Theorem ruclem12 13523

Description: Lemma for ruc 13525. 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 13522	. . . . 5 |- ( ph -> ( ran ( 1st o. G ) C_ RR /\ ran ( 1st o. G ) =/= (/) /\ A. z e. ran ( 1st o. G ) z <_ 1 ) )
7	6	simp1d 1000	. . . 4 |- ( ph -> ran ( 1st o. G ) C_ RR )
8	6	simp2d 1001	. . . 4 |- ( ph -> ran ( 1st o. G ) =/= (/) )
9		1re 9385	. . . . 5 |- 1 e. RR
10	6	simp3d 1002	. . . . 5 |- ( ph -> A. z e. ran ( 1st o. G ) z <_ 1 )
11		breq2 4296	. . . . . . 7 |- ( n = 1 -> ( z <_ n <-> z <_ 1 ) )
12	11	ralbidv 2735	. . . . . 6 |- ( n = 1 -> ( A. z e. ran ( 1st o. G ) z <_ n <-> A. z e. ran ( 1st o. G ) z <_ 1 ) )
13	12	rspcev 3073	. . . . 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 663	. . . 4 |- ( ph -> E. n e. RR A. z e. ran ( 1st o. G ) z <_ n )
15		suprcl 10290	. . . 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 1218	. . 3 |- ( ph -> sup ( ran ( 1st o. G ) , RR , < ) e. RR )
17	1, 16	syl5eqel 2527	. 2 |- ( ph -> S e. RR )
18	2	adantr 465	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> F : NN --> RR )
19	3	adantr 465	. . . . . . . . 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 13517	. . . . . . . . . . 11 |- ( ph -> G : NN0 --> ( RR X. RR ) )
21		nnm1nn0 10621	. . . . . . . . . . 11 |- ( n e. NN -> ( n - 1 ) e. NN0 )
22		ffvelrn 5841	. . . . . . . . . . 11 |- ( ( G : NN0 --> ( RR X. RR ) /\ ( n - 1 ) e. NN0 ) -> ( G ` ( n - 1 ) ) e. ( RR X. RR ) )
23	20, 21, 22	syl2an 477	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( G ` ( n - 1 ) ) e. ( RR X. RR ) )
24		xp1st 6606	. . . . . . . . . 10 |- ( ( G ` ( n - 1 ) ) e. ( RR X. RR ) -> ( 1st ` ( G ` ( n - 1 ) ) ) e. RR )
25	23, 24	syl 16	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` ( n - 1 ) ) ) e. RR )
26		xp2nd 6607	. . . . . . . . . 10 |- ( ( G ` ( n - 1 ) ) e. ( RR X. RR ) -> ( 2nd ` ( G ` ( n - 1 ) ) ) e. RR )
27	23, 26	syl 16	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( G ` ( n - 1 ) ) ) e. RR )
28	2	ffvelrnda 5843	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( F ` n ) e. RR )
29		eqid 2443	. . . . . . . . 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 2443	. . . . . . . . 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 13519	. . . . . . . . . 10 |- ( ( ph /\ ( n - 1 ) e. NN0 ) -> ( 1st ` ( G ` ( n - 1 ) ) ) < ( 2nd ` ( G ` ( n - 1 ) ) ) )
32	21, 31	sylan2 474	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` ( n - 1 ) ) ) < ( 2nd ` ( G ` ( n - 1 ) ) ) )
33	18, 19, 25, 27, 28, 29, 30, 32	ruclem3 13515	. . . . . . . 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 13518	. . . . . . . . . . . . 13 |- ( ( ph /\ ( n - 1 ) e. NN0 ) -> ( G ` ( ( n - 1 ) + 1 ) ) = ( ( G ` ( n - 1 ) ) D ( F ` ( ( n - 1 ) + 1 ) ) ) )
35	21, 34	sylan2 474	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( G ` ( ( n - 1 ) + 1 ) ) = ( ( G ` ( n - 1 ) ) D ( F ` ( ( n - 1 ) + 1 ) ) ) )
36		nncn 10330	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n e. CC )
37	36	adantl 466	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> n e. CC )
38		ax-1cn 9340	. . . . . . . . . . . . . 14 |- 1 e. CC
39		npcan 9619	. . . . . . . . . . . . . 14 |- ( ( n e. CC /\ 1 e. CC ) -> ( ( n - 1 ) + 1 ) = n )
40	37, 38, 39	sylancl 662	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( n - 1 ) + 1 ) = n )
41	40	fveq2d 5695	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( G ` ( ( n - 1 ) + 1 ) ) = ( G ` n ) )
42		1st2nd2 6613	. . . . . . . . . . . . . 14 |- ( ( G ` ( n - 1 ) ) e. ( RR X. RR ) -> ( G ` ( n - 1 ) ) = <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. )
43	23, 42	syl 16	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( G ` ( n - 1 ) ) = <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. )
44	40	fveq2d 5695	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( F ` ( ( n - 1 ) + 1 ) ) = ( F ` n ) )
45	43, 44	oveq12d 6109	. . . . . . . . . . . 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 2483	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( G ` n ) = ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) )
47	46	fveq2d 5695	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) = ( 1st ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) )
48	47	breq2d 4304	. . . . . . . . 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 5695	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) = ( 2nd ` ( <. ( 1st ` ( G ` ( n - 1 ) ) ) , ( 2nd ` ( G ` ( n - 1 ) ) ) >. D ( F ` n ) ) ) )
50	49	breq1d 4302	. . . . . . . . 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 709	. . . . . . . 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 465	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ran ( 1st o. G ) C_ RR )
54	8	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ran ( 1st o. G ) =/= (/) )
55	14	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> E. n e. RR A. z e. ran ( 1st o. G ) z <_ n )
56		nnnn0 10586	. . . . . . . . . . . . 13 |- ( n e. NN -> n e. NN0 )
57		fvco3 5768	. . . . . . . . . . . . 13 |- ( ( G : NN0 --> ( RR X. RR ) /\ n e. NN0 ) -> ( ( 1st o. G ) ` n ) = ( 1st ` ( G ` n ) ) )
58	20, 56, 57	syl2an 477	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( ( 1st o. G ) ` n ) = ( 1st ` ( G ` n ) ) )
59	20	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> G : NN0 --> ( RR X. RR ) )
60		1stcof 6604	. . . . . . . . . . . . . 14 |- ( G : NN0 --> ( RR X. RR ) -> ( 1st o. G ) : NN0 --> RR )
61		ffn 5559	. . . . . . . . . . . . . 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 466	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> n e. NN0 )
64		fnfvelrn 5840	. . . . . . . . . . . . 13 |- ( ( ( 1st o. G ) Fn NN0 /\ n e. NN0 ) -> ( ( 1st o. G ) ` n ) e. ran ( 1st o. G ) )
65	62, 63, 64	syl2anc 661	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( ( 1st o. G ) ` n ) e. ran ( 1st o. G ) )
66	58, 65	eqeltrrd 2518	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) e. ran ( 1st o. G ) )
67		suprub 10291	. . . . . . . . . . 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 1221	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) <_ sup ( ran ( 1st o. G ) , RR , < ) )
69	68, 1	syl6breqr 4332	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) <_ S )
70		ffvelrn 5841	. . . . . . . . . . . 12 |- ( ( G : NN0 --> ( RR X. RR ) /\ n e. NN0 ) -> ( G ` n ) e. ( RR X. RR ) )
71	20, 56, 70	syl2an 477	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( G ` n ) e. ( RR X. RR ) )
72		xp1st 6606	. . . . . . . . . . 11 |- ( ( G ` n ) e. ( RR X. RR ) -> ( 1st ` ( G ` n ) ) e. RR )
73	71, 72	syl 16	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) e. RR )
74	17	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> S e. RR )
75		ltletr 9466	. . . . . . . . . 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 1218	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( ( F ` n ) < ( 1st ` ( G ` n ) ) /\ ( 1st ` ( G ` n ) ) <_ S ) -> ( F ` n ) < S ) )
77	69, 76	mpan2d 674	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( F ` n ) < ( 1st ` ( G ` n ) ) -> ( F ` n ) < S ) )
78		fvco3 5768	. . . . . . . . . . . . . . 15 |- ( ( G : NN0 --> ( RR X. RR ) /\ k e. NN0 ) -> ( ( 1st o. G ) ` k ) = ( 1st ` ( G ` k ) ) )
79	59, 78	sylan 471	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( ( 1st o. G ) ` k ) = ( 1st ` ( G ` k ) ) )
80	59	ffvelrnda 5843	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( G ` k ) e. ( RR X. RR ) )
81		xp1st 6606	. . . . . . . . . . . . . . . 16 |- ( ( G ` k ) e. ( RR X. RR ) -> ( 1st ` ( G ` k ) ) e. RR )
82	80, 81	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( 1st ` ( G ` k ) ) e. RR )
83		xp2nd 6607	. . . . . . . . . . . . . . . . 17 |- ( ( G ` n ) e. ( RR X. RR ) -> ( 2nd ` ( G ` n ) ) e. RR )
84	71, 83	syl 16	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) e. RR )
85	84	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( 2nd ` ( G ` n ) ) e. RR )
86	18	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> F : NN --> RR )
87	19	adantr 465	. . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> k e. NN0 )
89	63	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> n e. NN0 )
90	86, 87, 4, 5, 88, 89	ruclem10 13521	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( 1st ` ( G ` k ) ) < ( 2nd ` ( G ` n ) ) )
91	82, 85, 90	ltled 9522	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( 1st ` ( G ` k ) ) <_ ( 2nd ` ( G ` n ) ) )
92	79, 91	eqbrtrd 4312	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. NN ) /\ k e. NN0 ) -> ( ( 1st o. G ) ` k ) <_ ( 2nd ` ( G ` n ) ) )
93	92	ralrimiva 2799	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> A. k e. NN0 ( ( 1st o. G ) ` k ) <_ ( 2nd ` ( G ` n ) ) )
94		breq1 4295	. . . . . . . . . . . . . 14 |- ( z = ( ( 1st o. G ) ` k ) -> ( z <_ ( 2nd ` ( G ` n ) ) <-> ( ( 1st o. G ) ` k ) <_ ( 2nd ` ( G ` n ) ) ) )
95	94	ralrn 5846	. . . . . . . . . . . . 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 16	. . . . . . . . . . . 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 10294	. . . . . . . . . . . 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 1221	. . . . . . . . . . 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 4325	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> S <_ ( 2nd ` ( G ` n ) ) )
102		lelttr 9465	. . . . . . . . . 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 1218	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( S <_ ( 2nd ` ( G ` n ) ) /\ ( 2nd ` ( G ` n ) ) < ( F ` n ) ) -> S < ( F ` n ) ) )
104	101, 103	mpand 675	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) < ( F ` n ) -> S < ( F ` n ) ) )
105	77, 104	orim12d 834	. . . . . . 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 9513	. . . . . 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 2624	. . . 4 |- ( ( ph /\ n e. NN ) -> -. ( F ` n ) = S )
110	109	nrexdv 2819	. . 3 |- ( ph -> -. E. n e. NN ( F ` n ) = S )
111		risset 2763	. . . 4 |- ( S e. ran F <-> E. z e. ran F z = S )
112		ffn 5559	. . . . 5 |- ( F : NN --> RR -> F Fn NN )
113		eqeq1 2449	. . . . . 6 |- ( z = ( F ` n ) -> ( z = S <-> ( F ` n ) = S ) )
114	113	rexrn 5845	. . . . 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 301	. 2 |- ( ph -> -. S e. ran F )
118	17, 117	eldifd 3339	1 |- ( ph -> S e. ( RR \ ran F ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2606   A.wral 2715   E.wrex 2716   [_csb 3288   \ cdif 3325   u. cun 3326   C_ wss 3328   (/)c0 3637   ifcif 3791   {csn 3877   <.cop 3883   class class class wbr 4292   X. cxp 4838   ran crn 4841   o. ccom 4844   Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   e. cmpt2 6093   1stc1st 6575   2ndc2nd 6576   supcsup 7690   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283   + caddc 9285   < clt 9418   <_ cle 9419   - cmin 9595   / cdiv 9993   NNcn 10322   2c2 10371   NN0cn0 10579   seqcseq 11806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-seq 11807

This theorem is referenced by:  ruclem13  13524