Theorem imo72b2 38475

Description: IMO 1972 B2. (14th International Mathemahics Olympiad in Poland, problem B2). (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypotheses

Ref	Expression
imo72b2.1	|- ( ph -> F : RR --> RR )
imo72b2.2	|- ( ph -> G : RR --> RR )
imo72b2.4	|- ( ph -> B e. RR )
imo72b2.5	|- ( ph -> A. u e. RR A. v e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) )
imo72b2.6	|- ( ph -> A. y e. RR ( abs ` ( F ` y ) ) <_ 1 )
imo72b2.7	|- ( ph -> E. x e. RR ( F ` x ) =/= 0 )

Assertion

Ref	Expression
imo72b2	|- ( ph -> ( abs ` ( G ` B ) ) <_ 1 )

Distinct variable groups:   u, B, v   x, B   y, B   u, F, v   x, F   y, F   u, G, v   x, G   y, G   ph, u, v   ph, x   ph, y, u

Proof of Theorem imo72b2

Dummy variables c t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imo72b2.2	. . . . 5 |- ( ph -> G : RR --> RR )
2		imo72b2.4	. . . . 5 |- ( ph -> B e. RR )
3	1, 2	ffvelrnd 6360	. . . 4 |- ( ph -> ( G ` B ) e. RR )
4	3	recnd 10068	. . 3 |- ( ph -> ( G ` B ) e. CC )
5	4	abscld 14175	. 2 |- ( ph -> ( abs ` ( G ` B ) ) e. RR )
6		1red 10055	. 2 |- ( ph -> 1 e. RR )
7		simpr 477	. . 3 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 1 < ( abs ` ( G ` B ) ) )
8	1	adantr 481	. . . . . . 7 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> G : RR --> RR )
9	2	adantr 481	. . . . . . 7 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> B e. RR )
10	8, 9	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( G ` B ) e. RR )
11	10	recnd 10068	. . . . 5 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( G ` B ) e. CC )
12	11	abscld 14175	. . . 4 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs ` ( G ` B ) ) e. RR )
13	6	adantr 481	. . . 4 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 1 e. RR )
14		ax-resscn 9993	. . . . . . . . 9 |- RR C_ CC
15		imaco 5640	. . . . . . . . . . . 12 |- ( ( abs o. F ) " RR ) = ( abs " ( F " RR ) )
16	15	eqcomi 2631	. . . . . . . . . . 11 |- ( abs " ( F " RR ) ) = ( ( abs o. F ) " RR )
17		imassrn 5477	. . . . . . . . . . . . 13 |- ( ( abs o. F ) " RR ) C_ ran ( abs o. F )
18	17	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( ( abs o. F ) " RR ) C_ ran ( abs o. F ) )
19		imo72b2.1	. . . . . . . . . . . . . . 15 |- ( ph -> F : RR --> RR )
20	19	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> F : RR --> RR )
21		absf 14077	. . . . . . . . . . . . . . . 16 |- abs : CC --> RR
22	21	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> abs : CC --> RR )
23	14	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> RR C_ CC )
24	22, 23	fssresd 6071	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs |` RR ) : RR --> RR )
25	20, 24	fco2d 38461	. . . . . . . . . . . . 13 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs o. F ) : RR --> RR )
26		frn 6053	. . . . . . . . . . . . 13 |- ( ( abs o. F ) : RR --> RR -> ran ( abs o. F ) C_ RR )
27	25, 26	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ran ( abs o. F ) C_ RR )
28	18, 27	sstrd 3613	. . . . . . . . . . 11 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( ( abs o. F ) " RR ) C_ RR )
29	16, 28	syl5eqss 3649	. . . . . . . . . 10 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs " ( F " RR ) ) C_ RR )
30		0re 10040	. . . . . . . . . . . . . . . 16 |- 0 e. RR
31	30	ne0ii 3923	. . . . . . . . . . . . . . 15 |- RR =/= (/)
32	31	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> RR =/= (/) )
33	32, 25	wnefimgd 38460	. . . . . . . . . . . . 13 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( ( abs o. F ) " RR ) =/= (/) )
34	33	necomd 2849	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> (/) =/= ( ( abs o. F ) " RR ) )
35	16	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs " ( F " RR ) ) = ( ( abs o. F ) " RR ) )
36	34, 35	neeqtrrd 2868	. . . . . . . . . . 11 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> (/) =/= ( abs " ( F " RR ) ) )
37	36	necomd 2849	. . . . . . . . . 10 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs " ( F " RR ) ) =/= (/) )
38		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ c = 1 ) -> c = 1 )
39	38	breq2d 4665	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ c = 1 ) -> ( t <_ c <-> t <_ 1 ) )
40	39	ralbidv 2986	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ c = 1 ) -> ( A. t e. ( abs " ( F " RR ) ) t <_ c <-> A. t e. ( abs " ( F " RR ) ) t <_ 1 ) )
41		imo72b2.6	. . . . . . . . . . . . 13 |- ( ph -> A. y e. RR ( abs ` ( F ` y ) ) <_ 1 )
42	19, 41	extoimad 38464	. . . . . . . . . . . 12 |- ( ph -> A. t e. ( abs " ( F " RR ) ) t <_ 1 )
43	42	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> A. t e. ( abs " ( F " RR ) ) t <_ 1 )
44	13, 40, 43	rspcedvd 3317	. . . . . . . . . 10 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> E. c e. RR A. t e. ( abs " ( F " RR ) ) t <_ c )
45	29, 37, 44	suprcld 10986	. . . . . . . . 9 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) e. RR )
46	14, 45	sseldi 3601	. . . . . . . 8 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) e. CC )
47	14, 12	sseldi 3601	. . . . . . . 8 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs ` ( G ` B ) ) e. CC )
48	46, 47	mulcomd 10061	. . . . . . 7 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( sup ( ( abs " ( F " RR ) ) , RR , < ) x. ( abs ` ( G ` B ) ) ) = ( ( abs ` ( G ` B ) ) x. sup ( ( abs " ( F " RR ) ) , RR , < ) ) )
49	30	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 0 e. RR )
50		0lt1 10550	. . . . . . . . . . . . 13 |- 0 < 1
51	50	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 0 < 1 )
52	49, 13, 12, 51, 7	lttrd 10198	. . . . . . . . . . 11 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 0 < ( abs ` ( G ` B ) ) )
53	52	gt0ne0d 10592	. . . . . . . . . 10 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs ` ( G ` B ) ) =/= 0 )
54	45, 12, 53	redivcld 10853	. . . . . . . . 9 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( sup ( ( abs " ( F " RR ) ) , RR , < ) / ( abs ` ( G ` B ) ) ) e. RR )
55	20	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> F : RR --> RR )
56	8	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> G : RR --> RR )
57		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> u e. RR )
58	9	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> B e. RR )
59		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ v = B ) -> v = B )
60	59	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ v = B ) -> ( u + v ) = ( u + B ) )
61	60	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ v = B ) -> ( F ` ( u + v ) ) = ( F ` ( u + B ) ) )
62	59	oveq2d 6666	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ v = B ) -> ( u - v ) = ( u - B ) )
63	62	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ v = B ) -> ( F ` ( u - v ) ) = ( F ` ( u - B ) ) )
64	61, 63	oveq12d 6668	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ v = B ) -> ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) )
65	59	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ v = B ) -> ( G ` v ) = ( G ` B ) )
66	65	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ v = B ) -> ( ( F ` u ) x. ( G ` v ) ) = ( ( F ` u ) x. ( G ` B ) ) )
67	66	oveq2d 6666	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ v = B ) -> ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) )
68	64, 67	eqeq12d 2637	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ v = B ) -> ( ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) <-> ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) ) )
69	68	ralbidv 2986	. . . . . . . . . . . . . . 15 |- ( ( ph /\ v = B ) -> ( A. u e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) <-> A. u e. RR ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) ) )
70		imo72b2.5	. . . . . . . . . . . . . . . 16 |- ( ph -> A. u e. RR A. v e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) )
71		ralcom2 3104	. . . . . . . . . . . . . . . . . 18 |- ( A. u e. RR A. v e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) -> A. v e. RR A. u e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) )
72	71	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( A. u e. RR A. v e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) -> A. v e. RR A. u e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) ) )
73	72	imp 445	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ A. u e. RR A. v e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) ) -> A. v e. RR A. u e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) )
74	70, 73	mpdan 702	. . . . . . . . . . . . . . 15 |- ( ph -> A. v e. RR A. u e. RR ( ( F ` ( u + v ) ) + ( F ` ( u - v ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` v ) ) ) )
75	69, 2, 74	rspcdvinvd 38474	. . . . . . . . . . . . . 14 |- ( ph -> A. u e. RR ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) )
76	75	r19.21bi 2932	. . . . . . . . . . . . 13 |- ( ( ph /\ u e. RR ) -> ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) )
77	76	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( ( F ` ( u + B ) ) + ( F ` ( u - B ) ) ) = ( 2 x. ( ( F ` u ) x. ( G ` B ) ) ) )
78	41	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> A. y e. RR ( abs ` ( F ` y ) ) <_ 1 )
79	55, 56, 57, 58, 77, 78	imo72b2lem0 38465	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( ( abs ` ( F ` u ) ) x. ( abs ` ( G ` B ) ) ) <_ sup ( ( abs " ( F " RR ) ) , RR , < ) )
80		0xr 10086	. . . . . . . . . . . . 13 |- 0 e. RR*
81	80	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> 0 e. RR* )
82		1re 10039	. . . . . . . . . . . . . 14 |- 1 e. RR
83	82	rexri 10097	. . . . . . . . . . . . 13 |- 1 e. RR*
84	83	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> 1 e. RR* )
85	12	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( abs ` ( G ` B ) ) e. RR )
86	85	rexrd 10089	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( abs ` ( G ` B ) ) e. RR* )
87	50	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> 0 < 1 )
88		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> 1 < ( abs ` ( G ` B ) ) )
89	81, 84, 86, 87, 88	xrlttrd 11990	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> 0 < ( abs ` ( G ` B ) ) )
90	20	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( F ` u ) e. RR )
91	90	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( F ` u ) e. CC )
92	91	abscld 14175	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( abs ` ( F ` u ) ) e. RR )
93	45	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) e. RR )
94	79, 89, 85, 92, 93	lemuldiv3d 38472	. . . . . . . . . 10 |- ( ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) /\ u e. RR ) -> ( abs ` ( F ` u ) ) <_ ( sup ( ( abs " ( F " RR ) ) , RR , < ) / ( abs ` ( G ` B ) ) ) )
95	94	ralrimiva 2966	. . . . . . . . 9 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> A. u e. RR ( abs ` ( F ` u ) ) <_ ( sup ( ( abs " ( F " RR ) ) , RR , < ) / ( abs ` ( G ` B ) ) ) )
96	20, 54, 95	imo72b2lem2 38467	. . . . . . . 8 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) <_ ( sup ( ( abs " ( F " RR ) ) , RR , < ) / ( abs ` ( G ` B ) ) ) )
97	96, 52, 12, 45, 45	lemuldiv4d 38473	. . . . . . 7 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( sup ( ( abs " ( F " RR ) ) , RR , < ) x. ( abs ` ( G ` B ) ) ) <_ sup ( ( abs " ( F " RR ) ) , RR , < ) )
98	48, 97	eqbrtrrd 4677	. . . . . 6 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( ( abs ` ( G ` B ) ) x. sup ( ( abs " ( F " RR ) ) , RR , < ) ) <_ sup ( ( abs " ( F " RR ) ) , RR , < ) )
99		imo72b2.7	. . . . . . . 8 |- ( ph -> E. x e. RR ( F ` x ) =/= 0 )
100	99	adantr 481	. . . . . . 7 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> E. x e. RR ( F ` x ) =/= 0 )
101	41	adantr 481	. . . . . . 7 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> A. y e. RR ( abs ` ( F ` y ) ) <_ 1 )
102	20, 100, 101	imo72b2lem1 38471	. . . . . 6 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 0 < sup ( ( abs " ( F " RR ) ) , RR , < ) )
103	98, 102, 45, 12, 45	lemuldiv3d 38472	. . . . 5 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs ` ( G ` B ) ) <_ ( sup ( ( abs " ( F " RR ) ) , RR , < ) / sup ( ( abs " ( F " RR ) ) , RR , < ) ) )
104	23, 45	sseldd 3604	. . . . . . 7 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) e. CC )
105	102	gt0ne0d 10592	. . . . . . 7 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> sup ( ( abs " ( F " RR ) ) , RR , < ) =/= 0 )
106	104, 105	dividd 10799	. . . . . 6 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( sup ( ( abs " ( F " RR ) ) , RR , < ) / sup ( ( abs " ( F " RR ) ) , RR , < ) ) = 1 )
107	106	eqcomd 2628	. . . . 5 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> 1 = ( sup ( ( abs " ( F " RR ) ) , RR , < ) / sup ( ( abs " ( F " RR ) ) , RR , < ) ) )
108	103, 107	breqtrrd 4681	. . . 4 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> ( abs ` ( G ` B ) ) <_ 1 )
109	12, 13, 108	lensymd 10188	. . 3 |- ( ( ph /\ 1 < ( abs ` ( G ` B ) ) ) -> -. 1 < ( abs ` ( G ` B ) ) )
110	7, 109	pm2.65da 600	. 2 |- ( ph -> -. 1 < ( abs ` ( G ` B ) ) )
111	5, 6, 110	nltled 10187	1 |- ( ph -> ( abs ` ( G ` B ) ) <_ 1 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ran crn 5115   "cima 5117   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   abscabs 13974

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  ax-pre-sup 10014

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-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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by: (None)