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Theorem efif1olem3 22012
Description: Lemma for efif1o 22014. (Contributed by Mario Carneiro, 8-May-2015.)
Hypotheses
Ref Expression
efif1o.1  |-  F  =  ( w  e.  D  |->  ( exp `  (
_i  x.  w )
) )
efif1o.2  |-  C  =  ( `' abs " {
1 } )
Assertion
Ref Expression
efif1olem3  |-  ( (
ph  /\  x  e.  C )  ->  (
Im `  ( sqr `  x ) )  e.  ( -u 1 [,] 1 ) )
Distinct variable groups:    x, w, C    x, F    ph, w, x   
w, D, x
Allowed substitution hint:    F( w)

Proof of Theorem efif1olem3
StepHypRef Expression
1 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  C )
2 efif1o.2 . . . . . . 7  |-  C  =  ( `' abs " {
1 } )
31, 2syl6eleq 2533 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  ( `' abs " {
1 } ) )
4 absf 12837 . . . . . . 7  |-  abs : CC
--> RR
5 ffn 5571 . . . . . . 7  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
6 fniniseg 5836 . . . . . . 7  |-  ( abs 
Fn  CC  ->  ( x  e.  ( `' abs " { 1 } )  <-> 
( x  e.  CC  /\  ( abs `  x
)  =  1 ) ) )
74, 5, 6mp2b 10 . . . . . 6  |-  ( x  e.  ( `' abs " { 1 } )  <-> 
( x  e.  CC  /\  ( abs `  x
)  =  1 ) )
83, 7sylib 196 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  (
x  e.  CC  /\  ( abs `  x )  =  1 ) )
98simpld 459 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  CC )
109sqrcld 12935 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  ( sqr `  x )  e.  CC )
1110imcld 12696 . 2  |-  ( (
ph  /\  x  e.  C )  ->  (
Im `  ( sqr `  x ) )  e.  RR )
12 absimle 12810 . . . . . 6  |-  ( ( sqr `  x )  e.  CC  ->  ( abs `  ( Im `  ( sqr `  x ) ) )  <_  ( abs `  ( sqr `  x
) ) )
1310, 12syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( Im `  ( sqr `  x ) ) )  <_  ( abs `  ( sqr `  x
) ) )
149sqsqrd 12937 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  (
( sqr `  x
) ^ 2 )  =  x )
1514fveq2d 5707 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( ( sqr `  x ) ^ 2 ) )  =  ( abs `  x ) )
16 2nn0 10608 . . . . . . . . 9  |-  2  e.  NN0
17 absexp 12805 . . . . . . . . 9  |-  ( ( ( sqr `  x
)  e.  CC  /\  2  e.  NN0 )  -> 
( abs `  (
( sqr `  x
) ^ 2 ) )  =  ( ( abs `  ( sqr `  x ) ) ^
2 ) )
1810, 16, 17sylancl 662 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( ( sqr `  x ) ^ 2 ) )  =  ( ( abs `  ( sqr `  x ) ) ^ 2 ) )
198simprd 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  x )  =  1 )
2015, 18, 193eqtr3d 2483 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( abs `  ( sqr `  x ) ) ^ 2 )  =  1 )
21 sq1 11972 . . . . . . 7  |-  ( 1 ^ 2 )  =  1
2220, 21syl6eqr 2493 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  (
( abs `  ( sqr `  x ) ) ^ 2 )  =  ( 1 ^ 2 ) )
2310abscld 12934 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( sqr `  x
) )  e.  RR )
2410absge0d 12942 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  0  <_  ( abs `  ( sqr `  x ) ) )
25 1re 9397 . . . . . . . 8  |-  1  e.  RR
26 0le1 9875 . . . . . . . 8  |-  0  <_  1
27 sq11 11950 . . . . . . . 8  |-  ( ( ( ( abs `  ( sqr `  x ) )  e.  RR  /\  0  <_  ( abs `  ( sqr `  x ) ) )  /\  ( 1  e.  RR  /\  0  <_  1 ) )  -> 
( ( ( abs `  ( sqr `  x
) ) ^ 2 )  =  ( 1 ^ 2 )  <->  ( abs `  ( sqr `  x
) )  =  1 ) )
2825, 26, 27mpanr12 685 . . . . . . 7  |-  ( ( ( abs `  ( sqr `  x ) )  e.  RR  /\  0  <_  ( abs `  ( sqr `  x ) ) )  ->  ( (
( abs `  ( sqr `  x ) ) ^ 2 )  =  ( 1 ^ 2 )  <->  ( abs `  ( sqr `  x ) )  =  1 ) )
2923, 24, 28syl2anc 661 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  (
( ( abs `  ( sqr `  x ) ) ^ 2 )  =  ( 1 ^ 2 )  <->  ( abs `  ( sqr `  x ) )  =  1 ) )
3022, 29mpbid 210 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( sqr `  x
) )  =  1 )
3113, 30breqtrd 4328 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( Im `  ( sqr `  x ) ) )  <_  1
)
32 absle 12815 . . . . 5  |-  ( ( ( Im `  ( sqr `  x ) )  e.  RR  /\  1  e.  RR )  ->  (
( abs `  (
Im `  ( sqr `  x ) ) )  <_  1  <->  ( -u 1  <_  ( Im `  ( sqr `  x ) )  /\  ( Im `  ( sqr `  x ) )  <_  1 ) ) )
3311, 25, 32sylancl 662 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
( abs `  (
Im `  ( sqr `  x ) ) )  <_  1  <->  ( -u 1  <_  ( Im `  ( sqr `  x ) )  /\  ( Im `  ( sqr `  x ) )  <_  1 ) ) )
3431, 33mpbid 210 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  ( -u 1  <_  ( Im `  ( sqr `  x
) )  /\  (
Im `  ( sqr `  x ) )  <_ 
1 ) )
3534simpld 459 . 2  |-  ( (
ph  /\  x  e.  C )  ->  -u 1  <_  ( Im `  ( sqr `  x ) ) )
3634simprd 463 . 2  |-  ( (
ph  /\  x  e.  C )  ->  (
Im `  ( sqr `  x ) )  <_ 
1 )
37 neg1rr 10438 . . 3  |-  -u 1  e.  RR
3837, 25elicc2i 11373 . 2  |-  ( ( Im `  ( sqr `  x ) )  e.  ( -u 1 [,] 1 )  <->  ( (
Im `  ( sqr `  x ) )  e.  RR  /\  -u 1  <_  ( Im `  ( sqr `  x ) )  /\  ( Im `  ( sqr `  x ) )  <_  1 ) )
3911, 35, 36, 38syl3anbrc 1172 1  |-  ( (
ph  /\  x  e.  C )  ->  (
Im `  ( sqr `  x ) )  e.  ( -u 1 [,] 1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3889   class class class wbr 4304    e. cmpt 4362   `'ccnv 4851   "cima 4855    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103   CCcc 9292   RRcr 9293   0cc0 9294   1c1 9295   _ici 9296    x. cmul 9299    <_ cle 9431   -ucneg 9608   2c2 10383   NN0cn0 10591   [,]cicc 11315   ^cexp 11877   Imcim 12599   sqrcsqr 12734   abscabs 12735   expce 13359
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 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-icc 11319  df-seq 11819  df-exp 11878  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737
This theorem is referenced by:  efif1olem4  22013
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