Theorem imo72b2lem1 38471

Description: Lemma for imo72b2 38475. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypotheses

Ref	Expression
imo72b2lem1.1	|- ( ph -> F : RR --> RR )
imo72b2lem1.7	|- ( ph -> E. x e. RR ( F ` x ) =/= 0 )
imo72b2lem1.6	|- ( ph -> A. y e. RR ( abs ` ( F ` y ) ) <_ 1 )

Assertion

Ref	Expression
imo72b2lem1	|- ( ph -> 0 < sup ( ( abs " ( F " RR ) ) , RR , < ) )

Distinct variable groups:   x, F   y, F   ph, x   ph, y

Proof of Theorem imo72b2lem1

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

Step	Hyp	Ref	Expression
1		imaco 5640	. . . 4 |- ( ( abs o. F ) " RR ) = ( abs " ( F " RR ) )
2	1	eqcomi 2631	. . 3 |- ( abs " ( F " RR ) ) = ( ( abs o. F ) " RR )
3		imassrn 5477	. . . . 5 |- ( ( abs o. F ) " RR ) C_ ran ( abs o. F )
4	3	a1i 11	. . . 4 |- ( ph -> ( ( abs o. F ) " RR ) C_ ran ( abs o. F ) )
5		imo72b2lem1.1	. . . . . 6 |- ( ph -> F : RR --> RR )
6		absf 14077	. . . . . . . 8 |- abs : CC --> RR
7	6	a1i 11	. . . . . . 7 |- ( ph -> abs : CC --> RR )
8		ax-resscn 9993	. . . . . . . 8 |- RR C_ CC
9	8	a1i 11	. . . . . . 7 |- ( ph -> RR C_ CC )
10	7, 9	fssresd 6071	. . . . . 6 |- ( ph -> ( abs |` RR ) : RR --> RR )
11	5, 10	fco2d 38461	. . . . 5 |- ( ph -> ( abs o. F ) : RR --> RR )
12		frn 6053	. . . . 5 |- ( ( abs o. F ) : RR --> RR -> ran ( abs o. F ) C_ RR )
13	11, 12	syl 17	. . . 4 |- ( ph -> ran ( abs o. F ) C_ RR )
14	4, 13	sstrd 3613	. . 3 |- ( ph -> ( ( abs o. F ) " RR ) C_ RR )
15	2, 14	syl5eqss 3649	. 2 |- ( ph -> ( abs " ( F " RR ) ) C_ RR )
16		0re 10040	. . . . . . . 8 |- 0 e. RR
17	16	ne0ii 3923	. . . . . . 7 |- RR =/= (/)
18	17	a1i 11	. . . . . 6 |- ( ph -> RR =/= (/) )
19	18, 11	wnefimgd 38460	. . . . 5 |- ( ph -> ( ( abs o. F ) " RR ) =/= (/) )
20	19	necomd 2849	. . . 4 |- ( ph -> (/) =/= ( ( abs o. F ) " RR ) )
21	2	a1i 11	. . . 4 |- ( ph -> ( abs " ( F " RR ) ) = ( ( abs o. F ) " RR ) )
22	20, 21	neeqtrrd 2868	. . 3 |- ( ph -> (/) =/= ( abs " ( F " RR ) ) )
23	22	necomd 2849	. 2 |- ( ph -> ( abs " ( F " RR ) ) =/= (/) )
24		1red 10055	. . 3 |- ( ph -> 1 e. RR )
25		simpr 477	. . . . 5 |- ( ( ph /\ c = 1 ) -> c = 1 )
26	25	breq2d 4665	. . . 4 |- ( ( ph /\ c = 1 ) -> ( t <_ c <-> t <_ 1 ) )
27	26	ralbidv 2986	. . 3 |- ( ( ph /\ c = 1 ) -> ( A. t e. ( abs " ( F " RR ) ) t <_ c <-> A. t e. ( abs " ( F " RR ) ) t <_ 1 ) )
28		imo72b2lem1.6	. . . 4 |- ( ph -> A. y e. RR ( abs ` ( F ` y ) ) <_ 1 )
29	5, 28	extoimad 38464	. . 3 |- ( ph -> A. t e. ( abs " ( F " RR ) ) t <_ 1 )
30	24, 27, 29	rspcedvd 3317	. 2 |- ( ph -> E. c e. RR A. t e. ( abs " ( F " RR ) ) t <_ c )
31		0red 10041	. 2 |- ( ph -> 0 e. RR )
32		imo72b2lem1.7	. . 3 |- ( ph -> E. x e. RR ( F ` x ) =/= 0 )
33	5	adantr 481	. . . . . 6 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> F : RR --> RR )
34		simprl 794	. . . . . 6 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> x e. RR )
35	33, 34	fvco3d 38462	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> ( ( abs o. F ) ` x ) = ( abs ` ( F ` x ) ) )
36	11	funfvima2d 38469	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( ( abs o. F ) ` x ) e. ( ( abs o. F ) " RR ) )
37	36	adantrr 753	. . . . . 6 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> ( ( abs o. F ) ` x ) e. ( ( abs o. F ) " RR ) )
38	37, 1	syl6eleq 2711	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> ( ( abs o. F ) ` x ) e. ( abs " ( F " RR ) ) )
39	35, 38	eqeltrrd 2702	. . . 4 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> ( abs ` ( F ` x ) ) e. ( abs " ( F " RR ) ) )
40		simpr 477	. . . . 5 |- ( ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) /\ z = ( abs ` ( F ` x ) ) ) -> z = ( abs ` ( F ` x ) ) )
41	40	breq2d 4665	. . . 4 |- ( ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) /\ z = ( abs ` ( F ` x ) ) ) -> ( 0 < z <-> 0 < ( abs ` ( F ` x ) ) ) )
42	5	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> ( F ` x ) e. RR )
43	42	adantrr 753	. . . . . . 7 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> ( F ` x ) e. RR )
44	43	recnd 10068	. . . . . 6 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> ( F ` x ) e. CC )
45		simprr 796	. . . . . 6 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> ( F ` x ) =/= 0 )
46	44, 45	absrpcld 14187	. . . . 5 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> ( abs ` ( F ` x ) ) e. RR+ )
47	46	rpgt0d 11875	. . . 4 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> 0 < ( abs ` ( F ` x ) ) )
48	39, 41, 47	rspcedvd 3317	. . 3 |- ( ( ph /\ ( x e. RR /\ ( F ` x ) =/= 0 ) ) -> E. z e. ( abs " ( F " RR ) ) 0 < z )
49	32, 48	rexlimddv 3035	. 2 |- ( ph -> E. z e. ( abs " ( F " RR ) ) 0 < z )
50	15, 23, 30, 31, 49	suprlubrd 38470	1 |- ( ph -> 0 < sup ( ( abs " ( F " RR ) ) , RR , < ) )

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   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   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:  imo72b2  38475