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Theorem sqeqd 11653
Description: A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.)
Hypotheses
Ref Expression
sqeqd.1  |-  ( ph  ->  A  e.  CC )
sqeqd.2  |-  ( ph  ->  B  e.  CC )
sqeqd.3  |-  ( ph  ->  ( A ^ 2 )  =  ( B ^ 2 ) )
sqeqd.4  |-  ( ph  ->  0  <_  ( Re `  A ) )
sqeqd.5  |-  ( ph  ->  0  <_  ( Re `  B ) )
sqeqd.6  |-  ( (
ph  /\  ( Re `  A )  =  0  /\  ( Re `  B )  =  0 )  ->  A  =  B )
Assertion
Ref Expression
sqeqd  |-  ( ph  ->  A  =  B )

Proof of Theorem sqeqd
StepHypRef Expression
1 sqeqd.3 . . . . 5  |-  ( ph  ->  ( A ^ 2 )  =  ( B ^ 2 ) )
2 sqeqd.1 . . . . . 6  |-  ( ph  ->  A  e.  CC )
3 sqeqd.2 . . . . . 6  |-  ( ph  ->  B  e.  CC )
4 sqeqor 11219 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( A  =  B  \/  A  =  -u B ) ) )
52, 3, 4syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( A ^
2 )  =  ( B ^ 2 )  <-> 
( A  =  B  \/  A  =  -u B ) ) )
61, 5mpbid 201 . . . 4  |-  ( ph  ->  ( A  =  B  \/  A  =  -u B ) )
76ord 366 . . 3  |-  ( ph  ->  ( -.  A  =  B  ->  A  =  -u B ) )
8 simpl 443 . . . . 5  |-  ( (
ph  /\  A  =  -u B )  ->  ph )
9 fveq2 5527 . . . . . . 7  |-  ( A  =  -u B  ->  (
Re `  A )  =  ( Re `  -u B ) )
10 reneg 11612 . . . . . . . 8  |-  ( B  e.  CC  ->  (
Re `  -u B )  =  -u ( Re `  B ) )
113, 10syl 15 . . . . . . 7  |-  ( ph  ->  ( Re `  -u B
)  =  -u (
Re `  B )
)
129, 11sylan9eqr 2339 . . . . . 6  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  A )  =  -u ( Re `  B ) )
13 sqeqd.4 . . . . . . . . . . . 12  |-  ( ph  ->  0  <_  ( Re `  A ) )
1413adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_  ( Re `  A
) )
1514, 12breqtrd 4049 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_ 
-u ( Re `  B ) )
163adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =  -u B )  ->  B  e.  CC )
17 recl 11597 . . . . . . . . . . . 12  |-  ( B  e.  CC  ->  (
Re `  B )  e.  RR )
1816, 17syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  e.  RR )
1918le0neg1d 9346 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  -u B )  ->  (
( Re `  B
)  <_  0  <->  0  <_  -u ( Re `  B ) ) )
2015, 19mpbird 223 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  <_  0 )
21 sqeqd.5 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( Re `  B ) )
2221adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  0  <_  ( Re `  B
) )
23 0re 8840 . . . . . . . . . 10  |-  0  e.  RR
24 letri3 8909 . . . . . . . . . 10  |-  ( ( ( Re `  B
)  e.  RR  /\  0  e.  RR )  ->  ( ( Re `  B )  =  0  <-> 
( ( Re `  B )  <_  0  /\  0  <_  ( Re
`  B ) ) ) )
2518, 23, 24sylancl 643 . . . . . . . . 9  |-  ( (
ph  /\  A  =  -u B )  ->  (
( Re `  B
)  =  0  <->  (
( Re `  B
)  <_  0  /\  0  <_  ( Re `  B ) ) ) )
2620, 22, 25mpbir2and 888 . . . . . . . 8  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  B )  =  0 )
2726negeqd 9048 . . . . . . 7  |-  ( (
ph  /\  A  =  -u B )  ->  -u (
Re `  B )  =  -u 0 )
28 neg0 9095 . . . . . . 7  |-  -u 0  =  0
2927, 28syl6eq 2333 . . . . . 6  |-  ( (
ph  /\  A  =  -u B )  ->  -u (
Re `  B )  =  0 )
3012, 29eqtrd 2317 . . . . 5  |-  ( (
ph  /\  A  =  -u B )  ->  (
Re `  A )  =  0 )
31 sqeqd.6 . . . . 5  |-  ( (
ph  /\  ( Re `  A )  =  0  /\  ( Re `  B )  =  0 )  ->  A  =  B )
328, 30, 26, 31syl3anc 1182 . . . 4  |-  ( (
ph  /\  A  =  -u B )  ->  A  =  B )
3332ex 423 . . 3  |-  ( ph  ->  ( A  =  -u B  ->  A  =  B ) )
347, 33syld 40 . 2  |-  ( ph  ->  ( -.  A  =  B  ->  A  =  B ) )
3534pm2.18d 103 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739    <_ cle 8870   -ucneg 9040   2c2 9797   ^cexp 11106   Recre 11584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-n0 9968  df-z 10027  df-uz 10233  df-seq 11049  df-exp 11107  df-cj 11586  df-re 11587  df-im 11588
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