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Theorem norm3lemt 28009
Description: Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
Assertion
Ref Expression
norm3lemt |- ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. RR ) ) -> ( ( ( normh ` ( A -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) -> ( normh ` ( A -h B ) ) < D ) )
Proof of Theorem norm3lemt
StepHypRef Expression
1 oveq1 6657 . . . . . 6 |- ( A = if ( A e. ~H , A , 0h ) -> ( A -h C ) = ( if ( A e. ~H , A , 0h ) -h C ) )
21fveq2d 6195 . . . . 5 |- ( A = if ( A e. ~H , A , 0h ) -> ( normh ` ( A -h C ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) )
32breq1d 4663 . . . 4 |- ( A = if ( A e. ~H , A , 0h ) -> ( ( normh ` ( A -h C ) ) < ( D / 2 ) <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) ) )
43anbi1d 741 . . 3 |- ( A = if ( A e. ~H , A , 0h ) -> ( ( ( normh ` ( A -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) <-> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) ) )
5 oveq1 6657 . . . . 5 |- ( A = if ( A e. ~H , A , 0h ) -> ( A -h B ) = ( if ( A e. ~H , A , 0h ) -h B ) )
65fveq2d 6195 . . . 4 |- ( A = if ( A e. ~H , A , 0h ) -> ( normh ` ( A -h B ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) )
76breq1d 4663 . . 3 |- ( A = if ( A e. ~H , A , 0h ) -> ( ( normh ` ( A -h B ) ) < D <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) < D ) )
84, 7imbi12d 334 . 2 |- ( A = if ( A e. ~H , A , 0h ) -> ( ( ( ( normh ` ( A -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) -> ( normh ` ( A -h B ) ) < D ) <-> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) < D ) ) )
9 oveq2 6658 . . . . . 6 |- ( B = if ( B e. ~H , B , 0h ) -> ( C -h B ) = ( C -h if ( B e. ~H , B , 0h ) ) )
109fveq2d 6195 . . . . 5 |- ( B = if ( B e. ~H , B , 0h ) -> ( normh ` ( C -h B ) ) = ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) )
1110breq1d 4663 . . . 4 |- ( B = if ( B e. ~H , B , 0h ) -> ( ( normh ` ( C -h B ) ) < ( D / 2 ) <-> ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) )
1211anbi2d 740 . . 3 |- ( B = if ( B e. ~H , B , 0h ) -> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) <-> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) ) )
13 oveq2 6658 . . . . 5 |- ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. ~H , A , 0h ) -h B ) = ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) )
1413fveq2d 6195 . . . 4 |- ( B = if ( B e. ~H , B , 0h ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) )
1514breq1d 4663 . . 3 |- ( B = if ( B e. ~H , B , 0h ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) < D <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D ) )
1612, 15imbi12d 334 . 2 |- ( B = if ( B e. ~H , B , 0h ) -> ( ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h B ) ) < D ) <-> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D ) ) )
17 oveq2 6658 . . . . . 6 |- ( C = if ( C e. ~H , C , 0h ) -> ( if ( A e. ~H , A , 0h ) -h C ) = ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) )
1817fveq2d 6195 . . . . 5 |- ( C = if ( C e. ~H , C , 0h ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) = ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) )
1918breq1d 4663 . . . 4 |- ( C = if ( C e. ~H , C , 0h ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) ) )
20 oveq1 6657 . . . . . 6 |- ( C = if ( C e. ~H , C , 0h ) -> ( C -h if ( B e. ~H , B , 0h ) ) = ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) )
2120fveq2d 6195 . . . . 5 |- ( C = if ( C e. ~H , C , 0h ) -> ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) = ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) )
2221breq1d 4663 . . . 4 |- ( C = if ( C e. ~H , C , 0h ) -> ( ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) <-> ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) )
2319, 22anbi12d 747 . . 3 |- ( C = if ( C e. ~H , C , 0h ) -> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) <-> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) ) )
2423imbi1d 331 . 2 |- ( C = if ( C e. ~H , C , 0h ) -> ( ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D ) <-> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D ) ) )
25 oveq1 6657 . . . . 5 |- ( D = if ( D e. RR , D , 2 ) -> ( D / 2 ) = ( if ( D e. RR , D , 2 ) / 2 ) )
2625breq2d 4665 . . . 4 |- ( D = if ( D e. RR , D , 2 ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) ) )
2725breq2d 4665 . . . 4 |- ( D = if ( D e. RR , D , 2 ) -> ( ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) <-> ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) ) )
2826, 27anbi12d 747 . . 3 |- ( D = if ( D e. RR , D , 2 ) -> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) <-> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) ) ) )
29 breq2 4657 . . 3 |- ( D = if ( D e. RR , D , 2 ) -> ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D <-> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < if ( D e. RR , D , 2 ) ) )
3028, 29imbi12d 334 . 2 |- ( D = if ( D e. RR , D , 2 ) -> ( ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( D / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( D / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < D ) <-> ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < if ( D e. RR , D , 2 ) ) ) )
31 ifhvhv0 27879 . . 3 |- if ( A e. ~H , A , 0h ) e. ~H
32 ifhvhv0 27879 . . 3 |- if ( B e. ~H , B , 0h ) e. ~H
33 ifhvhv0 27879 . . 3 |- if ( C e. ~H , C , 0h ) e. ~H
34 2re 11090 . . . 4 |- 2 e. RR
3534elimel 4150 . . 3 |- if ( D e. RR , D , 2 ) e. RR
3631, 32, 33, 35norm3lem 28006 . 2 |- ( ( ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) /\ ( normh ` ( if ( C e. ~H , C , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < ( if ( D e. RR , D , 2 ) / 2 ) ) -> ( normh ` ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) < if ( D e. RR , D , 2 ) )
378, 16, 24, 30, 36dedth4h 4142 1 |- ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. RR ) ) -> ( ( ( normh ` ( A -h C ) ) < ( D / 2 ) /\ ( normh ` ( C -h B ) ) < ( D / 2 ) ) -> ( normh ` ( A -h B ) ) < D ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   < clt 10074   / cdiv 10684   2c2 11070   ~Hchil 27776   normhcno 27780   0hc0v 27781   -h cmv 27782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-hnorm 27825  df-hvsub 27828
This theorem is referenced by: (None)
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