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Theorem pythagtriplem6 14221
Description: Lemma for pythagtrip 14234. Calculate  ( sqr `  ( C  -  B ) ). (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
pythagtriplem6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( C  -  B ) )  =  ( ( C  -  B )  gcd  A
) )

Proof of Theorem pythagtriplem6
StepHypRef Expression
1 nnz 10898 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  ZZ )
213ad2ant3 1019 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  ZZ )
3 nnz 10898 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  B  e.  ZZ )
433ad2ant2 1018 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  B  e.  ZZ )
52, 4zsubcld 10983 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B )  e.  ZZ )
653ad2ant1 1017 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  -  B )  e.  ZZ )
7 pythagtriplem10 14220 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  0  <  ( C  -  B )
)
873adant3 1016 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  0  <  ( C  -  B
) )
9 elnnz 10886 . . . . . . . 8  |-  ( ( C  -  B )  e.  NN  <->  ( ( C  -  B )  e.  ZZ  /\  0  < 
( C  -  B
) ) )
106, 8, 9sylanbrc 664 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  -  B )  e.  NN )
1110nnnn0d 10864 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  -  B )  e.  NN0 )
12 simp3 998 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  NN )
13 simp2 997 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  B  e.  NN )
1412, 13nnaddcld 10594 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  e.  NN )
1514nnzd 10977 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  e.  ZZ )
16153ad2ant1 1017 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  +  B )  e.  ZZ )
17 nnnn0 10814 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  NN0 )
18173ad2ant1 1017 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  NN0 )
19183ad2ant1 1017 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  A  e.  NN0 )
2011, 16, 193jca 1176 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  e.  NN0  /\  ( C  +  B
)  e.  ZZ  /\  A  e.  NN0 ) )
21 pythagtriplem4 14219 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  ( C  +  B ) )  =  1 )
2221oveq1d 6310 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( C  -  B )  gcd  ( C  +  B )
)  gcd  A )  =  ( 1  gcd 
A ) )
23 nnz 10898 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  ZZ )
24233ad2ant1 1017 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  ZZ )
25243ad2ant1 1017 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  A  e.  ZZ )
26 1gcd 14051 . . . . . . 7  |-  ( A  e.  ZZ  ->  (
1  gcd  A )  =  1 )
2725, 26syl 16 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
1  gcd  A )  =  1 )
2822, 27eqtrd 2508 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( C  -  B )  gcd  ( C  +  B )
)  gcd  A )  =  1 )
2920, 28jca 532 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( C  -  B )  e.  NN0  /\  ( C  +  B
)  e.  ZZ  /\  A  e.  NN0 )  /\  ( ( ( C  -  B )  gcd  ( C  +  B
) )  gcd  A
)  =  1 ) )
30 oveq1 6302 . . . . . 6  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  -  ( B ^ 2 ) )  =  ( ( C ^ 2 )  -  ( B ^ 2 ) ) )
31303ad2ant2 1018 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  -  ( B ^ 2 ) )  =  ( ( C ^ 2 )  -  ( B ^ 2 ) ) )
3224zcnd 10979 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  CC )
3332sqcld 12288 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( A ^ 2 )  e.  CC )
34 nncn 10556 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  CC )
35343ad2ant2 1018 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  B  e.  CC )
3635sqcld 12288 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( B ^ 2 )  e.  CC )
3733, 36pncand 9943 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  -  ( B ^ 2 ) )  =  ( A ^
2 ) )
38373ad2ant1 1017 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  -  ( B ^ 2 ) )  =  ( A ^
2 ) )
39 nncn 10556 . . . . . . . . 9  |-  ( C  e.  NN  ->  C  e.  CC )
40393ad2ant3 1019 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  CC )
41 subsq 12255 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C ^
2 )  -  ( B ^ 2 ) )  =  ( ( C  +  B )  x.  ( C  -  B
) ) )
4240, 35, 41syl2anc 661 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
( C ^ 2 )  -  ( B ^ 2 ) )  =  ( ( C  +  B )  x.  ( C  -  B
) ) )
4314nncnd 10564 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  e.  CC )
445zcnd 10979 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B )  e.  CC )
4543, 44mulcomd 9629 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
( C  +  B
)  x.  ( C  -  B ) )  =  ( ( C  -  B )  x.  ( C  +  B
) ) )
4642, 45eqtrd 2508 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
( C ^ 2 )  -  ( B ^ 2 ) )  =  ( ( C  -  B )  x.  ( C  +  B
) ) )
47463ad2ant1 1017 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C ^ 2 )  -  ( B ^ 2 ) )  =  ( ( C  -  B )  x.  ( C  +  B
) ) )
4831, 38, 473eqtr3d 2516 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( A ^ 2 )  =  ( ( C  -  B )  x.  ( C  +  B )
) )
49 coprimeprodsq 14209 . . . 4  |-  ( ( ( ( C  -  B )  e.  NN0  /\  ( C  +  B
)  e.  ZZ  /\  A  e.  NN0 )  /\  ( ( ( C  -  B )  gcd  ( C  +  B
) )  gcd  A
)  =  1 )  ->  ( ( A ^ 2 )  =  ( ( C  -  B )  x.  ( C  +  B )
)  ->  ( C  -  B )  =  ( ( ( C  -  B )  gcd  A
) ^ 2 ) ) )
5029, 48, 49sylc 60 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  -  B )  =  ( ( ( C  -  B )  gcd  A ) ^
2 ) )
5150fveq2d 5876 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( C  -  B ) )  =  ( sqr `  (
( ( C  -  B )  gcd  A
) ^ 2 ) ) )
526, 25gcdcld 14032 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  A )  e.  NN0 )
5352nn0red 10865 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( C  -  B
)  gcd  A )  e.  RR )
5452nn0ge0d 10867 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  0  <_  ( ( C  -  B )  gcd  A
) )
5553, 54sqrtsqd 13231 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( ( ( C  -  B )  gcd  A ) ^
2 ) )  =  ( ( C  -  B )  gcd  A
) )
5651, 55eqtrd 2508 1  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( C  -  B ) )  =  ( ( C  -  B )  gcd  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    < clt 9640    - cmin 9817   NNcn 10548   2c2 10597   NN0cn0 10807   ZZcz 10876   ^cexp 12146   sqrcsqrt 13046    || cdivides 13864    gcd cgcd 14020
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-prm 14094
This theorem is referenced by:  pythagtriplem8  14223  pythagtriplem11  14225  pythagtriplem13  14227
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