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Theorem efif1olem3 20438
Description: Lemma for efif1o 20440. (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 448 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  C )
2 efif1o.2 . . . . . . 7  |-  C  =  ( `' abs " {
1 } )
31, 2syl6eleq 2525 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  ( `' abs " {
1 } ) )
4 absf 12133 . . . . . . 7  |-  abs : CC
--> RR
5 ffn 5583 . . . . . . 7  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
6 fniniseg 5843 . . . . . . 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 189 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  (
x  e.  CC  /\  ( abs `  x )  =  1 ) )
98simpld 446 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  CC )
109sqrcld 12231 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  ( sqr `  x )  e.  CC )
1110imcld 11992 . 2  |-  ( (
ph  /\  x  e.  C )  ->  (
Im `  ( sqr `  x ) )  e.  RR )
12 absimle 12106 . . . . . 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 12233 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  (
( sqr `  x
) ^ 2 )  =  x )
1514fveq2d 5724 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( ( sqr `  x ) ^ 2 ) )  =  ( abs `  x ) )
16 2nn0 10230 . . . . . . . . 9  |-  2  e.  NN0
17 absexp 12101 . . . . . . . . 9  |-  ( ( ( sqr `  x
)  e.  CC  /\  2  e.  NN0 )  -> 
( abs `  (
( sqr `  x
) ^ 2 ) )  =  ( ( abs `  ( sqr `  x ) ) ^
2 ) )
1810, 16, 17sylancl 644 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( ( sqr `  x ) ^ 2 ) )  =  ( ( abs `  ( sqr `  x ) ) ^ 2 ) )
198simprd 450 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  x )  =  1 )
2015, 18, 193eqtr3d 2475 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( abs `  ( sqr `  x ) ) ^ 2 )  =  1 )
21 sq1 11468 . . . . . . 7  |-  ( 1 ^ 2 )  =  1
2220, 21syl6eqr 2485 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  (
( abs `  ( sqr `  x ) ) ^ 2 )  =  ( 1 ^ 2 ) )
2310abscld 12230 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( sqr `  x
) )  e.  RR )
2410absge0d 12238 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  0  <_  ( abs `  ( sqr `  x ) ) )
25 1re 9082 . . . . . . . 8  |-  1  e.  RR
26 0le1 9543 . . . . . . . 8  |-  0  <_  1
27 sq11 11446 . . . . . . . 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 667 . . . . . . 7  |-  ( ( ( abs `  ( sqr `  x ) )  e.  RR  /\  0  <_  ( abs `  ( sqr `  x ) ) )  ->  ( (
( abs `  ( sqr `  x ) ) ^ 2 )  =  ( 1 ^ 2 )  <->  ( abs `  ( sqr `  x ) )  =  1 ) )
2923, 24, 28syl2anc 643 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  (
( ( abs `  ( sqr `  x ) ) ^ 2 )  =  ( 1 ^ 2 )  <->  ( abs `  ( sqr `  x ) )  =  1 ) )
3022, 29mpbid 202 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( sqr `  x
) )  =  1 )
3113, 30breqtrd 4228 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( Im `  ( sqr `  x ) ) )  <_  1
)
32 absle 12111 . . . . 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 644 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
( abs `  (
Im `  ( sqr `  x ) ) )  <_  1  <->  ( -u 1  <_  ( Im `  ( sqr `  x ) )  /\  ( Im `  ( sqr `  x ) )  <_  1 ) ) )
3431, 33mpbid 202 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  ( -u 1  <_  ( Im `  ( sqr `  x
) )  /\  (
Im `  ( sqr `  x ) )  <_ 
1 ) )
3534simpld 446 . 2  |-  ( (
ph  /\  x  e.  C )  ->  -u 1  <_  ( Im `  ( sqr `  x ) ) )
3634simprd 450 . 2  |-  ( (
ph  /\  x  e.  C )  ->  (
Im `  ( sqr `  x ) )  <_ 
1 )
3725renegcli 9354 . . 3  |-  -u 1  e.  RR
3837, 25elicc2i 10968 . 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 1138 1  |-  ( (
ph  /\  x  e.  C )  ->  (
Im `  ( sqr `  x ) )  e.  ( -u 1 [,] 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3806   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983   _ici 8984    x. cmul 8987    <_ cle 9113   -ucneg 9284   2c2 10041   NN0cn0 10213   [,]cicc 10911   ^cexp 11374   Imcim 11895   sqrcsqr 12030   abscabs 12031   expce 12656
This theorem is referenced by:  efif1olem4  20439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-icc 10915  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033
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