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Theorem pythagtriplem10 13991
Description: Lemma for pythagtrip 14005. Show that  C  -  B is positive. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
pythagtriplem10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  0  <  ( C  -  B )
)

Proof of Theorem pythagtriplem10
StepHypRef Expression
1 nnre 10432 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  RR )
213ad2ant1 1009 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  RR )
3 nnne0 10457 . . . . . . . 8  |-  ( A  e.  NN  ->  A  =/=  0 )
433ad2ant1 1009 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  =/=  0 )
52, 4sqgt0d 12139 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  0  <  ( A ^ 2 ) )
62resqcld 12137 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( A ^ 2 )  e.  RR )
7 nnre 10432 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  RR )
873ad2ant2 1010 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  B  e.  RR )
98resqcld 12137 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( B ^ 2 )  e.  RR )
106, 9ltaddpos2d 10027 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
0  <  ( A ^ 2 )  <->  ( B ^ 2 )  < 
( ( A ^
2 )  +  ( B ^ 2 ) ) ) )
115, 10mpbid 210 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( B ^ 2 )  < 
( ( A ^
2 )  +  ( B ^ 2 ) ) )
1211adantr 465 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( B ^
2 )  <  (
( A ^ 2 )  +  ( B ^ 2 ) ) )
13 simpr 461 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )
1412, 13breqtrd 4416 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( B ^
2 )  <  ( C ^ 2 ) )
158adantr 465 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  B  e.  RR )
16 nnre 10432 . . . . . 6  |-  ( C  e.  NN  ->  C  e.  RR )
17163ad2ant3 1011 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  RR )
1817adantr 465 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  C  e.  RR )
19 nnnn0 10689 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  NN0 )
2019nn0ge0d 10742 . . . . . 6  |-  ( B  e.  NN  ->  0  <_  B )
21203ad2ant2 1010 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  0  <_  B )
2221adantr 465 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  0  <_  B
)
23 nnnn0 10689 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  NN0 )
2423nn0ge0d 10742 . . . . . 6  |-  ( C  e.  NN  ->  0  <_  C )
25243ad2ant3 1011 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  0  <_  C )
2625adantr 465 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  0  <_  C
)
2715, 18, 22, 26lt2sqd 12145 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( B  < 
C  <->  ( B ^
2 )  <  ( C ^ 2 ) ) )
2814, 27mpbird 232 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  B  <  C
)
2915, 18posdifd 10029 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( B  < 
C  <->  0  <  ( C  -  B )
) )
3028, 29mpbid 210 1  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  0  <  ( C  -  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4392  (class class class)co 6192   RRcr 9384   0cc0 9385    + caddc 9388    < clt 9521    <_ cle 9522    - cmin 9698   NNcn 10425   2c2 10474   ^cexp 11968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-n0 10683  df-z 10750  df-uz 10965  df-seq 11910  df-exp 11969
This theorem is referenced by:  pythagtriplem6  13992  pythagtriplem12  13997  pythagtriplem13  13998  pythagtriplem14  13999  pythagtriplem16  14001
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