Theorem pilem2 24206

Description: Lemma for pire 24210, pigt2lt4 24208 and sinpi 24209. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by AV, 14-Sep-2020.)

Hypotheses

Ref	Expression
pilem.1	|- ( ph -> A e. ( 2 (,) 4 ) )
pilem.2	|- ( ph -> B e. RR+ )
pilem.3	|- ( ph -> ( sin ` A ) = 0 )
pilem.4	|- ( ph -> ( sin ` B ) = 0 )
pilem.5	|- ( ph -> pi < A )

Assertion

Ref	Expression
pilem2	|- ( ph -> ( ( pi + A ) / 2 ) <_ B )

Proof of Theorem pilem2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pi 14803	. . . 4 |- pi = inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < )
2		inss1 3833	. . . . . . 7 |- ( RR+ i^i ( `' sin " { 0 } ) ) C_ RR+
3		rpssre 11843	. . . . . . 7 |- RR+ C_ RR
4	2, 3	sstri 3612	. . . . . 6 |- ( RR+ i^i ( `' sin " { 0 } ) ) C_ RR
5	4	a1i 11	. . . . 5 |- ( ph -> ( RR+ i^i ( `' sin " { 0 } ) ) C_ RR )
6		0re 10040	. . . . . . 7 |- 0 e. RR
7	2	sseli 3599	. . . . . . . . 9 |- ( y e. ( RR+ i^i ( `' sin " { 0 } ) ) -> y e. RR+ )
8	7	rpge0d 11876	. . . . . . . 8 |- ( y e. ( RR+ i^i ( `' sin " { 0 } ) ) -> 0 <_ y )
9	8	rgen 2922	. . . . . . 7 |- A. y e. ( RR+ i^i ( `' sin " { 0 } ) ) 0 <_ y
10		breq1 4656	. . . . . . . . 9 |- ( x = 0 -> ( x <_ y <-> 0 <_ y ) )
11	10	ralbidv 2986	. . . . . . . 8 |- ( x = 0 -> ( A. y e. ( RR+ i^i ( `' sin " { 0 } ) ) x <_ y <-> A. y e. ( RR+ i^i ( `' sin " { 0 } ) ) 0 <_ y ) )
12	11	rspcev 3309	. . . . . . 7 |- ( ( 0 e. RR /\ A. y e. ( RR+ i^i ( `' sin " { 0 } ) ) 0 <_ y ) -> E. x e. RR A. y e. ( RR+ i^i ( `' sin " { 0 } ) ) x <_ y )
13	6, 9, 12	mp2an 708	. . . . . 6 |- E. x e. RR A. y e. ( RR+ i^i ( `' sin " { 0 } ) ) x <_ y
14	13	a1i 11	. . . . 5 |- ( ph -> E. x e. RR A. y e. ( RR+ i^i ( `' sin " { 0 } ) ) x <_ y )
15		2re 11090	. . . . . . . . 9 |- 2 e. RR
16		pilem.2	. . . . . . . . . 10 |- ( ph -> B e. RR+ )
17	16	rpred 11872	. . . . . . . . 9 |- ( ph -> B e. RR )
18		remulcl 10021	. . . . . . . . 9 |- ( ( 2 e. RR /\ B e. RR ) -> ( 2 x. B ) e. RR )
19	15, 17, 18	sylancr 695	. . . . . . . 8 |- ( ph -> ( 2 x. B ) e. RR )
20		pilem.1	. . . . . . . . 9 |- ( ph -> A e. ( 2 (,) 4 ) )
21		elioore 12205	. . . . . . . . 9 |- ( A e. ( 2 (,) 4 ) -> A e. RR )
22	20, 21	syl 17	. . . . . . . 8 |- ( ph -> A e. RR )
23	19, 22	resubcld 10458	. . . . . . 7 |- ( ph -> ( ( 2 x. B ) - A ) e. RR )
24		4re 11097	. . . . . . . . . 10 |- 4 e. RR
25	24	a1i 11	. . . . . . . . 9 |- ( ph -> 4 e. RR )
26		eliooord 12233	. . . . . . . . . . 11 |- ( A e. ( 2 (,) 4 ) -> ( 2 < A /\ A < 4 ) )
27	20, 26	syl 17	. . . . . . . . . 10 |- ( ph -> ( 2 < A /\ A < 4 ) )
28	27	simprd 479	. . . . . . . . 9 |- ( ph -> A < 4 )
29		2t2e4 11177	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
30	15	a1i 11	. . . . . . . . . . . 12 |- ( ph -> 2 e. RR )
31		0red 10041	. . . . . . . . . . . . . . . . 17 |- ( ph -> 0 e. RR )
32		2pos 11112	. . . . . . . . . . . . . . . . . 18 |- 0 < 2
33	32	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> 0 < 2 )
34	27	simpld 475	. . . . . . . . . . . . . . . . 17 |- ( ph -> 2 < A )
35	31, 30, 22, 33, 34	lttrd 10198	. . . . . . . . . . . . . . . 16 |- ( ph -> 0 < A )
36	22, 35	elrpd 11869	. . . . . . . . . . . . . . 15 |- ( ph -> A e. RR+ )
37		pilem.3	. . . . . . . . . . . . . . 15 |- ( ph -> ( sin ` A ) = 0 )
38		pilem1 24205	. . . . . . . . . . . . . . 15 |- ( A e. ( RR+ i^i ( `' sin " { 0 } ) ) <-> ( A e. RR+ /\ ( sin ` A ) = 0 ) )
39	36, 37, 38	sylanbrc 698	. . . . . . . . . . . . . 14 |- ( ph -> A e. ( RR+ i^i ( `' sin " { 0 } ) ) )
40		ne0i 3921	. . . . . . . . . . . . . 14 |- ( A e. ( RR+ i^i ( `' sin " { 0 } ) ) -> ( RR+ i^i ( `' sin " { 0 } ) ) =/= (/) )
41	39, 40	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( RR+ i^i ( `' sin " { 0 } ) ) =/= (/) )
42		infrecl 11005	. . . . . . . . . . . . . 14 |- ( ( ( RR+ i^i ( `' sin " { 0 } ) ) C_ RR /\ ( RR+ i^i ( `' sin " { 0 } ) ) =/= (/) /\ E. x e. RR A. y e. ( RR+ i^i ( `' sin " { 0 } ) ) x <_ y ) -> inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < ) e. RR )
43	4, 13, 42	mp3an13 1415	. . . . . . . . . . . . 13 |- ( ( RR+ i^i ( `' sin " { 0 } ) ) =/= (/) -> inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < ) e. RR )
44	41, 43	syl 17	. . . . . . . . . . . 12 |- ( ph -> inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < ) e. RR )
45		pilem1 24205	. . . . . . . . . . . . . . 15 |- ( x e. ( RR+ i^i ( `' sin " { 0 } ) ) <-> ( x e. RR+ /\ ( sin ` x ) = 0 ) )
46		rpre 11839	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. RR+ -> x e. RR )
47	46	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. RR+ ) -> x e. RR )
48		letric 10137	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 2 e. RR /\ x e. RR ) -> ( 2 <_ x \/ x <_ 2 ) )
49	15, 47, 48	sylancr 695	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. RR+ ) -> ( 2 <_ x \/ x <_ 2 ) )
50	49	ord 392	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. RR+ ) -> ( -. 2 <_ x -> x <_ 2 ) )
51	46	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ x e. RR+ ) /\ x <_ 2 ) -> x e. RR )
52		rpgt0 11844	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( x e. RR+ -> 0 < x )
53	52	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ x e. RR+ ) /\ x <_ 2 ) -> 0 < x )
54		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ x e. RR+ ) /\ x <_ 2 ) -> x <_ 2 )
55		0xr 10086	. . . . . . . . . . . . . . . . . . . . . . 23 |- 0 e. RR*
56		elioc2 12236	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( 0 e. RR* /\ 2 e. RR ) -> ( x e. ( 0 (,] 2 ) <-> ( x e. RR /\ 0 < x /\ x <_ 2 ) ) )
57	55, 15, 56	mp2an 708	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x e. ( 0 (,] 2 ) <-> ( x e. RR /\ 0 < x /\ x <_ 2 ) )
58	51, 53, 54, 57	syl3anbrc 1246	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. RR+ ) /\ x <_ 2 ) -> x e. ( 0 (,] 2 ) )
59		sin02gt0 14922	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. ( 0 (,] 2 ) -> 0 < ( sin ` x ) )
60	58, 59	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. RR+ ) /\ x <_ 2 ) -> 0 < ( sin ` x ) )
61	60	gt0ne0d 10592	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. RR+ ) /\ x <_ 2 ) -> ( sin ` x ) =/= 0 )
62	61	ex 450	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. RR+ ) -> ( x <_ 2 -> ( sin ` x ) =/= 0 ) )
63	50, 62	syld 47	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. RR+ ) -> ( -. 2 <_ x -> ( sin ` x ) =/= 0 ) )
64	63	necon4bd 2814	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. RR+ ) -> ( ( sin ` x ) = 0 -> 2 <_ x ) )
65	64	expimpd 629	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( x e. RR+ /\ ( sin ` x ) = 0 ) -> 2 <_ x ) )
66	45, 65	syl5bi 232	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. ( RR+ i^i ( `' sin " { 0 } ) ) -> 2 <_ x ) )
67	66	ralrimiv 2965	. . . . . . . . . . . . 13 |- ( ph -> A. x e. ( RR+ i^i ( `' sin " { 0 } ) ) 2 <_ x )
68		infregelb 11007	. . . . . . . . . . . . . 14 |- ( ( ( ( RR+ i^i ( `' sin " { 0 } ) ) C_ RR /\ ( RR+ i^i ( `' sin " { 0 } ) ) =/= (/) /\ E. x e. RR A. y e. ( RR+ i^i ( `' sin " { 0 } ) ) x <_ y ) /\ 2 e. RR ) -> ( 2 <_ inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < ) <-> A. x e. ( RR+ i^i ( `' sin " { 0 } ) ) 2 <_ x ) )
69	5, 41, 14, 30, 68	syl31anc 1329	. . . . . . . . . . . . 13 |- ( ph -> ( 2 <_ inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < ) <-> A. x e. ( RR+ i^i ( `' sin " { 0 } ) ) 2 <_ x ) )
70	67, 69	mpbird 247	. . . . . . . . . . . 12 |- ( ph -> 2 <_ inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < ) )
71		pilem.4	. . . . . . . . . . . . . 14 |- ( ph -> ( sin ` B ) = 0 )
72		pilem1 24205	. . . . . . . . . . . . . 14 |- ( B e. ( RR+ i^i ( `' sin " { 0 } ) ) <-> ( B e. RR+ /\ ( sin ` B ) = 0 ) )
73	16, 71, 72	sylanbrc 698	. . . . . . . . . . . . 13 |- ( ph -> B e. ( RR+ i^i ( `' sin " { 0 } ) ) )
74		infrelb 11008	. . . . . . . . . . . . 13 |- ( ( ( RR+ i^i ( `' sin " { 0 } ) ) C_ RR /\ E. x e. RR A. y e. ( RR+ i^i ( `' sin " { 0 } ) ) x <_ y /\ B e. ( RR+ i^i ( `' sin " { 0 } ) ) ) -> inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < ) <_ B )
75	5, 14, 73, 74	syl3anc 1326	. . . . . . . . . . . 12 |- ( ph -> inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < ) <_ B )
76	30, 44, 17, 70, 75	letrd 10194	. . . . . . . . . . 11 |- ( ph -> 2 <_ B )
77	15, 32	pm3.2i 471	. . . . . . . . . . . . 13 |- ( 2 e. RR /\ 0 < 2 )
78	77	a1i 11	. . . . . . . . . . . 12 |- ( ph -> ( 2 e. RR /\ 0 < 2 ) )
79		lemul2 10876	. . . . . . . . . . . 12 |- ( ( 2 e. RR /\ B e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( 2 <_ B <-> ( 2 x. 2 ) <_ ( 2 x. B ) ) )
80	30, 17, 78, 79	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> ( 2 <_ B <-> ( 2 x. 2 ) <_ ( 2 x. B ) ) )
81	76, 80	mpbid 222	. . . . . . . . . 10 |- ( ph -> ( 2 x. 2 ) <_ ( 2 x. B ) )
82	29, 81	syl5eqbrr 4689	. . . . . . . . 9 |- ( ph -> 4 <_ ( 2 x. B ) )
83	22, 25, 19, 28, 82	ltletrd 10197	. . . . . . . 8 |- ( ph -> A < ( 2 x. B ) )
84	22, 19	posdifd 10614	. . . . . . . 8 |- ( ph -> ( A < ( 2 x. B ) <-> 0 < ( ( 2 x. B ) - A ) ) )
85	83, 84	mpbid 222	. . . . . . 7 |- ( ph -> 0 < ( ( 2 x. B ) - A ) )
86	23, 85	elrpd 11869	. . . . . 6 |- ( ph -> ( ( 2 x. B ) - A ) e. RR+ )
87	19	recnd 10068	. . . . . . . 8 |- ( ph -> ( 2 x. B ) e. CC )
88	22	recnd 10068	. . . . . . . 8 |- ( ph -> A e. CC )
89		sinsub 14898	. . . . . . . 8 |- ( ( ( 2 x. B ) e. CC /\ A e. CC ) -> ( sin ` ( ( 2 x. B ) - A ) ) = ( ( ( sin ` ( 2 x. B ) ) x. ( cos ` A ) ) - ( ( cos ` ( 2 x. B ) ) x. ( sin ` A ) ) ) )
90	87, 88, 89	syl2anc 693	. . . . . . 7 |- ( ph -> ( sin ` ( ( 2 x. B ) - A ) ) = ( ( ( sin ` ( 2 x. B ) ) x. ( cos ` A ) ) - ( ( cos ` ( 2 x. B ) ) x. ( sin ` A ) ) ) )
91	17	recnd 10068	. . . . . . . . . . . . 13 |- ( ph -> B e. CC )
92		sin2t 14907	. . . . . . . . . . . . 13 |- ( B e. CC -> ( sin ` ( 2 x. B ) ) = ( 2 x. ( ( sin ` B ) x. ( cos ` B ) ) ) )
93	91, 92	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( sin ` ( 2 x. B ) ) = ( 2 x. ( ( sin ` B ) x. ( cos ` B ) ) ) )
94	71	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( sin ` B ) x. ( cos ` B ) ) = ( 0 x. ( cos ` B ) ) )
95	91	coscld 14861	. . . . . . . . . . . . . . . 16 |- ( ph -> ( cos ` B ) e. CC )
96	95	mul02d 10234	. . . . . . . . . . . . . . 15 |- ( ph -> ( 0 x. ( cos ` B ) ) = 0 )
97	94, 96	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ph -> ( ( sin ` B ) x. ( cos ` B ) ) = 0 )
98	97	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ph -> ( 2 x. ( ( sin ` B ) x. ( cos ` B ) ) ) = ( 2 x. 0 ) )
99		2t0e0 11183	. . . . . . . . . . . . 13 |- ( 2 x. 0 ) = 0
100	98, 99	syl6eq 2672	. . . . . . . . . . . 12 |- ( ph -> ( 2 x. ( ( sin ` B ) x. ( cos ` B ) ) ) = 0 )
101	93, 100	eqtrd 2656	. . . . . . . . . . 11 |- ( ph -> ( sin ` ( 2 x. B ) ) = 0 )
102	101	oveq1d 6665	. . . . . . . . . 10 |- ( ph -> ( ( sin ` ( 2 x. B ) ) x. ( cos ` A ) ) = ( 0 x. ( cos ` A ) ) )
103	88	coscld 14861	. . . . . . . . . . 11 |- ( ph -> ( cos ` A ) e. CC )
104	103	mul02d 10234	. . . . . . . . . 10 |- ( ph -> ( 0 x. ( cos ` A ) ) = 0 )
105	102, 104	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( sin ` ( 2 x. B ) ) x. ( cos ` A ) ) = 0 )
106	37	oveq2d 6666	. . . . . . . . . 10 |- ( ph -> ( ( cos ` ( 2 x. B ) ) x. ( sin ` A ) ) = ( ( cos ` ( 2 x. B ) ) x. 0 ) )
107	87	coscld 14861	. . . . . . . . . . 11 |- ( ph -> ( cos ` ( 2 x. B ) ) e. CC )
108	107	mul01d 10235	. . . . . . . . . 10 |- ( ph -> ( ( cos ` ( 2 x. B ) ) x. 0 ) = 0 )
109	106, 108	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( cos ` ( 2 x. B ) ) x. ( sin ` A ) ) = 0 )
110	105, 109	oveq12d 6668	. . . . . . . 8 |- ( ph -> ( ( ( sin ` ( 2 x. B ) ) x. ( cos ` A ) ) - ( ( cos ` ( 2 x. B ) ) x. ( sin ` A ) ) ) = ( 0 - 0 ) )
111		0m0e0 11130	. . . . . . . 8 |- ( 0 - 0 ) = 0
112	110, 111	syl6eq 2672	. . . . . . 7 |- ( ph -> ( ( ( sin ` ( 2 x. B ) ) x. ( cos ` A ) ) - ( ( cos ` ( 2 x. B ) ) x. ( sin ` A ) ) ) = 0 )
113	90, 112	eqtrd 2656	. . . . . 6 |- ( ph -> ( sin ` ( ( 2 x. B ) - A ) ) = 0 )
114		pilem1 24205	. . . . . 6 |- ( ( ( 2 x. B ) - A ) e. ( RR+ i^i ( `' sin " { 0 } ) ) <-> ( ( ( 2 x. B ) - A ) e. RR+ /\ ( sin ` ( ( 2 x. B ) - A ) ) = 0 ) )
115	86, 113, 114	sylanbrc 698	. . . . 5 |- ( ph -> ( ( 2 x. B ) - A ) e. ( RR+ i^i ( `' sin " { 0 } ) ) )
116		infrelb 11008	. . . . 5 |- ( ( ( RR+ i^i ( `' sin " { 0 } ) ) C_ RR /\ E. x e. RR A. y e. ( RR+ i^i ( `' sin " { 0 } ) ) x <_ y /\ ( ( 2 x. B ) - A ) e. ( RR+ i^i ( `' sin " { 0 } ) ) ) -> inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < ) <_ ( ( 2 x. B ) - A ) )
117	5, 14, 115, 116	syl3anc 1326	. . . 4 |- ( ph -> inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < ) <_ ( ( 2 x. B ) - A ) )
118	1, 117	syl5eqbr 4688	. . 3 |- ( ph -> pi <_ ( ( 2 x. B ) - A ) )
119	1, 44	syl5eqel 2705	. . . 4 |- ( ph -> pi e. RR )
120		leaddsub 10504	. . . 4 |- ( ( pi e. RR /\ A e. RR /\ ( 2 x. B ) e. RR ) -> ( ( pi + A ) <_ ( 2 x. B ) <-> pi <_ ( ( 2 x. B ) - A ) ) )
121	119, 22, 19, 120	syl3anc 1326	. . 3 |- ( ph -> ( ( pi + A ) <_ ( 2 x. B ) <-> pi <_ ( ( 2 x. B ) - A ) ) )
122	118, 121	mpbird 247	. 2 |- ( ph -> ( pi + A ) <_ ( 2 x. B ) )
123	119, 22	readdcld 10069	. . 3 |- ( ph -> ( pi + A ) e. RR )
124		ledivmul 10899	. . 3 |- ( ( ( pi + A ) e. RR /\ B e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( ( pi + A ) / 2 ) <_ B <-> ( pi + A ) <_ ( 2 x. B ) ) )
125	123, 17, 78, 124	syl3anc 1326	. 2 |- ( ph -> ( ( ( pi + A ) / 2 ) <_ B <-> ( pi + A ) <_ ( 2 x. B ) ) )
126	122, 125	mpbird 247	1 |- ( ph -> ( ( pi + A ) / 2 ) <_ B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650  infcinf 8347   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   4c4 11072   RR+crp 11832   (,)cioo 12175   (,]cioc 12176   sincsin 14794   cosccos 14795   picpi 14797

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ioo 12179  df-ioc 12180  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803

This theorem is referenced by:  pilem3  24207