Theorem norm1exi 24604

Description: A normalized vector exists in a subspace iff the subspace has a nonzero vector. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
norm1ex.1	|- H e. SH

Assertion

Ref	Expression
norm1exi	|- ( E. x e. H x =/= 0h <-> E. y e. H ( normh ` y ) = 1 )

Distinct variable groups:   x, H   y, H

Proof of Theorem norm1exi

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neeq1 2611	. . 3 |- ( x = z -> ( x =/= 0h <-> z =/= 0h ) )
2	1	cbvrexv 2943	. 2 |- ( E. x e. H x =/= 0h <-> E. z e. H z =/= 0h )
3		norm1ex.1	. . . . . . . . . . 11 |- H e. SH
4	3	sheli 24567	. . . . . . . . . 10 |- ( z e. H -> z e. ~H )
5		normcl 24478	. . . . . . . . . 10 |- ( z e. ~H -> ( normh ` z ) e. RR )
6	4, 5	syl 16	. . . . . . . . 9 |- ( z e. H -> ( normh ` z ) e. RR )
7	6	adantr 465	. . . . . . . 8 |- ( ( z e. H /\ z =/= 0h ) -> ( normh ` z ) e. RR )
8		normne0 24483	. . . . . . . . . 10 |- ( z e. ~H -> ( ( normh ` z ) =/= 0 <-> z =/= 0h ) )
9	4, 8	syl 16	. . . . . . . . 9 |- ( z e. H -> ( ( normh ` z ) =/= 0 <-> z =/= 0h ) )
10	9	biimpar 485	. . . . . . . 8 |- ( ( z e. H /\ z =/= 0h ) -> ( normh ` z ) =/= 0 )
11	7, 10	rereccld 10150	. . . . . . 7 |- ( ( z e. H /\ z =/= 0h ) -> ( 1 / ( normh ` z ) ) e. RR )
12	11	recnd 9404	. . . . . 6 |- ( ( z e. H /\ z =/= 0h ) -> ( 1 / ( normh ` z ) ) e. CC )
13		simpl 457	. . . . . 6 |- ( ( z e. H /\ z =/= 0h ) -> z e. H )
14		shmulcl 24571	. . . . . . 7 |- ( ( H e. SH /\ ( 1 / ( normh ` z ) ) e. CC /\ z e. H ) -> ( ( 1 / ( normh ` z ) ) .h z ) e. H )
15	3, 14	mp3an1 1301	. . . . . 6 |- ( ( ( 1 / ( normh ` z ) ) e. CC /\ z e. H ) -> ( ( 1 / ( normh ` z ) ) .h z ) e. H )
16	12, 13, 15	syl2anc 661	. . . . 5 |- ( ( z e. H /\ z =/= 0h ) -> ( ( 1 / ( normh ` z ) ) .h z ) e. H )
17		norm1 24603	. . . . . 6 |- ( ( z e. ~H /\ z =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` z ) ) .h z ) ) = 1 )
18	4, 17	sylan 471	. . . . 5 |- ( ( z e. H /\ z =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` z ) ) .h z ) ) = 1 )
19		fveq2 5686	. . . . . . 7 |- ( y = ( ( 1 / ( normh ` z ) ) .h z ) -> ( normh ` y ) = ( normh ` ( ( 1 / ( normh ` z ) ) .h z ) ) )
20	19	eqeq1d 2446	. . . . . 6 |- ( y = ( ( 1 / ( normh ` z ) ) .h z ) -> ( ( normh ` y ) = 1 <-> ( normh ` ( ( 1 / ( normh ` z ) ) .h z ) ) = 1 ) )
21	20	rspcev 3068	. . . . 5 |- ( ( ( ( 1 / ( normh ` z ) ) .h z ) e. H /\ ( normh ` ( ( 1 / ( normh ` z ) ) .h z ) ) = 1 ) -> E. y e. H ( normh ` y ) = 1 )
22	16, 18, 21	syl2anc 661	. . . 4 |- ( ( z e. H /\ z =/= 0h ) -> E. y e. H ( normh ` y ) = 1 )
23	22	rexlimiva 2831	. . 3 |- ( E. z e. H z =/= 0h -> E. y e. H ( normh ` y ) = 1 )
24		ax-1ne0 9343	. . . . . . . 8 |- 1 =/= 0
25		df-ne 2603	. . . . . . . 8 |- ( 1 =/= 0 <-> -. 1 = 0 )
26	24, 25	mpbi 208	. . . . . . 7 |- -. 1 = 0
27		eqeq1 2444	. . . . . . 7 |- ( ( normh ` y ) = 1 -> ( ( normh ` y ) = 0 <-> 1 = 0 ) )
28	26, 27	mtbiri 303	. . . . . 6 |- ( ( normh ` y ) = 1 -> -. ( normh ` y ) = 0 )
29	3	sheli 24567	. . . . . . . 8 |- ( y e. H -> y e. ~H )
30		norm-i 24482	. . . . . . . 8 |- ( y e. ~H -> ( ( normh ` y ) = 0 <-> y = 0h ) )
31	29, 30	syl 16	. . . . . . 7 |- ( y e. H -> ( ( normh ` y ) = 0 <-> y = 0h ) )
32	31	necon3bbid 2637	. . . . . 6 |- ( y e. H -> ( -. ( normh ` y ) = 0 <-> y =/= 0h ) )
33	28, 32	syl5ib 219	. . . . 5 |- ( y e. H -> ( ( normh ` y ) = 1 -> y =/= 0h ) )
34	33	reximia 2816	. . . 4 |- ( E. y e. H ( normh ` y ) = 1 -> E. y e. H y =/= 0h )
35		neeq1 2611	. . . . 5 |- ( y = z -> ( y =/= 0h <-> z =/= 0h ) )
36	35	cbvrexv 2943	. . . 4 |- ( E. y e. H y =/= 0h <-> E. z e. H z =/= 0h )
37	34, 36	sylib 196	. . 3 |- ( E. y e. H ( normh ` y ) = 1 -> E. z e. H z =/= 0h )
38	23, 37	impbii 188	. 2 |- ( E. z e. H z =/= 0h <-> E. y e. H ( normh ` y ) = 1 )
39	2, 38	bitri 249	1 |- ( E. x e. H x =/= 0h <-> E. y e. H ( normh ` y ) = 1 )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2601   E.wrex 2711   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275   / cdiv 9985   ~Hchil 24272   .h csm 24274   normhcno 24276   0hc0v 24277   SHcsh 24281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-hilex 24352  ax-hfvadd 24353  ax-hv0cl 24356  ax-hfvmul 24358  ax-hvmul0 24363  ax-hfi 24432  ax-his1 24435  ax-his3 24437  ax-his4 24438

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-hnorm 24321  df-sh 24560

This theorem is referenced by:  norm1hex  24605  pjnmopi  25503