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Theorem pythagtriplem7 13188
Description: Lemma for pythagtrip 13200. Calculate  ( sqr `  ( C  +  B ) ). (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
pythagtriplem7  |-  ( ( ( 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 pythagtriplem7
StepHypRef Expression
1 simp3 959 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  NN )
21nnzd 10366 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  ZZ )
3 simp2 958 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  B  e.  NN )
43nnzd 10366 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  B  e.  ZZ )
52, 4zsubcld 10372 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B )  e.  ZZ )
653ad2ant1 978 . . . . . 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 )
71, 3nnaddcld 10038 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  e.  NN )
87nnnn0d 10266 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  e.  NN0 )
983ad2ant1 978 . . . . . 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 )
10 nnnn0 10220 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  NN0 )
11103ad2ant1 978 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  NN0 )
12113ad2ant1 978 . . . . . 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 )
136, 9, 123jca 1134 . . . . 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.  ZZ  /\  ( C  +  B
)  e.  NN0  /\  A  e.  NN0 ) )
14 pythagtriplem4 13185 . . . . . . 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 )
1514oveq1d 6088 . . . . . 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 ) )
16 nnz 10295 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  ZZ )
17163ad2ant1 978 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  ZZ )
18173ad2ant1 978 . . . . . . 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 )
19 1gcd 13029 . . . . . . 7  |-  ( A  e.  ZZ  ->  (
1  gcd  A )  =  1 )
2018, 19syl 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 )
2115, 20eqtrd 2467 . . . . 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 )
2213, 21jca 519 . . . 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.  ZZ  /\  ( C  +  B
)  e.  NN0  /\  A  e.  NN0 )  /\  ( ( ( C  -  B )  gcd  ( C  +  B
) )  gcd  A
)  =  1 ) )
23 oveq1 6080 . . . . . 6  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  -  ( B ^ 2 ) )  =  ( ( C ^ 2 )  -  ( B ^ 2 ) ) )
24233ad2ant2 979 . . . . 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 ) ) )
25 nncn 10000 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  CC )
26253ad2ant1 978 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  CC )
2726sqcld 11513 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( A ^ 2 )  e.  CC )
283nncnd 10008 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  B  e.  CC )
2928sqcld 11513 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( B ^ 2 )  e.  CC )
3027, 29pncand 9404 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
( ( A ^
2 )  +  ( B ^ 2 ) )  -  ( B ^ 2 ) )  =  ( A ^
2 ) )
31303ad2ant1 978 . . . . 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 ) )
321nncnd 10008 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  CC )
33 subsq 11480 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C ^
2 )  -  ( B ^ 2 ) )  =  ( ( C  +  B )  x.  ( C  -  B
) ) )
3432, 28, 33syl2anc 643 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
( C ^ 2 )  -  ( B ^ 2 ) )  =  ( ( C  +  B )  x.  ( C  -  B
) ) )
357nncnd 10008 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  e.  CC )
365zcnd 10368 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B )  e.  CC )
3735, 36mulcomd 9101 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
( C  +  B
)  x.  ( C  -  B ) )  =  ( ( C  -  B )  x.  ( C  +  B
) ) )
3834, 37eqtrd 2467 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
( C ^ 2 )  -  ( B ^ 2 ) )  =  ( ( C  -  B )  x.  ( C  +  B
) ) )
39383ad2ant1 978 . . . . 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
) ) )
4024, 31, 393eqtr3d 2475 . . . 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 )
) )
41 coprimeprodsq2 13176 . . . 4  |-  ( ( ( ( C  -  B )  e.  ZZ  /\  ( C  +  B
)  e.  NN0  /\  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 ) ) )
4222, 40, 41sylc 58 . . 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 ) )
4342fveq2d 5724 . 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 ) ) )
447nnzd 10366 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  e.  ZZ )
45443ad2ant1 978 . . . . 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.  ZZ )
4645, 18gcdcld 13010 . . . 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 )
4746nn0red 10267 . . 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 )
4846nn0ge0d 10269 . . 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
) )
4947, 48sqrsqd 12214 . 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
) )
5043, 49eqtrd 2467 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283   NNcn 9992   2c2 10041   NN0cn0 10213   ZZcz 10274   ^cexp 11374   sqrcsqr 12030    || cdivides 12844    gcd cgcd 12998
This theorem is referenced by:  pythagtriplem9  13190  pythagtriplem11  13191  pythagtriplem13  13193
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-int 4043  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-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  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-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072
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