Theorem lnon0 27653

Description: The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnon0.1	|- X = ( BaseSet ` U )
lnon0.6	|- Z = ( 0vec ` U )
lnon0.0	|- O = ( U 0op W )
lnon0.7	|- L = ( U LnOp W )

Assertion

Ref	Expression
lnon0	|- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ T =/= O ) -> E. x e. X x =/= Z )

Distinct variable groups:   x, L   x, T   x, U   x, W   x, X

Allowed substitution hints:   O( x)   Z( x)

Proof of Theorem lnon0

Step	Hyp	Ref	Expression
1		ralnex 2992	. . . . 5 |- ( A. x e. X -. x =/= Z <-> -. E. x e. X x =/= Z )
2		nne 2798	. . . . . 6 |- ( -. x =/= Z <-> x = Z )
3	2	ralbii 2980	. . . . 5 |- ( A. x e. X -. x =/= Z <-> A. x e. X x = Z )
4	1, 3	bitr3i 266	. . . 4 |- ( -. E. x e. X x =/= Z <-> A. x e. X x = Z )
5		fveq2 6191	. . . . . . . . . 10 |- ( x = Z -> ( T ` x ) = ( T ` Z ) )
6		lnon0.1	. . . . . . . . . . 11 |- X = ( BaseSet ` U )
7		eqid 2622	. . . . . . . . . . 11 |- ( BaseSet ` W ) = ( BaseSet ` W )
8		lnon0.6	. . . . . . . . . . 11 |- Z = ( 0vec ` U )
9		eqid 2622	. . . . . . . . . . 11 |- ( 0vec ` W ) = ( 0vec ` W )
10		lnon0.7	. . . . . . . . . . 11 |- L = ( U LnOp W )
11	6, 7, 8, 9, 10	lno0 27611	. . . . . . . . . 10 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> ( T ` Z ) = ( 0vec ` W ) )
12	5, 11	sylan9eqr 2678	. . . . . . . . 9 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ x = Z ) -> ( T ` x ) = ( 0vec ` W ) )
13	12	ex 450	. . . . . . . 8 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> ( x = Z -> ( T ` x ) = ( 0vec ` W ) ) )
14	13	ralimdv 2963	. . . . . . 7 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> ( A. x e. X x = Z -> A. x e. X ( T ` x ) = ( 0vec ` W ) ) )
15	6, 7, 10	lnof 27610	. . . . . . . 8 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> T : X --> ( BaseSet ` W ) )
16		ffn 6045	. . . . . . . 8 |- ( T : X --> ( BaseSet ` W ) -> T Fn X )
17	15, 16	syl 17	. . . . . . 7 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> T Fn X )
18	14, 17	jctild 566	. . . . . 6 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> ( A. x e. X x = Z -> ( T Fn X /\ A. x e. X ( T ` x ) = ( 0vec ` W ) ) ) )
19		fconstfv 6476	. . . . . . 7 |- ( T : X --> { ( 0vec ` W ) } <-> ( T Fn X /\ A. x e. X ( T ` x ) = ( 0vec ` W ) ) )
20		fvex 6201	. . . . . . . 8 |- ( 0vec ` W ) e. _V
21	20	fconst2 6470	. . . . . . 7 |- ( T : X --> { ( 0vec ` W ) } <-> T = ( X X. { ( 0vec ` W ) } ) )
22	19, 21	bitr3i 266	. . . . . 6 |- ( ( T Fn X /\ A. x e. X ( T ` x ) = ( 0vec ` W ) ) <-> T = ( X X. { ( 0vec ` W ) } ) )
23	18, 22	syl6ib 241	. . . . 5 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> ( A. x e. X x = Z -> T = ( X X. { ( 0vec ` W ) } ) ) )
24		lnon0.0	. . . . . . . 8 |- O = ( U 0op W )
25	6, 9, 24	0ofval 27642	. . . . . . 7 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> O = ( X X. { ( 0vec ` W ) } ) )
26	25	3adant3 1081	. . . . . 6 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> O = ( X X. { ( 0vec ` W ) } ) )
27	26	eqeq2d 2632	. . . . 5 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> ( T = O <-> T = ( X X. { ( 0vec ` W ) } ) ) )
28	23, 27	sylibrd 249	. . . 4 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> ( A. x e. X x = Z -> T = O ) )
29	4, 28	syl5bi 232	. . 3 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> ( -. E. x e. X x =/= Z -> T = O ) )
30	29	necon1ad 2811	. 2 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> ( T =/= O -> E. x e. X x =/= Z ) )
31	30	imp 445	1 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ T =/= O ) -> E. x e. X x =/= Z )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {csn 4177   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   NrmCVeccnv 27439   BaseSetcba 27441   0veccn0v 27443   LnOp clno 27595   0op c0o 27598

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-lno 27599  df-0o 27602

This theorem is referenced by: (None)