Theorem ruclem12 14181

Description: Lemma for ruc 14183. The supremum of the increasing sequence 1st o. G is a real number that is not in the range of F. (Contributed by Mario Carneiro, 28-May-2014.)

Hypotheses

Ref	Expression
ruc.1	|- ( ph -> F : NN --> RR )
ruc.2	|- ( ph -> D = ( x e. ( RR X. RR ) , y e. RR |-> [_ ( ( ( 1st ` x ) + ( 2nd ` x ) ) / 2 ) / m ]_ if ( m < y , <. ( 1st ` x ) , m >. , <. ( ( m + ( 2nd ` x ) ) / 2 ) , ( 2nd ` x ) >. ) ) )
ruc.4	|- C = ( { <. 0 , <. 0 , 1 >. >. } u. F )
ruc.5	|- G = seq 0 ( D , C )
ruc.6	|- S = sup ( ran ( 1st o. G ) , RR , < )

Assertion

Ref	Expression
ruclem12	|- ( ph -> S e. ( RR \ ran F ) )

Distinct variable groups:   x, m, y, F   m, G, x, y

Allowed substitution hints:   ph( x, y, m)   C( x, y, m)   D( x, y, m)   S( x, y, m)

Proof of Theorem ruclem12

Dummy variables z n k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ruc.6	. . 3 |- S = sup ( ran ( 1st o. G ) , RR , < )
2		ruc.1	. . . . . 6 |- ( ph -> F : NN --> RR )
3		ruc.2	. . . . . 6 |- ( ph -> D = ( x e. ( RR X. RR ) , y e. RR |-> [_ ( ( ( 1st ` x ) + ( 2nd ` x ) ) / 2 ) / m ]_ if ( m < y , <. ( 1st ` x ) , m >. , <. ( ( m + ( 2nd ` x ) ) / 2 ) , ( 2nd ` x ) >. ) ) )
4		ruc.4	. . . . . 6 |- C = ( { <. 0 , <. 0 , 1 >. >. } u. F )
5		ruc.5	. . . . . 6 |- G = seq 0 ( D , C )
6	2, 3, 4, 5	ruclem11 14180	. . . . 5 |- ( ph -> ( ran ( 1st o. G ) C_ RR /\ ran ( 1st o. G ) =/= (/) /\ A. z e. ran ( 1st o. G ) z <_ 1 ) )
7	6	simp1d 1009	. . . 4 |- ( ph -> ran ( 1st o. G ) C_ RR )
8	6	simp2d 1010	. . . 4 |- ( ph -> ran ( 1st o. G ) =/= (/) )
9		1re 9624	. . . . 5 |- 1 e. RR
10	6	simp3d 1011	. . . . 5 |- ( ph -> A. z e. ran ( 1st o. G ) z <_ 1 )
11		breq2 4398	. . . . . . 7 |- ( n = 1 -> ( z <_ n <-> z <_ 1 ) )
12	11	ralbidv 2842	. . . . . 6 |- ( n = 1 -> ( A. z e. ran ( 1st o. G ) z <_ n <-> A. z e. ran ( 1st o. G ) z <_ 1 ) )
13	12	rspcev 3159	. . . . 5 |- ( ( 1 e. RR /\ A. z e. ran ( 1st o. G ) z <_ 1 ) -> E. n e. RR A. z e. ran ( 1st o. G ) z <_ n )
14	9, 10, 13	sylancr 661	. . . 4 |- ( ph -> E. n e. RR A. z e. ran ( 1st o. G ) z <_ n )
15		suprcl 10542	. . . 4 |- ( ( ran ( 1st o. G ) C_ RR /\ ran ( 1st o. G ) =/= (/) /\ E. n e. RR A. z e. ran ( 1st o. G ) z <_ n ) -> sup ( ran ( 1st o. G ) , RR , < ) e. RR )
16	7, 8, 14, 15	syl3anc 1230	. . 3 |- ( ph -> sup ( ran ( 1st o. G ) , RR , < ) e. RR )
17	1, 16	syl5eqel 2494	. 2 |- ( ph -> S e. RR )
18	2	adantr 463	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> F : NN --> RR )
19	3	adantr 463	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> D = ( x e. ( RR X. RR ) , y e. RR |-> [_ ( ( ( 1st ` x ) + ( 2nd ` x ) ) / 2 ) / m ]_ if ( m < y , <. ( 1st ` x ) , m >. , <. ( ( m + ( 2nd ` x ) ) / 2 ) , ( 2nd ` x ) >. ) ) )
20	2, 3, 4, 5	ruclem6 14175	. . . . . . . . . . 11 |- ( ph -> G : NN0 --> ( RR X. RR ) )
21		nnm1nn0 10877	. . . . . . . . . . 11 |- ( n e. NN -> ( n - 1 ) e. NN0 )
22		ffvelrn 6006	. . . . . . . . . . 11 |- ( ( G : NN0 --> ( RR X. RR ) /\ ( n - 1 ) e. NN0 ) -> ( G ` ( n - 1 ) ) e. ( RR X. RR ) )
23	20, 21, 22	syl2an 475	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( G ` ( n - 1 ) ) e. ( RR X. RR ) )
24		xp1st 6813	. . . . . . . . . 10 |- ( ( G ` ( n - 1 ) ) e. ( RR X. RR ) -> ( 1st ` ( G ` ( n - 1 ) ) ) e. RR )
25	23, 24	syl 17	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` ( n - 1 ) ) ) e. RR )
26		xp2nd 6814	. . . . . . . . . 10 |- ( ( G ` ( n - 1 ) ) e. ( RR X. RR ) -> ( 2nd ` ( G ` ( n - 1 ) ) ) e. RR )
27	23, 26	syl 17	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( G ` ( n - 1 ) ) ) e. RR )
28	2	ffvelrnda 6008	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( F ` n ) e. RR )
29		eqid 2402	. . . . . . . . 9 |- ( 1st ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) = ( 1st ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) )
30		eqid 2402	. . . . . . . . 9 |- ( 2nd ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) = ( 2nd ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) )
31	2, 3, 4, 5	ruclem8 14177	. . . . . . . . . 10 |- ( ( ph /\ ( n - 1 ) e. NN0 ) -> ( 1st ` ( G ` ( n - 1 ) ) ) < ( 2nd ` ( G ` ( n - 1 ) ) ) )
32	21, 31	sylan2 472	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` ( n - 1 ) ) ) < ( 2nd ` ( G ` ( n - 1 ) ) ) )
33	18, 19, 25, 27, 28, 29, 30, 32	ruclem3 14173	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( F ` n ) < ( 1st ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) \/ ( 2nd ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) < ( F ` n ) ) )
34	2, 3, 4, 5	ruclem7 14176	. . . . . . . . . . . . 13 |- ( ( ph /\ ( n - 1 ) e. NN0 ) -> ( G ` ( ( n - 1 ) + 1 ) ) = ( ( G ` ( n - 1 ) ) D ( F ` ( ( n - 1 ) + 1 ) ) ) )
35	21, 34	sylan2 472	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( G ` ( ( n - 1 ) + 1 ) ) = ( ( G ` ( n - 1 ) ) D ( F ` ( ( n - 1 ) + 1 ) ) ) )
36		nncn 10583	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n e. CC )
37	36	adantl 464	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> n e. CC )
38		ax-1cn 9579	. . . . . . . . . . . . . 14 |- 1 e. CC
39		npcan 9864	. . . . . . . . . . . . . 14 |- ( ( n e. CC /\ 1 e. CC ) -> ( ( n - 1 ) + 1 ) = n )
40	37, 38, 39	sylancl 660	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( n - 1 ) + 1 ) = n )
41	40	fveq2d 5852	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( G ` ( ( n - 1 ) + 1 ) ) = ( G ` n ) )
42		1st2nd2 6820	. . . . . . . . . . . . . 14 |- ( ( G ` ( n - 1 ) ) e. ( RR X. RR ) -> ( G ` ( n - 1 ) ) = <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. )
43	23, 42	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( G ` ( n - 1 ) ) = <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. )
44	40	fveq2d 5852	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( F ` ( ( n - 1 ) + 1 ) ) = ( F ` n ) )
45	43, 44	oveq12d 6295	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( ( G ` ( n - 1 ) ) D ( F ` ( ( n - 1 ) + 1 ) ) ) = ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) )
46	35, 41, 45	3eqtr3d 2451	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( G ` n ) = ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) )
47	46	fveq2d 5852	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) = ( 1st ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) )
48	47	breq2d 4406	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( F ` n ) < ( 1st ` ( G ` n ) ) <-> ( F ` n ) < ( 1st ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) ) )
49	46	fveq2d 5852	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) = ( 2nd ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) )
50	49	breq1d 4404	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) < ( F ` n ) <-> ( 2nd ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) < ( F ` n ) ) )
51	48, 50	orbi12d 708	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( ( F ` n ) < ( 1st ` ( G ` n ) ) \/ ( 2nd ` ( G ` n ) ) < ( F ` n ) ) <-> ( ( F ` n ) < ( 1st ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) \/ ( 2nd ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) < ( F ` n ) ) ) )
52	33, 51	mpbird 232	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( ( F ` n ) < ( 1st ` ( G ` n ) ) \/ ( 2nd ` ( G ` n ) ) < ( F ` n ) ) )
53	7	adantr 463	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ran ( 1st o. G ) C_ RR )
54	8	adantr 463	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ran ( 1st o. G ) =/= (/) )
55	14	adantr 463	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> E. n e. RR A. z e. ran ( 1st o. G ) z <_ n )
56		nnnn0 10842	. . . . . . . . . . . . 13 |- ( n e. NN -> n e. NN0 )
57		fvco3 5925	. . . . . . . . . . . . 13 |- ( ( G : NN0 --> ( RR X. RR ) /\ n e. NN0 ) -> ( ( 1st o. G ) ` n ) = ( 1st ` ( G ` n ) ) )
58	20, 56, 57	syl2an 475	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( ( 1st o. G ) ` n ) = ( 1st ` ( G ` n ) ) )
59	20	adantr 463	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> G : NN0 --> ( RR X. RR ) )
60		1stcof 6811	. . . . . . . . . . . . . 14 |- ( G : NN0 --> ( RR X. RR ) -> ( 1st o. G ) : NN0 --> RR )
61		ffn 5713	. . . . . . . . . . . . . 14 |- ( ( 1st o. G ) : NN0 --> RR -> ( 1st o. G ) Fn NN0 )
62	59, 60, 61	3syl 20	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( 1st o. G ) Fn NN0 )
63	56	adantl 464	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> n e. NN0 )
64		fnfvelrn 6005	. . . . . . . . . . . . 13 |- ( ( ( 1st o. G ) Fn NN0 /\ n e. NN0 ) -> ( ( 1st o. G ) ` n ) e. ran ( 1st o. G ) )
65	62, 63, 64	syl2anc 659	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( ( 1st o. G ) ` n ) e. ran ( 1st o. G ) )
66	58, 65	eqeltrrd 2491	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) e. ran ( 1st o. G ) )
67		suprub 10543	. . . . . . . . . . 11 |- ( ( ( ran ( 1st o. G ) C_ RR /\ ran ( 1st o. G ) =/= (/) /\ E. n e. RR A. z e. ran ( 1st o. G ) z <_ n ) /\ ( 1st ` ( G ` n ) ) e. ran ( 1st o. G ) ) -> ( 1st ` ( G ` n ) ) <_ sup ( ran ( 1st o. G ) , RR , < ) )
68	53, 54, 55, 66, 67	syl31anc 1233	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) <_ sup ( ran ( 1st o. G ) , RR , < ) )
69	68, 1	syl6breqr 4434	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) <_ S )
70		ffvelrn 6006	. . . . . . . . . . . 12 |- ( ( G : NN0 --> ( RR X. RR ) /\ n e. NN0 ) -> ( G ` n ) e. ( RR X. RR ) )
71	20, 56, 70	syl2an 475	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( G ` n ) e. ( RR X. RR ) )
72		xp1st 6813	. . . . . . . . . . 11 |- ( ( G ` n ) e. ( RR X. RR ) -> ( 1st ` ( G ` n ) ) e. RR )
73	71, 72	syl 17	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) e. RR )
74	17	adantr 463	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> S e. RR )
75		ltletr 9706	. . . . . . . . . 10 |- ( ( ( F ` n ) e. RR /\ ( 1st ` ( G ` n ) ) e. RR /\ S e. RR ) -> ( ( ( F ` n ) < ( 1st ` ( G ` n ) ) /\ ( 1st ` ( G ` n ) ) <_ S ) -> ( F ` n ) < S ) )
76	28, 73, 74, 75	syl3anc 1230	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( ( F ` n ) < ( 1st ` ( G ` n ) ) /\ ( 1st ` ( G ` n ) ) <_ S ) -> ( F ` n ) < S ) )
77	69, 76	mpan2d 672	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( F ` n ) < ( 1st ` ( G ` n ) ) -> ( F ` n ) < S ) )
78		fvco3 5925	. . . . . . . . . . . . . . 15 |- ( ( G : NN0 --> ( RR X. RR ) /\ k e. NN0 ) -> ( ( 1st o. G ) ` k ) = ( 1st ` ( G ` k ) ) )
79	59, 78	sylan 469	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( ( 1st o. G ) ` k ) = ( 1st ` ( G ` k ) ) )
80	59	ffvelrnda 6008	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( G ` k ) e. ( RR X. RR ) )
81		xp1st 6813	. . . . . . . . . . . . . . . 16 |- ( ( G ` k ) e. ( RR X. RR ) -> ( 1st ` ( G ` k ) ) e. RR )
82	80, 81	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( 1st ` ( G ` k ) ) e. RR )
83		xp2nd 6814	. . . . . . . . . . . . . . . . 17 |- ( ( G ` n ) e. ( RR X. RR ) -> ( 2nd ` ( G ` n ) ) e. RR )
84	71, 83	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) e. RR )
85	84	adantr 463	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( 2nd ` ( G ` n ) ) e. RR )
86	18	adantr 463	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> F : NN --> RR )
87	19	adantr 463	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> D = ( x e. ( RR X. RR ) , y e. RR |-> [_ ( ( ( 1st ` x ) + ( 2nd ` x ) ) / 2 ) / m ]_ if ( m < y , <. ( 1st ` x ) , m >. , <. ( ( m + ( 2nd ` x ) ) / 2 ) , ( 2nd ` x ) >. ) ) )
88		simpr 459	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> k e. NN0 )
89	63	adantr 463	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> n e. NN0 )
90	86, 87, 4, 5, 88, 89	ruclem10 14179	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( 1st ` ( G ` k ) ) < ( 2nd ` ( G ` n ) ) )
91	82, 85, 90	ltled 9764	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( 1st ` ( G ` k ) ) <_ ( 2nd ` ( G ` n ) ) )
92	79, 91	eqbrtrd 4414	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( ( 1st o. G ) ` k ) <_ ( 2nd ` ( G ` n ) ) )
93	92	ralrimiva 2817	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> A. k e. NN0 ( ( 1st o. G ) ` k ) <_ ( 2nd ` ( G ` n ) ) )
94		breq1 4397	. . . . . . . . . . . . . 14 |- ( z = ( ( 1st o. G ) ` k ) -> ( z <_ ( 2nd ` ( G ` n ) ) <-> ( ( 1st o. G ) ` k ) <_ ( 2nd ` ( G ` n ) ) ) )
95	94	ralrn 6011	. . . . . . . . . . . . 13 |- ( ( 1st o. G ) Fn NN0 -> ( A. z e. ran ( 1st o. G ) z <_ ( 2nd ` ( G ` n ) ) <-> A. k e. NN0 ( ( 1st o. G ) ` k ) <_ ( 2nd ` ( G ` n ) ) ) )
96	62, 95	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( A. z e. ran ( 1st o. G ) z <_ ( 2nd ` ( G ` n ) ) <-> A. k e. NN0 ( ( 1st o. G ) ` k ) <_ ( 2nd ` ( G ` n ) ) ) )
97	93, 96	mpbird 232	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> A. z e. ran ( 1st o. G ) z <_ ( 2nd ` ( G ` n ) ) )
98		suprleub 10546	. . . . . . . . . . . 12 |- ( ( ( ran ( 1st o. G ) C_ RR /\ ran ( 1st o. G ) =/= (/) /\ E. n e. RR A. z e. ran ( 1st o. G ) z <_ n ) /\ ( 2nd ` ( G ` n ) ) e. RR ) -> ( sup ( ran ( 1st o. G ) , RR , < ) <_ ( 2nd ` ( G ` n ) ) <-> A. z e. ran ( 1st o. G ) z <_ ( 2nd ` ( G ` n ) ) ) )
99	53, 54, 55, 84, 98	syl31anc 1233	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( sup ( ran ( 1st o. G ) , RR , < ) <_ ( 2nd ` ( G ` n ) ) <-> A. z e. ran ( 1st o. G ) z <_ ( 2nd ` ( G ` n ) ) ) )
100	97, 99	mpbird 232	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> sup ( ran ( 1st o. G ) , RR , < ) <_ ( 2nd ` ( G ` n ) ) )
101	1, 100	syl5eqbr 4427	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> S <_ ( 2nd ` ( G ` n ) ) )
102		lelttr 9705	. . . . . . . . . 10 |- ( ( S e. RR /\ ( 2nd ` ( G ` n ) ) e. RR /\ ( F ` n ) e. RR ) -> ( ( S <_ ( 2nd ` ( G ` n ) ) /\ ( 2nd ` ( G ` n ) ) < ( F ` n ) ) -> S < ( F ` n ) ) )
103	74, 84, 28, 102	syl3anc 1230	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( S <_ ( 2nd ` ( G ` n ) ) /\ ( 2nd ` ( G ` n ) ) < ( F ` n ) ) -> S < ( F ` n ) ) )
104	101, 103	mpand 673	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) < ( F ` n ) -> S < ( F ` n ) ) )
105	77, 104	orim12d 839	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( ( ( F ` n ) < ( 1st ` ( G ` n ) ) \/ ( 2nd ` ( G ` n ) ) < ( F ` n ) ) -> ( ( F ` n ) < S \/ S < ( F ` n ) ) ) )
106	52, 105	mpd 15	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( ( F ` n ) < S \/ S < ( F ` n ) ) )
107	28, 74	lttri2d 9755	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( ( F ` n ) =/= S <-> ( ( F ` n ) < S \/ S < ( F ` n ) ) ) )
108	106, 107	mpbird 232	. . . . 5 |- ( ( ph /\ n e. NN ) -> ( F ` n ) =/= S )
109	108	neneqd 2605	. . . 4 |- ( ( ph /\ n e. NN ) -> -. ( F ` n ) = S )
110	109	nrexdv 2859	. . 3 |- ( ph -> -. E. n e. NN ( F ` n ) = S )
111		risset 2931	. . . 4 |- ( S e. ran F <-> E. z e. ran F z = S )
112		ffn 5713	. . . . 5 |- ( F : NN --> RR -> F Fn NN )
113		eqeq1 2406	. . . . . 6 |- ( z = ( F ` n ) -> ( z = S <-> ( F ` n ) = S ) )
114	113	rexrn 6010	. . . . 5 |- ( F Fn NN -> ( E. z e. ran F z = S <-> E. n e. NN ( F ` n ) = S ) )
115	2, 112, 114	3syl 20	. . . 4 |- ( ph -> ( E. z e. ran F z = S <-> E. n e. NN ( F ` n ) = S ) )
116	111, 115	syl5bb 257	. . 3 |- ( ph -> ( S e. ran F <-> E. n e. NN ( F ` n ) = S ) )
117	110, 116	mtbird 299	. 2 |- ( ph -> -. S e. ran F )
118	17, 117	eldifd 3424	1 |- ( ph -> S e. ( RR \ ran F ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   = wceq 1405   e. wcel 1842   =/= wne 2598   A.wral 2753   E.wrex 2754   [_csb 3372   \ cdif 3410   u. cun 3411   C_ wss 3413   (/)c0 3737   ifcif 3884   {csn 3971   <.cop 3977   class class class wbr 4394   X. cxp 4820   ran crn 4823   o. ccom 4826   Fn wfn 5563   -->wf 5564   ` cfv 5568  (class class class)co 6277   |-> cmpt2 6279   1stc1st 6781   2ndc2nd 6782   supcsup 7933   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522   + caddc 9524   < clt 9657   <_ cle 9658   - cmin 9840   / cdiv 10246   NNcn 10575   2c2 10625   NN0cn0 10835   seqcseq 12149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-seq 12150

This theorem is referenced by:  ruclem13  14182