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Theorem efif1olem3 22692
Description: Lemma for efif1o 22694. (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 2565 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  ( `' abs " {
1 } ) )
4 absf 13133 . . . . . . 7  |-  abs : CC
--> RR
5 ffn 5731 . . . . . . 7  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
6 fniniseg 6002 . . . . . . 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 )
109sqrtcld 13231 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  ( sqr `  x )  e.  CC )
1110imcld 12991 . 2  |-  ( (
ph  /\  x  e.  C )  ->  (
Im `  ( sqr `  x ) )  e.  RR )
12 absimle 13105 . . . . . 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
) ) )
149sqsqrtd 13233 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  (
( sqr `  x
) ^ 2 )  =  x )
1514fveq2d 5870 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( ( sqr `  x ) ^ 2 ) )  =  ( abs `  x ) )
16 2nn0 10812 . . . . . . . . 9  |-  2  e.  NN0
17 absexp 13100 . . . . . . . . 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 2516 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( abs `  ( sqr `  x ) ) ^ 2 )  =  1 )
21 sq1 12230 . . . . . . 7  |-  ( 1 ^ 2 )  =  1
2220, 21syl6eqr 2526 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  (
( abs `  ( sqr `  x ) ) ^ 2 )  =  ( 1 ^ 2 ) )
2310abscld 13230 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( sqr `  x
) )  e.  RR )
2410absge0d 13238 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  0  <_  ( abs `  ( sqr `  x ) ) )
25 1re 9595 . . . . . . . 8  |-  1  e.  RR
26 0le1 10076 . . . . . . . 8  |-  0  <_  1
27 sq11 12208 . . . . . . . 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 4471 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  ( abs `  ( Im `  ( sqr `  x ) ) )  <_  1
)
32 absle 13111 . . . . 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 10640 . . 3  |-  -u 1  e.  RR
3837, 25elicc2i 11590 . 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 1180 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 1379    e. wcel 1767   {csn 4027   class class class wbr 4447    |-> cmpt 4505   `'ccnv 4998   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493   _ici 9494    x. cmul 9497    <_ cle 9629   -ucneg 9806   2c2 10585   NN0cn0 10795   [,]cicc 11532   ^cexp 12134   Imcim 12894   sqrcsqrt 13029   abscabs 13030   expce 13659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-icc 11536  df-seq 12076  df-exp 12135  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032
This theorem is referenced by:  efif1olem4  22693
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