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Theorem pythagtriplem10 13186
Description: Lemma for pythagtrip 13200. 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 9999 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  RR )
213ad2ant1 978 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  RR )
3 nnne0 10024 . . . . . . . 8  |-  ( A  e.  NN  ->  A  =/=  0 )
433ad2ant1 978 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  =/=  0 )
52, 4sqgt0d 11543 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  0  <  ( A ^ 2 ) )
62resqcld 11541 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( A ^ 2 )  e.  RR )
7 nnre 9999 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  RR )
873ad2ant2 979 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  B  e.  RR )
98resqcld 11541 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( B ^ 2 )  e.  RR )
106, 9ltaddpos2d 9603 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  (
0  <  ( A ^ 2 )  <->  ( B ^ 2 )  < 
( ( A ^
2 )  +  ( B ^ 2 ) ) ) )
115, 10mpbid 202 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( B ^ 2 )  < 
( ( A ^
2 )  +  ( B ^ 2 ) ) )
1211adantr 452 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( B ^
2 )  <  (
( A ^ 2 )  +  ( B ^ 2 ) ) )
13 simpr 448 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( ( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )
1412, 13breqtrd 4228 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( B ^
2 )  <  ( C ^ 2 ) )
158adantr 452 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  B  e.  RR )
16 nnre 9999 . . . . . 6  |-  ( C  e.  NN  ->  C  e.  RR )
17163ad2ant3 980 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  RR )
1817adantr 452 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  C  e.  RR )
19 nnnn0 10220 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  NN0 )
2019nn0ge0d 10269 . . . . . 6  |-  ( B  e.  NN  ->  0  <_  B )
21203ad2ant2 979 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  0  <_  B )
2221adantr 452 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  0  <_  B
)
23 nnnn0 10220 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  NN0 )
2423nn0ge0d 10269 . . . . . 6  |-  ( C  e.  NN  ->  0  <_  C )
25243ad2ant3 980 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  0  <_  C )
2625adantr 452 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  0  <_  C
)
2715, 18, 22, 26lt2sqd 11549 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( B  < 
C  <->  ( B ^
2 )  <  ( C ^ 2 ) ) )
2814, 27mpbird 224 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  B  <  C
)
2915, 18posdifd 9605 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  ( B  < 
C  <->  0  <  ( C  -  B )
) )
3028, 29mpbid 202 1  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 ) )  ->  0  <  ( C  -  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204  (class class class)co 6073   RRcr 8981   0cc0 8982    + caddc 8985    < clt 9112    <_ cle 9113    - cmin 9283   NNcn 9992   2c2 10041   ^cexp 11374
This theorem is referenced by:  pythagtriplem6  13187  pythagtriplem12  13192  pythagtriplem13  13193  pythagtriplem14  13194  pythagtriplem16  13196
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
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-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-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-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-seq 11316  df-exp 11375
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