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Theorem normpyc 21727
Description: Corollary to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
normpyc  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  -> 
( normh `  A )  <_  ( normh `  ( A  +h  B ) ) ) )

Proof of Theorem normpyc
StepHypRef Expression
1 normcl 21706 . . . . . . . . . 10  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
21resqcld 11273 . . . . . . . . 9  |-  ( A  e.  ~H  ->  (
( normh `  A ) ^ 2 )  e.  RR )
32recnd 8863 . . . . . . . 8  |-  ( A  e.  ~H  ->  (
( normh `  A ) ^ 2 )  e.  CC )
43addid1d 9014 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( ( normh `  A
) ^ 2 )  +  0 )  =  ( ( normh `  A
) ^ 2 ) )
54adantr 451 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( normh `  A ) ^ 2 )  +  0 )  =  ( ( normh `  A ) ^ 2 ) )
6 normcl 21706 . . . . . . . . 9  |-  ( B  e.  ~H  ->  ( normh `  B )  e.  RR )
76sqge0d 11274 . . . . . . . 8  |-  ( B  e.  ~H  ->  0  <_  ( ( normh `  B
) ^ 2 ) )
87adantl 452 . . . . . . 7  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  0  <_  ( ( normh `  B ) ^
2 ) )
96resqcld 11273 . . . . . . . 8  |-  ( B  e.  ~H  ->  (
( normh `  B ) ^ 2 )  e.  RR )
10 0re 8840 . . . . . . . . 9  |-  0  e.  RR
11 leadd2 9245 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( ( normh `  B
) ^ 2 )  e.  RR  /\  (
( normh `  A ) ^ 2 )  e.  RR )  ->  (
0  <_  ( ( normh `  B ) ^
2 )  <->  ( (
( normh `  A ) ^ 2 )  +  0 )  <_  (
( ( normh `  A
) ^ 2 )  +  ( ( normh `  B ) ^ 2 ) ) ) )
1210, 11mp3an1 1264 . . . . . . . 8  |-  ( ( ( ( normh `  B
) ^ 2 )  e.  RR  /\  (
( normh `  A ) ^ 2 )  e.  RR )  ->  (
0  <_  ( ( normh `  B ) ^
2 )  <->  ( (
( normh `  A ) ^ 2 )  +  0 )  <_  (
( ( normh `  A
) ^ 2 )  +  ( ( normh `  B ) ^ 2 ) ) ) )
139, 2, 12syl2anr 464 . . . . . . 7  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( 0  <_  (
( normh `  B ) ^ 2 )  <->  ( (
( normh `  A ) ^ 2 )  +  0 )  <_  (
( ( normh `  A
) ^ 2 )  +  ( ( normh `  B ) ^ 2 ) ) ) )
148, 13mpbid 201 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( normh `  A ) ^ 2 )  +  0 )  <_  ( ( (
normh `  A ) ^
2 )  +  ( ( normh `  B ) ^ 2 ) ) )
155, 14eqbrtrrd 4047 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( normh `  A
) ^ 2 )  <_  ( ( (
normh `  A ) ^
2 )  +  ( ( normh `  B ) ^ 2 ) ) )
1615adantr 451 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( A  .ih  B
)  =  0 )  ->  ( ( normh `  A ) ^ 2 )  <_  ( (
( normh `  A ) ^ 2 )  +  ( ( normh `  B
) ^ 2 ) ) )
17 normpyth 21726 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  -> 
( ( normh `  ( A  +h  B ) ) ^ 2 )  =  ( ( ( normh `  A ) ^ 2 )  +  ( (
normh `  B ) ^
2 ) ) ) )
1817imp 418 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( A  .ih  B
)  =  0 )  ->  ( ( normh `  ( A  +h  B
) ) ^ 2 )  =  ( ( ( normh `  A ) ^ 2 )  +  ( ( normh `  B
) ^ 2 ) ) )
1916, 18breqtrrd 4051 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( A  .ih  B
)  =  0 )  ->  ( ( normh `  A ) ^ 2 )  <_  ( ( normh `  ( A  +h  B ) ) ^
2 ) )
2019ex 423 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  -> 
( ( normh `  A
) ^ 2 )  <_  ( ( normh `  ( A  +h  B
) ) ^ 2 ) ) )
211adantr 451 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( normh `  A )  e.  RR )
22 hvaddcl 21594 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
23 normcl 21706 . . . 4  |-  ( ( A  +h  B )  e.  ~H  ->  ( normh `  ( A  +h  B ) )  e.  RR )
2422, 23syl 15 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( normh `  ( A  +h  B ) )  e.  RR )
25 normge0 21707 . . . 4  |-  ( A  e.  ~H  ->  0  <_  ( normh `  A )
)
2625adantr 451 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  0  <_  ( normh `  A ) )
27 normge0 21707 . . . 4  |-  ( ( A  +h  B )  e.  ~H  ->  0  <_  ( normh `  ( A  +h  B ) ) )
2822, 27syl 15 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  0  <_  ( normh `  ( A  +h  B
) ) )
2921, 24, 26, 28le2sqd 11282 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( normh `  A
)  <_  ( normh `  ( A  +h  B
) )  <->  ( ( normh `  A ) ^
2 )  <_  (
( normh `  ( A  +h  B ) ) ^
2 ) ) )
3020, 29sylibrd 225 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  -> 
( normh `  A )  <_  ( normh `  ( A  +h  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   RRcr 8738   0cc0 8739    + caddc 8742    <_ cle 8870   2c2 9797   ^cexp 11106   ~Hchil 21501    +h cva 21502    .ih csp 21504   normhcno 21505
This theorem is referenced by:  pjnormi  22302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-hfvadd 21582  ax-hv0cl 21585  ax-hvmul0 21592  ax-hfi 21660  ax-his1 21663  ax-his2 21664  ax-his3 21665  ax-his4 21666
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-sup 7196  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-seq 11049  df-exp 11107  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-hnorm 21550
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