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Theorem normgt0 22631
Description: The norm of nonzero vector is positive. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
normgt0  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )

Proof of Theorem normgt0
StepHypRef Expression
1 hiidrcl 22599 . . . . . 6  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
21adantr 453 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( A  .ih  A
)  e.  RR )
3 ax-his4 22589 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
4 sqrgt0 12066 . . . . 5  |-  ( ( ( A  .ih  A
)  e.  RR  /\  0  <  ( A  .ih  A ) )  ->  0  <  ( sqr `  ( A  .ih  A ) ) )
52, 3, 4syl2anc 644 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( sqr `  ( A  .ih  A
) ) )
65ex 425 . . 3  |-  ( A  e.  ~H  ->  ( A  =/=  0h  ->  0  <  ( sqr `  ( A  .ih  A ) ) ) )
7 oveq1 6090 . . . . . . . . 9  |-  ( A  =  0h  ->  ( A  .ih  A )  =  ( 0h  .ih  A
) )
8 hi01 22600 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( 0h  .ih  A )  =  0 )
97, 8sylan9eqr 2492 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( A  .ih  A
)  =  0 )
109fveq2d 5734 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( sqr `  ( A  .ih  A ) )  =  ( sqr `  0
) )
11 sqr0 12049 . . . . . . 7  |-  ( sqr `  0 )  =  0
1210, 11syl6eq 2486 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( sqr `  ( A  .ih  A ) )  =  0 )
1312ex 425 . . . . 5  |-  ( A  e.  ~H  ->  ( A  =  0h  ->  ( sqr `  ( A 
.ih  A ) )  =  0 ) )
14 hiidge0 22602 . . . . . . . 8  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
151, 14resqrcld 12222 . . . . . . 7  |-  ( A  e.  ~H  ->  ( sqr `  ( A  .ih  A ) )  e.  RR )
16 0re 9093 . . . . . . 7  |-  0  e.  RR
17 lttri3 9160 . . . . . . 7  |-  ( ( ( sqr `  ( A  .ih  A ) )  e.  RR  /\  0  e.  RR )  ->  (
( sqr `  ( A  .ih  A ) )  =  0  <->  ( -.  ( sqr `  ( A 
.ih  A ) )  <  0  /\  -.  0  <  ( sqr `  ( A  .ih  A ) ) ) ) )
1815, 16, 17sylancl 645 . . . . . 6  |-  ( A  e.  ~H  ->  (
( sqr `  ( A  .ih  A ) )  =  0  <->  ( -.  ( sqr `  ( A 
.ih  A ) )  <  0  /\  -.  0  <  ( sqr `  ( A  .ih  A ) ) ) ) )
19 simpr 449 . . . . . 6  |-  ( ( -.  ( sqr `  ( A  .ih  A ) )  <  0  /\  -.  0  <  ( sqr `  ( A  .ih  A ) ) )  ->  -.  0  <  ( sqr `  ( A  .ih  A ) ) )
2018, 19syl6bi 221 . . . . 5  |-  ( A  e.  ~H  ->  (
( sqr `  ( A  .ih  A ) )  =  0  ->  -.  0  <  ( sqr `  ( A  .ih  A ) ) ) )
2113, 20syld 43 . . . 4  |-  ( A  e.  ~H  ->  ( A  =  0h  ->  -.  0  <  ( sqr `  ( A  .ih  A
) ) ) )
2221necon2ad 2654 . . 3  |-  ( A  e.  ~H  ->  (
0  <  ( sqr `  ( A  .ih  A
) )  ->  A  =/=  0h ) )
236, 22impbid 185 . 2  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  ( sqr `  ( A 
.ih  A ) ) ) )
24 normval 22628 . . 3  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
2524breq2d 4226 . 2  |-  ( A  e.  ~H  ->  (
0  <  ( normh `  A )  <->  0  <  ( sqr `  ( A 
.ih  A ) ) ) )
2623, 25bitr4d 249 1  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   RRcr 8991   0cc0 8992    < clt 9122   sqrcsqr 12040   ~Hchil 22424    .ih csp 22427   normhcno 22428   0hc0v 22429
This theorem is referenced by:  norm-i  22633  norm1  22753  nmlnop0iALT  23500  nmbdoplbi  23529  nmcoplbi  23533  nmbdfnlbi  23554  nmcfnlbi  23557  branmfn  23610  strlem1  23755
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  ax-hv0cl 22508  ax-hvmul0 22515  ax-hfi 22583  ax-his1 22586  ax-his3 22588  ax-his4 22589
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-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-hnorm 22473
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