Theorem evth 22758

Description: The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)

Hypotheses

Ref	Expression
bndth.1	|- X = U. J
bndth.2	|- K = ( topGen ` ran (,) )
bndth.3	|- ( ph -> J e. Comp )
bndth.4	|- ( ph -> F e. ( J Cn K ) )
evth.5	|- ( ph -> X =/= (/) )

Assertion

Ref	Expression
evth	|- ( ph -> E. x e. X A. y e. X ( F ` y ) <_ ( F ` x ) )

Distinct variable groups:   x, y, F   y, K   ph, x, y   x, X, y   x, J, y

Allowed substitution hint:   K( x)

Proof of Theorem evth

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bndth.1	. . . . 5 |- X = U. J
2		bndth.2	. . . . 5 |- K = ( topGen ` ran (,) )
3		bndth.3	. . . . . 6 |- ( ph -> J e. Comp )
4	3	adantr 481	. . . . 5 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> J e. Comp )
5		cmptop 21198	. . . . . . . . . 10 |- ( J e. Comp -> J e. Top )
6	4, 5	syl 17	. . . . . . . . 9 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> J e. Top )
7	1	toptopon 20722	. . . . . . . . 9 |- ( J e. Top <-> J e. (TopOn ` X ) )
8	6, 7	sylib 208	. . . . . . . 8 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> J e. (TopOn ` X ) )
9		eqid 2622	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
10	9	cnfldtopon 22586	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
11	10	a1i 11	. . . . . . . . 9 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
12		1cnd 10056	. . . . . . . . 9 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> 1 e. CC )
13	8, 11, 12	cnmptc 21465	. . . . . . . 8 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( z e. X |-> 1 ) e. ( J Cn ( TopOpen ` ℂfld ) ) )
14		bndth.4	. . . . . . . . . . . . . . . . 17 |- ( ph -> F e. ( J Cn K ) )
15		uniretop 22566	. . . . . . . . . . . . . . . . . . 19 |- RR = U. ( topGen ` ran (,) )
16	2	unieqi 4445	. . . . . . . . . . . . . . . . . . 19 |- U. K = U. ( topGen ` ran (,) )
17	15, 16	eqtr4i 2647	. . . . . . . . . . . . . . . . . 18 |- RR = U. K
18	1, 17	cnf 21050	. . . . . . . . . . . . . . . . 17 |- ( F e. ( J Cn K ) -> F : X --> RR )
19	14, 18	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> F : X --> RR )
20		frn 6053	. . . . . . . . . . . . . . . 16 |- ( F : X --> RR -> ran F C_ RR )
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ran F C_ RR )
22		fdm 6051	. . . . . . . . . . . . . . . . . 18 |- ( F : X --> RR -> dom F = X )
23	19, 22	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> dom F = X )
24		evth.5	. . . . . . . . . . . . . . . . 17 |- ( ph -> X =/= (/) )
25	23, 24	eqnetrd 2861	. . . . . . . . . . . . . . . 16 |- ( ph -> dom F =/= (/) )
26		dm0rn0 5342	. . . . . . . . . . . . . . . . 17 |- ( dom F = (/) <-> ran F = (/) )
27	26	necon3bii 2846	. . . . . . . . . . . . . . . 16 |- ( dom F =/= (/) <-> ran F =/= (/) )
28	25, 27	sylib 208	. . . . . . . . . . . . . . 15 |- ( ph -> ran F =/= (/) )
29	1, 2, 3, 14	bndth 22757	. . . . . . . . . . . . . . . 16 |- ( ph -> E. x e. RR A. y e. X ( F ` y ) <_ x )
30		ffn 6045	. . . . . . . . . . . . . . . . . . 19 |- ( F : X --> RR -> F Fn X )
31	19, 30	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ph -> F Fn X )
32		breq1 4656	. . . . . . . . . . . . . . . . . . 19 |- ( z = ( F ` y ) -> ( z <_ x <-> ( F ` y ) <_ x ) )
33	32	ralrn 6362	. . . . . . . . . . . . . . . . . 18 |- ( F Fn X -> ( A. z e. ran F z <_ x <-> A. y e. X ( F ` y ) <_ x ) )
34	31, 33	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( A. z e. ran F z <_ x <-> A. y e. X ( F ` y ) <_ x ) )
35	34	rexbidv 3052	. . . . . . . . . . . . . . . 16 |- ( ph -> ( E. x e. RR A. z e. ran F z <_ x <-> E. x e. RR A. y e. X ( F ` y ) <_ x ) )
36	29, 35	mpbird 247	. . . . . . . . . . . . . . 15 |- ( ph -> E. x e. RR A. z e. ran F z <_ x )
37	21, 28, 36	3jca 1242	. . . . . . . . . . . . . 14 |- ( ph -> ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. z e. ran F z <_ x ) )
38		suprcl 10983	. . . . . . . . . . . . . 14 |- ( ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. z e. ran F z <_ x ) -> sup ( ran F , RR , < ) e. RR )
39	37, 38	syl 17	. . . . . . . . . . . . 13 |- ( ph -> sup ( ran F , RR , < ) e. RR )
40	39	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> sup ( ran F , RR , < ) e. CC )
41	40	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> sup ( ran F , RR , < ) e. CC )
42	8, 11, 41	cnmptc 21465	. . . . . . . . . 10 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( z e. X |-> sup ( ran F , RR , < ) ) e. ( J Cn ( TopOpen ` ℂfld ) ) )
43	19	feqmptd 6249	. . . . . . . . . . . 12 |- ( ph -> F = ( z e. X |-> ( F ` z ) ) )
44	9	cnfldtop 22587	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) e. Top
45		cnrest2r 21091	. . . . . . . . . . . . . 14 |- ( ( TopOpen ` ℂfld ) e. Top -> ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) C_ ( J Cn ( TopOpen ` ℂfld ) ) )
46	44, 45	ax-mp 5	. . . . . . . . . . . . 13 |- ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) C_ ( J Cn ( TopOpen ` ℂfld ) )
47	9	tgioo2 22606	. . . . . . . . . . . . . . . 16 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
48	2, 47	eqtri 2644	. . . . . . . . . . . . . . 15 |- K = ( ( TopOpen ` ℂfld ) ↾t RR )
49	48	oveq2i 6661	. . . . . . . . . . . . . 14 |- ( J Cn K ) = ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) )
50	14, 49	syl6eleq 2711	. . . . . . . . . . . . 13 |- ( ph -> F e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) )
51	46, 50	sseldi 3601	. . . . . . . . . . . 12 |- ( ph -> F e. ( J Cn ( TopOpen ` ℂfld ) ) )
52	43, 51	eqeltrrd 2702	. . . . . . . . . . 11 |- ( ph -> ( z e. X |-> ( F ` z ) ) e. ( J Cn ( TopOpen ` ℂfld ) ) )
53	52	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( z e. X |-> ( F ` z ) ) e. ( J Cn ( TopOpen ` ℂfld ) ) )
54	9	subcn 22669	. . . . . . . . . . 11 |- - e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
55	54	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> - e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) ) )
56	8, 42, 53, 55	cnmpt12f 21469	. . . . . . . . 9 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) e. ( J Cn ( TopOpen ` ℂfld ) ) )
57	39	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> sup ( ran F , RR , < ) e. RR )
58		ffvelrn 6357	. . . . . . . . . . . . . . . . . 18 |- ( ( F : X --> ( RR \ { sup ( ran F , RR , < ) } ) /\ z e. X ) -> ( F ` z ) e. ( RR \ { sup ( ran F , RR , < ) } ) )
59	58	adantll 750	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> ( F ` z ) e. ( RR \ { sup ( ran F , RR , < ) } ) )
60		eldifsn 4317	. . . . . . . . . . . . . . . . 17 |- ( ( F ` z ) e. ( RR \ { sup ( ran F , RR , < ) } ) <-> ( ( F ` z ) e. RR /\ ( F ` z ) =/= sup ( ran F , RR , < ) ) )
61	59, 60	sylib 208	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> ( ( F ` z ) e. RR /\ ( F ` z ) =/= sup ( ran F , RR , < ) ) )
62	61	simpld 475	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> ( F ` z ) e. RR )
63	57, 62	resubcld 10458	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> ( sup ( ran F , RR , < ) - ( F ` z ) ) e. RR )
64	63	recnd 10068	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> ( sup ( ran F , RR , < ) - ( F ` z ) ) e. CC )
65	57	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> sup ( ran F , RR , < ) e. CC )
66	62	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> ( F ` z ) e. CC )
67	61	simprd 479	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> ( F ` z ) =/= sup ( ran F , RR , < ) )
68	67	necomd 2849	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> sup ( ran F , RR , < ) =/= ( F ` z ) )
69	65, 66, 68	subne0d 10401	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> ( sup ( ran F , RR , < ) - ( F ` z ) ) =/= 0 )
70		eldifsn 4317	. . . . . . . . . . . . 13 |- ( ( sup ( ran F , RR , < ) - ( F ` z ) ) e. ( CC \ { 0 } ) <-> ( ( sup ( ran F , RR , < ) - ( F ` z ) ) e. CC /\ ( sup ( ran F , RR , < ) - ( F ` z ) ) =/= 0 ) )
71	64, 69, 70	sylanbrc 698	. . . . . . . . . . . 12 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> ( sup ( ran F , RR , < ) - ( F ` z ) ) e. ( CC \ { 0 } ) )
72		eqid 2622	. . . . . . . . . . . 12 |- ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) = ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) )
73	71, 72	fmptd 6385	. . . . . . . . . . 11 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) : X --> ( CC \ { 0 } ) )
74		frn 6053	. . . . . . . . . . 11 |- ( ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) : X --> ( CC \ { 0 } ) -> ran ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) C_ ( CC \ { 0 } ) )
75	73, 74	syl 17	. . . . . . . . . 10 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ran ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) C_ ( CC \ { 0 } ) )
76		difssd 3738	. . . . . . . . . 10 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( CC \ { 0 } ) C_ CC )
77		cnrest2 21090	. . . . . . . . . 10 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ran ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) C_ ( CC \ { 0 } ) /\ ( CC \ { 0 } ) C_ CC ) -> ( ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) e. ( J Cn ( TopOpen ` ℂfld ) ) <-> ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t ( CC \ { 0 } ) ) ) ) )
78	11, 75, 76, 77	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) e. ( J Cn ( TopOpen ` ℂfld ) ) <-> ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t ( CC \ { 0 } ) ) ) ) )
79	56, 78	mpbid 222	. . . . . . . 8 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( z e. X |-> ( sup ( ran F , RR , < ) - ( F ` z ) ) ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t ( CC \ { 0 } ) ) ) )
80		eqid 2622	. . . . . . . . . 10 |- ( ( TopOpen ` ℂfld ) ↾t ( CC \ { 0 } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( CC \ { 0 } ) )
81	9, 80	divcn 22671	. . . . . . . . 9 |- / e. ( ( ( TopOpen ` ℂfld ) tX ( ( TopOpen ` ℂfld ) ↾t ( CC \ { 0 } ) ) ) Cn ( TopOpen ` ℂfld ) )
82	81	a1i 11	. . . . . . . 8 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> / e. ( ( ( TopOpen ` ℂfld ) tX ( ( TopOpen ` ℂfld ) ↾t ( CC \ { 0 } ) ) ) Cn ( TopOpen ` ℂfld ) ) )
83	8, 13, 79, 82	cnmpt12f 21469	. . . . . . 7 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) e. ( J Cn ( TopOpen ` ℂfld ) ) )
84	63, 69	rereccld 10852	. . . . . . . . . 10 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ z e. X ) -> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) e. RR )
85		eqid 2622	. . . . . . . . . 10 |- ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) = ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) )
86	84, 85	fmptd 6385	. . . . . . . . 9 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) : X --> RR )
87		frn 6053	. . . . . . . . 9 |- ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) : X --> RR -> ran ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) C_ RR )
88	86, 87	syl 17	. . . . . . . 8 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ran ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) C_ RR )
89		ax-resscn 9993	. . . . . . . . 9 |- RR C_ CC
90	89	a1i 11	. . . . . . . 8 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> RR C_ CC )
91		cnrest2 21090	. . . . . . . 8 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ran ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) C_ RR /\ RR C_ CC ) -> ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) e. ( J Cn ( TopOpen ` ℂfld ) ) <-> ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) ) )
92	11, 88, 90, 91	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) e. ( J Cn ( TopOpen ` ℂfld ) ) <-> ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) ) )
93	83, 92	mpbid 222	. . . . . 6 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) e. ( J Cn ( ( TopOpen ` ℂfld ) ↾t RR ) ) )
94	93, 49	syl6eleqr 2712	. . . . 5 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) e. ( J Cn K ) )
95	1, 2, 4, 94	bndth 22757	. . . 4 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> E. x e. RR A. y e. X ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) ` y ) <_ x )
96	39	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> sup ( ran F , RR , < ) e. RR )
97		simpr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> x e. RR )
98		1re 10039	. . . . . . . . . . 11 |- 1 e. RR
99		ifcl 4130	. . . . . . . . . . 11 |- ( ( x e. RR /\ 1 e. RR ) -> if ( 1 <_ x , x , 1 ) e. RR )
100	97, 98, 99	sylancl 694	. . . . . . . . . 10 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> if ( 1 <_ x , x , 1 ) e. RR )
101		0red 10041	. . . . . . . . . . . 12 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> 0 e. RR )
102	98	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> 1 e. RR )
103		0lt1 10550	. . . . . . . . . . . . 13 |- 0 < 1
104	103	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> 0 < 1 )
105		max1 12016	. . . . . . . . . . . . 13 |- ( ( 1 e. RR /\ x e. RR ) -> 1 <_ if ( 1 <_ x , x , 1 ) )
106	98, 97, 105	sylancr 695	. . . . . . . . . . . 12 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> 1 <_ if ( 1 <_ x , x , 1 ) )
107	101, 102, 100, 104, 106	ltletrd 10197	. . . . . . . . . . 11 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> 0 < if ( 1 <_ x , x , 1 ) )
108	107	gt0ne0d 10592	. . . . . . . . . 10 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> if ( 1 <_ x , x , 1 ) =/= 0 )
109	100, 108	rereccld 10852	. . . . . . . . 9 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> ( 1 / if ( 1 <_ x , x , 1 ) ) e. RR )
110	100, 107	recgt0d 10958	. . . . . . . . 9 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> 0 < ( 1 / if ( 1 <_ x , x , 1 ) ) )
111	109, 110	elrpd 11869	. . . . . . . 8 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> ( 1 / if ( 1 <_ x , x , 1 ) ) e. RR+ )
112	96, 111	ltsubrpd 11904	. . . . . . 7 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) < sup ( ran F , RR , < ) )
113	96, 109	resubcld 10458	. . . . . . . 8 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) e. RR )
114	113, 96	ltnled 10184	. . . . . . 7 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> ( ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) < sup ( ran F , RR , < ) <-> -. sup ( ran F , RR , < ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
115	112, 114	mpbid 222	. . . . . 6 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> -. sup ( ran F , RR , < ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) )
116		simprl 794	. . . . . . . . . . . 12 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> x e. RR )
117		max2 12018	. . . . . . . . . . . 12 |- ( ( 1 e. RR /\ x e. RR ) -> x <_ if ( 1 <_ x , x , 1 ) )
118	98, 116, 117	sylancr 695	. . . . . . . . . . 11 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> x <_ if ( 1 <_ x , x , 1 ) )
119	39	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> sup ( ran F , RR , < ) e. RR )
120		ffvelrn 6357	. . . . . . . . . . . . . . . . 17 |- ( ( F : X --> ( RR \ { sup ( ran F , RR , < ) } ) /\ y e. X ) -> ( F ` y ) e. ( RR \ { sup ( ran F , RR , < ) } ) )
121	120	ad2ant2l 782	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( F ` y ) e. ( RR \ { sup ( ran F , RR , < ) } ) )
122		eldifsn 4317	. . . . . . . . . . . . . . . 16 |- ( ( F ` y ) e. ( RR \ { sup ( ran F , RR , < ) } ) <-> ( ( F ` y ) e. RR /\ ( F ` y ) =/= sup ( ran F , RR , < ) ) )
123	121, 122	sylib 208	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( ( F ` y ) e. RR /\ ( F ` y ) =/= sup ( ran F , RR , < ) ) )
124	123	simpld 475	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( F ` y ) e. RR )
125	119, 124	resubcld 10458	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( sup ( ran F , RR , < ) - ( F ` y ) ) e. RR )
126	37	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ y e. X ) -> ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. z e. ran F z <_ x ) )
127		fnfvelrn 6356	. . . . . . . . . . . . . . . . . . 19 |- ( ( F Fn X /\ y e. X ) -> ( F ` y ) e. ran F )
128	31, 127	sylan 488	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ y e. X ) -> ( F ` y ) e. ran F )
129		suprub 10984	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. z e. ran F z <_ x ) /\ ( F ` y ) e. ran F ) -> ( F ` y ) <_ sup ( ran F , RR , < ) )
130	126, 128, 129	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. X ) -> ( F ` y ) <_ sup ( ran F , RR , < ) )
131	130	ad2ant2rl 785	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( F ` y ) <_ sup ( ran F , RR , < ) )
132	123	simprd 479	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( F ` y ) =/= sup ( ran F , RR , < ) )
133	132	necomd 2849	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> sup ( ran F , RR , < ) =/= ( F ` y ) )
134	124, 119	ltlend 10182	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( ( F ` y ) < sup ( ran F , RR , < ) <-> ( ( F ` y ) <_ sup ( ran F , RR , < ) /\ sup ( ran F , RR , < ) =/= ( F ` y ) ) ) )
135	131, 133, 134	mpbir2and 957	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( F ` y ) < sup ( ran F , RR , < ) )
136	124, 119	posdifd 10614	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( ( F ` y ) < sup ( ran F , RR , < ) <-> 0 < ( sup ( ran F , RR , < ) - ( F ` y ) ) ) )
137	135, 136	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> 0 < ( sup ( ran F , RR , < ) - ( F ` y ) ) )
138	137	gt0ne0d 10592	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( sup ( ran F , RR , < ) - ( F ` y ) ) =/= 0 )
139	125, 138	rereccld 10852	. . . . . . . . . . . 12 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) e. RR )
140	116, 98, 99	sylancl 694	. . . . . . . . . . . 12 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> if ( 1 <_ x , x , 1 ) e. RR )
141		letr 10131	. . . . . . . . . . . 12 |- ( ( ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) e. RR /\ x e. RR /\ if ( 1 <_ x , x , 1 ) e. RR ) -> ( ( ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ x /\ x <_ if ( 1 <_ x , x , 1 ) ) -> ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ if ( 1 <_ x , x , 1 ) ) )
142	139, 116, 140, 141	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( ( ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ x /\ x <_ if ( 1 <_ x , x , 1 ) ) -> ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ if ( 1 <_ x , x , 1 ) ) )
143	118, 142	mpan2d 710	. . . . . . . . . 10 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ x -> ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ if ( 1 <_ x , x , 1 ) ) )
144		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( z = y -> ( F ` z ) = ( F ` y ) )
145	144	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( z = y -> ( sup ( ran F , RR , < ) - ( F ` z ) ) = ( sup ( ran F , RR , < ) - ( F ` y ) ) )
146	145	oveq2d 6666	. . . . . . . . . . . . 13 |- ( z = y -> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) = ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) )
147		ovex 6678	. . . . . . . . . . . . 13 |- ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) e. _V
148	146, 85, 147	fvmpt 6282	. . . . . . . . . . . 12 |- ( y e. X -> ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) ` y ) = ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) )
149	148	breq1d 4663	. . . . . . . . . . 11 |- ( y e. X -> ( ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) ` y ) <_ x <-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ x ) )
150	149	ad2antll 765	. . . . . . . . . 10 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) ` y ) <_ x <-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ x ) )
151	109	adantrr 753	. . . . . . . . . . . 12 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( 1 / if ( 1 <_ x , x , 1 ) ) e. RR )
152	107	adantrr 753	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> 0 < if ( 1 <_ x , x , 1 ) )
153	140, 152	recgt0d 10958	. . . . . . . . . . . 12 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> 0 < ( 1 / if ( 1 <_ x , x , 1 ) ) )
154		lerec 10906	. . . . . . . . . . . 12 |- ( ( ( ( 1 / if ( 1 <_ x , x , 1 ) ) e. RR /\ 0 < ( 1 / if ( 1 <_ x , x , 1 ) ) ) /\ ( ( sup ( ran F , RR , < ) - ( F ` y ) ) e. RR /\ 0 < ( sup ( ran F , RR , < ) - ( F ` y ) ) ) ) -> ( ( 1 / if ( 1 <_ x , x , 1 ) ) <_ ( sup ( ran F , RR , < ) - ( F ` y ) ) <-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ ( 1 / ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
155	151, 153, 125, 137, 154	syl22anc 1327	. . . . . . . . . . 11 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( ( 1 / if ( 1 <_ x , x , 1 ) ) <_ ( sup ( ran F , RR , < ) - ( F ` y ) ) <-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ ( 1 / ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
156		lesub 10507	. . . . . . . . . . . 12 |- ( ( ( 1 / if ( 1 <_ x , x , 1 ) ) e. RR /\ sup ( ran F , RR , < ) e. RR /\ ( F ` y ) e. RR ) -> ( ( 1 / if ( 1 <_ x , x , 1 ) ) <_ ( sup ( ran F , RR , < ) - ( F ` y ) ) <-> ( F ` y ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
157	151, 119, 124, 156	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( ( 1 / if ( 1 <_ x , x , 1 ) ) <_ ( sup ( ran F , RR , < ) - ( F ` y ) ) <-> ( F ` y ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
158	140	recnd 10068	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> if ( 1 <_ x , x , 1 ) e. CC )
159	108	adantrr 753	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> if ( 1 <_ x , x , 1 ) =/= 0 )
160	158, 159	recrecd 10798	. . . . . . . . . . . 12 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( 1 / ( 1 / if ( 1 <_ x , x , 1 ) ) ) = if ( 1 <_ x , x , 1 ) )
161	160	breq2d 4665	. . . . . . . . . . 11 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ ( 1 / ( 1 / if ( 1 <_ x , x , 1 ) ) ) <-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ if ( 1 <_ x , x , 1 ) ) )
162	155, 157, 161	3bitr3d 298	. . . . . . . . . 10 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( ( F ` y ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) <-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` y ) ) ) <_ if ( 1 <_ x , x , 1 ) ) )
163	143, 150, 162	3imtr4d 283	. . . . . . . . 9 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ ( x e. RR /\ y e. X ) ) -> ( ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) ` y ) <_ x -> ( F ` y ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
164	163	anassrs 680	. . . . . . . 8 |- ( ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) /\ y e. X ) -> ( ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) ` y ) <_ x -> ( F ` y ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
165	164	ralimdva 2962	. . . . . . 7 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> ( A. y e. X ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) ` y ) <_ x -> A. y e. X ( F ` y ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
166	37	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. z e. ran F z <_ x ) )
167		suprleub 10989	. . . . . . . . 9 |- ( ( ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. z e. ran F z <_ x ) /\ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) e. RR ) -> ( sup ( ran F , RR , < ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) <-> A. z e. ran F z <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
168	166, 113, 167	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> ( sup ( ran F , RR , < ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) <-> A. z e. ran F z <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
169	31	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> F Fn X )
170		breq1 4656	. . . . . . . . . 10 |- ( z = ( F ` y ) -> ( z <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) <-> ( F ` y ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
171	170	ralrn 6362	. . . . . . . . 9 |- ( F Fn X -> ( A. z e. ran F z <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) <-> A. y e. X ( F ` y ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
172	169, 171	syl 17	. . . . . . . 8 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> ( A. z e. ran F z <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) <-> A. y e. X ( F ` y ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
173	168, 172	bitrd 268	. . . . . . 7 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> ( sup ( ran F , RR , < ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) <-> A. y e. X ( F ` y ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
174	165, 173	sylibrd 249	. . . . . 6 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> ( A. y e. X ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) ` y ) <_ x -> sup ( ran F , RR , < ) <_ ( sup ( ran F , RR , < ) - ( 1 / if ( 1 <_ x , x , 1 ) ) ) ) )
175	115, 174	mtod 189	. . . . 5 |- ( ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) /\ x e. RR ) -> -. A. y e. X ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) ` y ) <_ x )
176	175	nrexdv 3001	. . . 4 |- ( ( ph /\ F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) -> -. E. x e. RR A. y e. X ( ( z e. X |-> ( 1 / ( sup ( ran F , RR , < ) - ( F ` z ) ) ) ) ` y ) <_ x )
177	95, 176	pm2.65da 600	. . 3 |- ( ph -> -. F : X --> ( RR \ { sup ( ran F , RR , < ) } ) )
178	130	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. y e. X ( F ` y ) <_ sup ( ran F , RR , < ) )
179		breq2 4657	. . . . . . . . . 10 |- ( ( F ` x ) = sup ( ran F , RR , < ) -> ( ( F ` y ) <_ ( F ` x ) <-> ( F ` y ) <_ sup ( ran F , RR , < ) ) )
180	179	ralbidv 2986	. . . . . . . . 9 |- ( ( F ` x ) = sup ( ran F , RR , < ) -> ( A. y e. X ( F ` y ) <_ ( F ` x ) <-> A. y e. X ( F ` y ) <_ sup ( ran F , RR , < ) ) )
181	178, 180	syl5ibrcom 237	. . . . . . . 8 |- ( ph -> ( ( F ` x ) = sup ( ran F , RR , < ) -> A. y e. X ( F ` y ) <_ ( F ` x ) ) )
182	181	necon3bd 2808	. . . . . . 7 |- ( ph -> ( -. A. y e. X ( F ` y ) <_ ( F ` x ) -> ( F ` x ) =/= sup ( ran F , RR , < ) ) )
183	182	adantr 481	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( -. A. y e. X ( F ` y ) <_ ( F ` x ) -> ( F ` x ) =/= sup ( ran F , RR , < ) ) )
184	19	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. RR )
185		eldifsn 4317	. . . . . . . 8 |- ( ( F ` x ) e. ( RR \ { sup ( ran F , RR , < ) } ) <-> ( ( F ` x ) e. RR /\ ( F ` x ) =/= sup ( ran F , RR , < ) ) )
186	185	baib 944	. . . . . . 7 |- ( ( F ` x ) e. RR -> ( ( F ` x ) e. ( RR \ { sup ( ran F , RR , < ) } ) <-> ( F ` x ) =/= sup ( ran F , RR , < ) ) )
187	184, 186	syl 17	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( F ` x ) e. ( RR \ { sup ( ran F , RR , < ) } ) <-> ( F ` x ) =/= sup ( ran F , RR , < ) ) )
188	183, 187	sylibrd 249	. . . . 5 |- ( ( ph /\ x e. X ) -> ( -. A. y e. X ( F ` y ) <_ ( F ` x ) -> ( F ` x ) e. ( RR \ { sup ( ran F , RR , < ) } ) ) )
189	188	ralimdva 2962	. . . 4 |- ( ph -> ( A. x e. X -. A. y e. X ( F ` y ) <_ ( F ` x ) -> A. x e. X ( F ` x ) e. ( RR \ { sup ( ran F , RR , < ) } ) ) )
190		ffnfv 6388	. . . . . 6 |- ( F : X --> ( RR \ { sup ( ran F , RR , < ) } ) <-> ( F Fn X /\ A. x e. X ( F ` x ) e. ( RR \ { sup ( ran F , RR , < ) } ) ) )
191	190	baib 944	. . . . 5 |- ( F Fn X -> ( F : X --> ( RR \ { sup ( ran F , RR , < ) } ) <-> A. x e. X ( F ` x ) e. ( RR \ { sup ( ran F , RR , < ) } ) ) )
192	31, 191	syl 17	. . . 4 |- ( ph -> ( F : X --> ( RR \ { sup ( ran F , RR , < ) } ) <-> A. x e. X ( F ` x ) e. ( RR \ { sup ( ran F , RR , < ) } ) ) )
193	189, 192	sylibrd 249	. . 3 |- ( ph -> ( A. x e. X -. A. y e. X ( F ` y ) <_ ( F ` x ) -> F : X --> ( RR \ { sup ( ran F , RR , < ) } ) ) )
194	177, 193	mtod 189	. 2 |- ( ph -> -. A. x e. X -. A. y e. X ( F ` y ) <_ ( F ` x ) )
195		dfrex2 2996	. 2 |- ( E. x e. X A. y e. X ( F ` y ) <_ ( F ` x ) <-> -. A. x e. X -. A. y e. X ( F ` y ) <_ ( F ` x ) )
196	194, 195	sylibr 224	1 |- ( ph -> E. x e. X A. y e. X ( F ` y ) <_ ( F ` x ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   (,)cioo 12175   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   Compccmp 21189   tX ctx 21363

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-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-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-iin 4523  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127

This theorem is referenced by:  evth2  22759  evthicc  23228  evthf  39186  cncmpmax  39191