -> ph )
14		infleinflem2.x	. . . . . . . 8 |- ( ph -> X e. B )
15		infleinflem2.b	. . . . . . . . 9 |- ( ph -> B C_ RR* )
16	15	sselda 3603	. . . . . . . 8 |- ( ( ph /\ X e. B ) -> X e. RR* )
17	13, 14, 16	syl2anc 693	. . . . . . 7 |- ( ph -> X e. RR* )
18	17	adantr 481	. . . . . 6 |- ( ( ph /\ Z =/= -oo ) -> X e. RR* )
19		infleinflem2.z	. . . . . . . . . 10 |- ( ph -> Z e. A )
20		infleinflem2.a	. . . . . . . . . . 11 |- ( ph -> A C_ RR* )
21	20	sselda 3603	. . . . . . . . . 10 |- ( ( ph /\ Z e. A ) -> Z e. RR* )
22	13, 19, 21	syl2anc 693	. . . . . . . . 9 |- ( ph -> Z e. RR* )
23	22	adantr 481	. . . . . . . 8 |- ( ( ph /\ Z =/= -oo ) -> Z e. RR* )
24		simpr 477	. . . . . . . 8 |- ( ( ph /\ Z =/= -oo ) -> Z =/= -oo )
25		pnfxr 10092	. . . . . . . . . . 11 |- +oo e. RR*
26	25	a1i 11	. . . . . . . . . 10 |- ( ph -> +oo e. RR* )
27		peano2rem 10348	. . . . . . . . . . . . 13 |- ( R e. RR -> ( R - 1 ) e. RR )
28	27	rexrd 10089	. . . . . . . . . . . 12 |- ( R e. RR -> ( R - 1 ) e. RR* )
29	1, 28	syl 17	. . . . . . . . . . 11 |- ( ph -> ( R - 1 ) e. RR* )
30	15, 14	sseldd 3604	. . . . . . . . . . . . 13 |- ( ph -> X e. RR* )
31		id 22	. . . . . . . . . . . . . 14 |- ( X e. RR* -> X e. RR* )
32		1re 10039	. . . . . . . . . . . . . . . 16 |- 1 e. RR
33	32	rexri 10097	. . . . . . . . . . . . . . 15 |- 1 e. RR*
34	33	a1i 11	. . . . . . . . . . . . . 14 |- ( X e. RR* -> 1 e. RR* )
35	31, 34	xaddcld 12131	. . . . . . . . . . . . 13 |- ( X e. RR* -> ( X +e 1 ) e. RR* )
36	30, 35	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( X +e 1 ) e. RR* )
37		infleinflem2.l	. . . . . . . . . . . 12 |- ( ph -> Z <_ ( X +e 1 ) )
38		infleinflem2.t	. . . . . . . . . . . . 13 |- ( ph -> X < ( R - 2 ) )
39		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 |- ( X = -oo -> ( X +e 1 ) = ( -oo +e 1 ) )
40		renepnf 10087	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 1 e. RR -> 1 =/= +oo )
41	32, 40	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 |- 1 =/= +oo
42		xaddmnf2 12060	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( 1 e. RR* /\ 1 =/= +oo ) -> ( -oo +e 1 ) = -oo )
43	33, 41, 42	mp2an 708	. . . . . . . . . . . . . . . . . . . 20 |- ( -oo +e 1 ) = -oo
44	43	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( X = -oo -> ( -oo +e 1 ) = -oo )
45	39, 44	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 |- ( X = -oo -> ( X +e 1 ) = -oo )
46	45	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( R e. RR /\ X = -oo ) -> ( X +e 1 ) = -oo )
47	27	mnfltd 11958	. . . . . . . . . . . . . . . . . 18 |- ( R e. RR -> -oo < ( R - 1 ) )
48	47	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( R e. RR /\ X = -oo ) -> -oo < ( R - 1 ) )
49	46, 48	eqbrtrd 4675	. . . . . . . . . . . . . . . 16 |- ( ( R e. RR /\ X = -oo ) -> ( X +e 1 ) < ( R - 1 ) )
50	49	adantlr 751	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RR /\ X e. RR* ) /\ X = -oo ) -> ( X +e 1 ) < ( R - 1 ) )
51	50	3adantl3 1219	. . . . . . . . . . . . . 14 |- ( ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) /\ X = -oo ) -> ( X +e 1 ) < ( R - 1 ) )
52		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) /\ -. X = -oo ) -> ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) )
53		simpl2 1065	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) /\ -. X = -oo ) -> X e. RR* )
54		neqne 2802	. . . . . . . . . . . . . . . . 17 |- ( -. X = -oo -> X =/= -oo )
55	54	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) /\ -. X = -oo ) -> X =/= -oo )
56		simp2 1062	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) -> X e. RR* )
57	25	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) -> +oo e. RR* )
58		id 22	. . . . . . . . . . . . . . . . . . . . . 22 |- ( R e. RR -> R e. RR )
59		2re 11090	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 e. RR
60	59	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( R e. RR -> 2 e. RR )
61	58, 60	resubcld 10458	. . . . . . . . . . . . . . . . . . . . 21 |- ( R e. RR -> ( R - 2 ) e. RR )
62	61	rexrd 10089	. . . . . . . . . . . . . . . . . . . 20 |- ( R e. RR -> ( R - 2 ) e. RR* )
63	62	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . 19 |- ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) -> ( R - 2 ) e. RR* )
64		simp3 1063	. . . . . . . . . . . . . . . . . . 19 |- ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) -> X < ( R - 2 ) )
65	61	ltpnfd 11955	. . . . . . . . . . . . . . . . . . . 20 |- ( R e. RR -> ( R - 2 ) < +oo )
66	65	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . 19 |- ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) -> ( R - 2 ) < +oo )
67	56, 63, 57, 64, 66	xrlttrd 11990	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) -> X < +oo )
68	56, 57, 67	xrltned 39573	. . . . . . . . . . . . . . . . 17 |- ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) -> X =/= +oo )
69	68	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) /\ -. X = -oo ) -> X =/= +oo )
70	53, 55, 69	xrred 39581	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) /\ -. X = -oo ) -> X e. RR )
71		id 22	. . . . . . . . . . . . . . . . . . . . 21 |- ( X e. RR -> X e. RR )
72	71	ad2antlr 763	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) -> X e. RR )
73	61	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) -> ( R - 2 ) e. RR )
74		1red 10055	. . . . . . . . . . . . . . . . . . . . 21 |- ( X e. RR -> 1 e. RR )
75	72, 74	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) -> 1 e. RR )
76		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) -> X < ( R - 2 ) )
77	72, 73, 75, 76	ltadd1dd 10638	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) -> ( X + 1 ) < ( ( R - 2 ) + 1 ) )
78		recn 10026	. . . . . . . . . . . . . . . . . . . . 21 |- ( R e. RR -> R e. CC )
79		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( R e. CC -> R e. CC )
80		2cnd 11093	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( R e. CC -> 2 e. CC )
81		1cnd 10056	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( R e. CC -> 1 e. CC )
82	79, 80, 81	subsubd 10420	. . . . . . . . . . . . . . . . . . . . . 22 |- ( R e. CC -> ( R - ( 2 - 1 ) ) = ( ( R - 2 ) + 1 ) )
83		2m1e1 11135	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 2 - 1 ) = 1
84	83	oveq2i 6661	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( R - ( 2 - 1 ) ) = ( R - 1 )
85	84	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( R e. CC -> ( R - ( 2 - 1 ) ) = ( R - 1 ) )
86	82, 85	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . . 21 |- ( R e. CC -> ( ( R - 2 ) + 1 ) = ( R - 1 ) )
87	78, 86	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( R e. RR -> ( ( R - 2 ) + 1 ) = ( R - 1 ) )
88	87	ad2antrr 762	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) -> ( ( R - 2 ) + 1 ) = ( R - 1 ) )
89	77, 88	breqtrd 4679	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) -> ( X + 1 ) < ( R - 1 ) )
90	71, 74	rexaddd 12065	. . . . . . . . . . . . . . . . . . . 20 |- ( X e. RR -> ( X +e 1 ) = ( X + 1 ) )
91	90	breq1d 4663	. . . . . . . . . . . . . . . . . . 19 |- ( X e. RR -> ( ( X +e 1 ) < ( R - 1 ) <-> ( X + 1 ) < ( R - 1 ) ) )
92	91	ad2antlr 763	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) -> ( ( X +e 1 ) < ( R - 1 ) <-> ( X + 1 ) < ( R - 1 ) ) )
93	89, 92	mpbird 247	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) -> ( X +e 1 ) < ( R - 1 ) )
94	93	an32s 846	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. RR /\ X < ( R - 2 ) ) /\ X e. RR ) -> ( X +e 1 ) < ( R - 1 ) )
95	94	3adantl2 1218	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) /\ X e. RR ) -> ( X +e 1 ) < ( R - 1 ) )
96	52, 70, 95	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) /\ -. X = -oo ) -> ( X +e 1 ) < ( R - 1 ) )
97	51, 96	pm2.61dan 832	. . . . . . . . . . . . 13 |- ( ( R e. RR /\ X e. RR* /\ X < ( R - 2 ) ) -> ( X +e 1 ) < ( R - 1 ) )
98	1, 30, 38, 97	syl3anc 1326	. . . . . . . . . . . 12 |- ( ph -> ( X +e 1 ) < ( R - 1 ) )
99	22, 36, 29, 37, 98	xrlelttrd 11991	. . . . . . . . . . 11 |- ( ph -> Z < ( R - 1 ) )
100	27	ltpnfd 11955	. . . . . . . . . . . 12 |- ( R e. RR -> ( R - 1 ) < +oo )
101	1, 100	syl 17	. . . . . . . . . . 11 |- ( ph -> ( R - 1 ) < +oo )
102	22, 29, 26, 99, 101	xrlttrd 11990	. . . . . . . . . 10 |- ( ph -> Z < +oo )
103	22, 26, 102	xrltned 39573	. . . . . . . . 9 |- ( ph -> Z =/= +oo )
104	103	adantr 481	. . . . . . . 8 |- ( ( ph /\ Z =/= -oo ) -> Z =/= +oo )
105	23, 24, 104	xrred 39581	. . . . . . 7 |- ( ( ph /\ Z =/= -oo ) -> Z e. RR )
106	37	adantr 481	. . . . . . 7 |- ( ( ph /\ Z =/= -oo ) -> Z <_ ( X +e 1 ) )
107		simpl3 1066	. . . . . . . . 9 |- ( ( ( Z e. RR /\ X e. RR* /\ Z <_ ( X +e 1 ) ) /\ X = -oo ) -> Z <_ ( X +e 1 ) )
108	45	adantl 482	. . . . . . . . . . . 12 |- ( ( Z e. RR /\ X = -oo ) -> ( X +e 1 ) = -oo )
109		mnflt 11957	. . . . . . . . . . . . 13 |- ( Z e. RR -> -oo < Z )
110	109	adantr 481	. . . . . . . . . . . 12 |- ( ( Z e. RR /\ X = -oo ) -> -oo < Z )
111	108, 110	eqbrtrd 4675	. . . . . . . . . . 11 |- ( ( Z e. RR /\ X = -oo ) -> ( X +e 1 ) < Z )
112		mnfxr 10096	. . . . . . . . . . . . 13 |- -oo e. RR*
113	108, 112	syl6eqel 2709	. . . . . . . . . . . 12 |- ( ( Z e. RR /\ X = -oo ) -> ( X +e 1 ) e. RR* )
114		rexr 10085	. . . . . . . . . . . . 13 |- ( Z e. RR -> Z e. RR* )
115	114	adantr 481	. . . . . . . . . . . 12 |- ( ( Z e. RR /\ X = -oo ) -> Z e. RR* )
116	113, 115	xrltnled 39579	. . . . . . . . . . 11 |- ( ( Z e. RR /\ X = -oo ) -> ( ( X +e 1 ) < Z <-> -. Z <_ ( X +e 1 ) ) )
117	111, 116	mpbid 222	. . . . . . . . . 10 |- ( ( Z e. RR /\ X = -oo ) -> -. Z <_ ( X +e 1 ) )
118	117	3ad2antl1 1223	. . . . . . . . 9 |- ( ( ( Z e. RR /\ X e. RR* /\ Z <_ ( X +e 1 ) ) /\ X = -oo ) -> -. Z <_ ( X +e 1 ) )
119	107, 118	pm2.65da 600	. . . . . . . 8 |- ( ( Z e. RR /\ X e. RR* /\ Z <_ ( X +e 1 ) ) -> -. X = -oo )
120	119	neqned 2801	. . . . . . 7 |- ( ( Z e. RR /\ X e. RR* /\ Z <_ ( X +e 1 ) ) -> X =/= -oo )
121	105, 18, 106, 120	syl3anc 1326	. . . . . 6 |- ( ( ph /\ Z =/= -oo ) -> X =/= -oo )
122	1, 17, 38, 68	syl3anc 1326	. . . . . . 7 |- ( ph -> X =/= +oo )
123	122	adantr 481	. . . . . 6 |- ( ( ph /\ Z =/= -oo ) -> X =/= +oo )
124	18, 121, 123	xrred 39581	. . . . 5 |- ( ( ph /\ Z =/= -oo ) -> X e. RR )
125	38	adantr 481	. . . . 5 |- ( ( ph /\ Z =/= -oo ) -> X < ( R - 2 ) )
126	12, 124, 125	jca31 557	. . . 4 |- ( ( ph /\ Z =/= -oo ) -> ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) )
127		simplr 792	. . . . 5 |- ( ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) /\ Z <_ ( X +e 1 ) ) -> Z e. RR )
128		simp-4r 807	. . . . . 6 |- ( ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) /\ Z <_ ( X +e 1 ) ) -> X e. RR )
129	71, 74	readdcld 10069	. . . . . . 7 |- ( X e. RR -> ( X + 1 ) e. RR )
130	90, 129	eqeltrd 2701	. . . . . 6 |- ( X e. RR -> ( X +e 1 ) e. RR )
131	128, 130	syl 17	. . . . 5 |- ( ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) /\ Z <_ ( X +e 1 ) ) -> ( X +e 1 ) e. RR )
132	58	ad4antr 768	. . . . 5 |- ( ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) /\ Z <_ ( X +e 1 ) ) -> R e. RR )
133		simpr 477	. . . . 5 |- ( ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) /\ Z <_ ( X +e 1 ) ) -> Z <_ ( X +e 1 ) )
134	130	ad3antlr 767	. . . . . . 7 |- ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) -> ( X +e 1 ) e. RR )
135	27	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) -> ( R - 1 ) e. RR )
136	58	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) -> R e. RR )
137	93	adantr 481	. . . . . . 7 |- ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) -> ( X +e 1 ) < ( R - 1 ) )
138	136	ltm1d 10956	. . . . . . 7 |- ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) -> ( R - 1 ) < R )
139	134, 135, 136, 137, 138	lttrd 10198	. . . . . 6 |- ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) -> ( X +e 1 ) < R )
140	139	adantr 481	. . . . 5 |- ( ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) /\ Z <_ ( X +e 1 ) ) -> ( X +e 1 ) < R )
141	127, 131, 132, 133, 140	lelttrd 10195	. . . 4 |- ( ( ( ( ( R e. RR /\ X e. RR ) /\ X < ( R - 2 ) ) /\ Z e. RR ) /\ Z <_ ( X +e 1 ) ) -> Z < R )
142	126, 105, 106, 141	syl21anc 1325	. . 3 |- ( ( ph /\ Z =/= -oo ) -> Z < R )
143	9, 11, 142	syl2anc 693	. 2 |- ( ( ph /\ -. Z = -oo ) -> Z < R )
144	8, 143	pm2.61dan 832	1 |- ( ph -> Z < R )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   2c2 11070   +ecxad 11944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-2 11079  df-xadd 11947

This theorem is referenced by:  infleinf  39588