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Theorem pythagtriplem3 13194
Description: Lemma for pythagtrip 13210. Show that  C and 
B are relatively prime under some conditions. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
pythagtriplem3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( B  gcd  C )  =  1 )

Proof of Theorem pythagtriplem3
StepHypRef Expression
1 oveq2 6091 . . . . . . 7  |-  ( ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  ->  (
( B ^ 2 )  gcd  ( ( A ^ 2 )  +  ( B ^
2 ) ) )  =  ( ( B ^ 2 )  gcd  ( C ^ 2 ) ) )
21adantl 454 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( ( B ^ 2 )  gcd  ( ( A ^
2 )  +  ( B ^ 2 ) ) )  =  ( ( B ^ 2 )  gcd  ( C ^ 2 ) ) )
3 nnz 10305 . . . . . . . . . . 11  |-  ( B  e.  NN  ->  B  e.  ZZ )
4 zsqcl 11454 . . . . . . . . . . 11  |-  ( B  e.  ZZ  ->  ( B ^ 2 )  e.  ZZ )
53, 4syl 16 . . . . . . . . . 10  |-  ( B  e.  NN  ->  ( B ^ 2 )  e.  ZZ )
653ad2ant2 980 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( B ^ 2 )  e.  ZZ )
7 nnz 10305 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  e.  ZZ )
8 zsqcl 11454 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
97, 8syl 16 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( A ^ 2 )  e.  ZZ )
1093ad2ant1 979 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( A ^ 2 )  e.  ZZ )
11 gcdadd 13032 . . . . . . . . 9  |-  ( ( ( B ^ 2 )  e.  ZZ  /\  ( A ^ 2 )  e.  ZZ )  -> 
( ( B ^
2 )  gcd  ( A ^ 2 ) )  =  ( ( B ^ 2 )  gcd  ( ( A ^
2 )  +  ( B ^ 2 ) ) ) )
126, 10, 11syl2anc 644 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
( B ^ 2 )  gcd  ( A ^ 2 ) )  =  ( ( B ^ 2 )  gcd  ( ( A ^
2 )  +  ( B ^ 2 ) ) ) )
13 gcdcom 13022 . . . . . . . . 9  |-  ( ( ( B ^ 2 )  e.  ZZ  /\  ( A ^ 2 )  e.  ZZ )  -> 
( ( B ^
2 )  gcd  ( A ^ 2 ) )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
146, 10, 13syl2anc 644 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
( B ^ 2 )  gcd  ( A ^ 2 ) )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
1512, 14eqtr3d 2472 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
( B ^ 2 )  gcd  ( ( A ^ 2 )  +  ( B ^
2 ) ) )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
1615adantr 453 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( ( B ^ 2 )  gcd  ( ( A ^
2 )  +  ( B ^ 2 ) ) )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
172, 16eqtr3d 2472 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( ( B ^ 2 )  gcd  ( C ^ 2 ) )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
18 simpl2 962 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  B  e.  NN )
19 simpl3 963 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  C  e.  NN )
20 sqgcd 13060 . . . . . 6  |-  ( ( B  e.  NN  /\  C  e.  NN )  ->  ( ( B  gcd  C ) ^ 2 )  =  ( ( B ^ 2 )  gcd  ( C ^ 2 ) ) )
2118, 19, 20syl2anc 644 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( ( B  gcd  C ) ^
2 )  =  ( ( B ^ 2 )  gcd  ( C ^ 2 ) ) )
22 simpl1 961 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  A  e.  NN )
23 sqgcd 13060 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B ) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
2422, 18, 23syl2anc 644 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( ( A  gcd  B ) ^
2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
2517, 21, 243eqtr4d 2480 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( ( B  gcd  C ) ^
2 )  =  ( ( A  gcd  B
) ^ 2 ) )
26253adant3 978 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( B  gcd  C
) ^ 2 )  =  ( ( A  gcd  B ) ^
2 ) )
27 simp3l 986 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( A  gcd  B )  =  1 )
2827oveq1d 6098 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( A  gcd  B
) ^ 2 )  =  ( 1 ^ 2 ) )
2926, 28eqtrd 2470 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( B  gcd  C
) ^ 2 )  =  ( 1 ^ 2 ) )
3033ad2ant2 980 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  B  e.  ZZ )
31 nnz 10305 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  ZZ )
32313ad2ant3 981 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  ZZ )
3330, 32gcdcld 13020 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( B  gcd  C )  e. 
NN0 )
3433nn0red 10277 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( B  gcd  C )  e.  RR )
35343ad2ant1 979 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( B  gcd  C )  e.  RR )
3633nn0ge0d 10279 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  0  <_  ( B  gcd  C
) )
37363ad2ant1 979 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  0  <_  ( B  gcd  C
) )
38 1re 9092 . . . 4  |-  1  e.  RR
39 0le1 9553 . . . 4  |-  0  <_  1
40 sq11 11456 . . . 4  |-  ( ( ( ( B  gcd  C )  e.  RR  /\  0  <_  ( B  gcd  C ) )  /\  (
1  e.  RR  /\  0  <_  1 ) )  ->  ( ( ( B  gcd  C ) ^ 2 )  =  ( 1 ^ 2 )  <->  ( B  gcd  C )  =  1 ) )
4138, 39, 40mpanr12 668 . . 3  |-  ( ( ( B  gcd  C
)  e.  RR  /\  0  <_  ( B  gcd  C ) )  ->  (
( ( B  gcd  C ) ^ 2 )  =  ( 1 ^ 2 )  <->  ( B  gcd  C )  =  1 ) )
4235, 37, 41syl2anc 644 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( B  gcd  C ) ^ 2 )  =  ( 1 ^ 2 )  <->  ( B  gcd  C )  =  1 ) )
4329, 42mpbid 203 1  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( B  gcd  C )  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214  (class class class)co 6083   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    <_ cle 9123   NNcn 10002   2c2 10051   ZZcz 10284   ^cexp 11384    || cdivides 12854    gcd cgcd 13008
This theorem is referenced by:  pythagtriplem4  13195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-gcd 13009
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