Theorem ruclem12 14970

Description: Lemma for ruc 14972. 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 14969	. . . . 5 |- ( ph -> ( ran ( 1st o. G ) C_ RR /\ ran ( 1st o. G ) =/= (/) /\ A. z e. ran ( 1st o. G ) z <_ 1 ) )
7	6	simp1d 1073	. . . 4 |- ( ph -> ran ( 1st o. G ) C_ RR )
8	6	simp2d 1074	. . . 4 |- ( ph -> ran ( 1st o. G ) =/= (/) )
9		1re 10039	. . . . 5 |- 1 e. RR
10	6	simp3d 1075	. . . . 5 |- ( ph -> A. z e. ran ( 1st o. G ) z <_ 1 )
11		breq2 4657	. . . . . . 7 |- ( n = 1 -> ( z <_ n <-> z <_ 1 ) )
12	11	ralbidv 2986	. . . . . 6 |- ( n = 1 -> ( A. z e. ran ( 1st o. G ) z <_ n <-> A. z e. ran ( 1st o. G ) z <_ 1 ) )
13	12	rspcev 3309	. . . . 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 695	. . . 4 |- ( ph -> E. n e. RR A. z e. ran ( 1st o. G ) z <_ n )
15		suprcl 10983	. . . 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 1326	. . 3 |- ( ph -> sup ( ran ( 1st o. G ) , RR , < ) e. RR )
17	1, 16	syl5eqel 2705	. 2 |- ( ph -> S e. RR )
18	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> F : NN --> RR )
19	3	adantr 481	. . . . . . . . 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 14964	. . . . . . . . . . 11 |- ( ph -> G : NN0 --> ( RR X. RR ) )
21		nnm1nn0 11334	. . . . . . . . . . 11 |- ( n e. NN -> ( n - 1 ) e. NN0 )
22		ffvelrn 6357	. . . . . . . . . . 11 |- ( ( G : NN0 --> ( RR X. RR ) /\ ( n - 1 ) e. NN0 ) -> ( G ` ( n - 1 ) ) e. ( RR X. RR ) )
23	20, 21, 22	syl2an 494	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( G ` ( n - 1 ) ) e. ( RR X. RR ) )
24		xp1st 7198	. . . . . . . . . 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 7199	. . . . . . . . . 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 6359	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( F ` n ) e. RR )
29		eqid 2622	. . . . . . . . 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 2622	. . . . . . . . 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 14966	. . . . . . . . . 10 |- ( ( ph /\ ( n - 1 ) e. NN0 ) -> ( 1st ` ( G ` ( n - 1 ) ) ) < ( 2nd ` ( G ` ( n - 1 ) ) ) )
32	21, 31	sylan2 491	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` ( n - 1 ) ) ) < ( 2nd ` ( G ` ( n - 1 ) ) ) )
33	18, 19, 25, 27, 28, 29, 30, 32	ruclem3 14962	. . . . . . . 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 14965	. . . . . . . . . . . . 13 |- ( ( ph /\ ( n - 1 ) e. NN0 ) -> ( G ` ( ( n - 1 ) + 1 ) ) = ( ( G ` ( n - 1 ) ) D ( F ` ( ( n - 1 ) + 1 ) ) ) )
35	21, 34	sylan2 491	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( G ` ( ( n - 1 ) + 1 ) ) = ( ( G ` ( n - 1 ) ) D ( F ` ( ( n - 1 ) + 1 ) ) ) )
36		nncn 11028	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n e. CC )
37	36	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> n e. CC )
38		ax-1cn 9994	. . . . . . . . . . . . . 14 |- 1 e. CC
39		npcan 10290	. . . . . . . . . . . . . 14 |- ( ( n e. CC /\ 1 e. CC ) -> ( ( n - 1 ) + 1 ) = n )
40	37, 38, 39	sylancl 694	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( n - 1 ) + 1 ) = n )
41	40	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( G ` ( ( n - 1 ) + 1 ) ) = ( G ` n ) )
42		1st2nd2 7205	. . . . . . . . . . . . . 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 6195	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( F ` ( ( n - 1 ) + 1 ) ) = ( F ` n ) )
45	43, 44	oveq12d 6668	. . . . . . . . . . . 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 2664	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( G ` n ) = ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) )
47	46	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) = ( 1st ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) )
48	47	breq2d 4665	. . . . . . . . 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 6195	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) = ( 2nd ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) )
50	49	breq1d 4663	. . . . . . . . 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 746	. . . . . . . 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 247	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( ( F ` n ) < ( 1st ` ( G ` n ) ) \/ ( 2nd ` ( G ` n ) ) < ( F ` n ) ) )
53	7	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ran ( 1st o. G ) C_ RR )
54	8	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ran ( 1st o. G ) =/= (/) )
55	14	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> E. n e. RR A. z e. ran ( 1st o. G ) z <_ n )
56		nnnn0 11299	. . . . . . . . . . . . 13 |- ( n e. NN -> n e. NN0 )
57		fvco3 6275	. . . . . . . . . . . . 13 |- ( ( G : NN0 --> ( RR X. RR ) /\ n e. NN0 ) -> ( ( 1st o. G ) ` n ) = ( 1st ` ( G ` n ) ) )
58	20, 56, 57	syl2an 494	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( ( 1st o. G ) ` n ) = ( 1st ` ( G ` n ) ) )
59	20	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> G : NN0 --> ( RR X. RR ) )
60		1stcof 7196	. . . . . . . . . . . . . 14 |- ( G : NN0 --> ( RR X. RR ) -> ( 1st o. G ) : NN0 --> RR )
61		ffn 6045	. . . . . . . . . . . . . 14 |- ( ( 1st o. G ) : NN0 --> RR -> ( 1st o. G ) Fn NN0 )
62	59, 60, 61	3syl 18	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( 1st o. G ) Fn NN0 )
63	56	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> n e. NN0 )
64		fnfvelrn 6356	. . . . . . . . . . . . 13 |- ( ( ( 1st o. G ) Fn NN0 /\ n e. NN0 ) -> ( ( 1st o. G ) ` n ) e. ran ( 1st o. G ) )
65	62, 63, 64	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( ( 1st o. G ) ` n ) e. ran ( 1st o. G ) )
66	58, 65	eqeltrrd 2702	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) e. ran ( 1st o. G ) )
67		suprub 10984	. . . . . . . . . . 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 1329	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) <_ sup ( ran ( 1st o. G ) , RR , < ) )
69	68, 1	syl6breqr 4695	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) <_ S )
70		ffvelrn 6357	. . . . . . . . . . . 12 |- ( ( G : NN0 --> ( RR X. RR ) /\ n e. NN0 ) -> ( G ` n ) e. ( RR X. RR ) )
71	20, 56, 70	syl2an 494	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( G ` n ) e. ( RR X. RR ) )
72		xp1st 7198	. . . . . . . . . . 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 481	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> S e. RR )
75		ltletr 10129	. . . . . . . . . 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 1326	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( ( F ` n ) < ( 1st ` ( G ` n ) ) /\ ( 1st ` ( G ` n ) ) <_ S ) -> ( F ` n ) < S ) )
77	69, 76	mpan2d 710	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( F ` n ) < ( 1st ` ( G ` n ) ) -> ( F ` n ) < S ) )
78		fvco3 6275	. . . . . . . . . . . . . . 15 |- ( ( G : NN0 --> ( RR X. RR ) /\ k e. NN0 ) -> ( ( 1st o. G ) ` k ) = ( 1st ` ( G ` k ) ) )
79	59, 78	sylan 488	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( ( 1st o. G ) ` k ) = ( 1st ` ( G ` k ) ) )
80	59	ffvelrnda 6359	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( G ` k ) e. ( RR X. RR ) )
81		xp1st 7198	. . . . . . . . . . . . . . . 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 7199	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( 2nd ` ( G ` n ) ) e. RR )
86	18	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> F : NN --> RR )
87	19	adantr 481	. . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> k e. NN0 )
89	63	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> n e. NN0 )
90	86, 87, 4, 5, 88, 89	ruclem10 14968	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( 1st ` ( G ` k ) ) < ( 2nd ` ( G ` n ) ) )
91	82, 85, 90	ltled 10185	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( 1st ` ( G ` k ) ) <_ ( 2nd ` ( G ` n ) ) )
92	79, 91	eqbrtrd 4675	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( ( 1st o. G ) ` k ) <_ ( 2nd ` ( G ` n ) ) )
93	92	ralrimiva 2966	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> A. k e. NN0 ( ( 1st o. G ) ` k ) <_ ( 2nd ` ( G ` n ) ) )
94		breq1 4656	. . . . . . . . . . . . . 14 |- ( z = ( ( 1st o. G ) ` k ) -> ( z <_ ( 2nd ` ( G ` n ) ) <-> ( ( 1st o. G ) ` k ) <_ ( 2nd ` ( G ` n ) ) ) )
95	94	ralrn 6362	. . . . . . . . . . . . 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 247	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> A. z e. ran ( 1st o. G ) z <_ ( 2nd ` ( G ` n ) ) )
98		suprleub 10989	. . . . . . . . . . . 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 1329	. . . . . . . . . . 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 247	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> sup ( ran ( 1st o. G ) , RR , < ) <_ ( 2nd ` ( G ` n ) ) )
101	1, 100	syl5eqbr 4688	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> S <_ ( 2nd ` ( G ` n ) ) )
102		lelttr 10128	. . . . . . . . . 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 1326	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( S <_ ( 2nd ` ( G ` n ) ) /\ ( 2nd ` ( G ` n ) ) < ( F ` n ) ) -> S < ( F ` n ) ) )
104	101, 103	mpand 711	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) < ( F ` n ) -> S < ( F ` n ) ) )
105	77, 104	orim12d 883	. . . . . . 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 10176	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( ( F ` n ) =/= S <-> ( ( F ` n ) < S \/ S < ( F ` n ) ) ) )
108	106, 107	mpbird 247	. . . . 5 |- ( ( ph /\ n e. NN ) -> ( F ` n ) =/= S )
109	108	neneqd 2799	. . . 4 |- ( ( ph /\ n e. NN ) -> -. ( F ` n ) = S )
110	109	nrexdv 3001	. . 3 |- ( ph -> -. E. n e. NN ( F ` n ) = S )
111		risset 3062	. . . 4 |- ( S e. ran F <-> E. z e. ran F z = S )
112		ffn 6045	. . . . 5 |- ( F : NN --> RR -> F Fn NN )
113		eqeq1 2626	. . . . . 6 |- ( z = ( F ` n ) -> ( z = S <-> ( F ` n ) = S ) )
114	113	rexrn 6361	. . . . 5 |- ( F Fn NN -> ( E. z e. ran F z = S <-> E. n e. NN ( F ` n ) = S ) )
115	2, 112, 114	3syl 18	. . . 4 |- ( ph -> ( E. z e. ran F z = S <-> E. n e. NN ( F ` n ) = S ) )
116	111, 115	syl5bb 272	. . 3 |- ( ph -> ( S e. ran F <-> E. n e. NN ( F ` n ) = S ) )
117	110, 116	mtbird 315	. 2 |- ( ph -> -. S e. ran F )
118	17, 117	eldifd 3585	1 |- ( ph -> S e. ( RR \ ran F ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   [_csb 3533   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   ran crn 5115   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   seqcseq 12801

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-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-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-1st 7168  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  ruclem13  14971