Theorem 0ofval 9787

Description: The zero operator between two normed complex vector spaces.

Hypotheses

Ref	Expression
0oval.1	|- X = (BaseSet` U)
0oval.6	|- Z = (0v` W)
0oval.0	|- O = (U 0op W)

Assertion

Ref	Expression
0ofval	|- ((U e. NrmCVec /\ W e. NrmCVec) -> O = (X X. {Z}))

Proof of Theorem 0ofval

Step	Hyp	Ref	Expression
1		0oval.1	. . . . 5 |- X = (BaseSet` U)
2		fvex 4689	. . . . 5 |- (BaseSet` U) e. _V
3	1, 2	eqeltri 1967	. . . 4 |- X e. _V
4		snex 3492	. . . 4 |- {Z} e. _V
5	3, 4	xpex 4096	. . 3 |- (X X. {Z}) e. _V
6		fveq2 4681	. . . . 5 |- (u = U -> (BaseSet` u) = (BaseSet` U))
7	6, 1	syl6eqr 1946	. . . 4 |- (u = U -> (BaseSet` u) = X)
8		xpeq1 4016	. . . 4 |- ((BaseSet` u) = X -> ((BaseSet` u) X. {(0v` w)}) = (X X. {(0v` w)}))
9	7, 8	syl 12	. . 3 |- (u = U -> ((BaseSet` u) X. {(0v` w)}) = (X X. {(0v` w)}))
10		fveq2 4681	. . . . . 6 |- (w = W -> (0v` w) = (0v` W))
11		0oval.6	. . . . . 6 |- Z = (0v` W)
12	10, 11	syl6eqr 1946	. . . . 5 |- (w = W -> (0v` w) = Z)
13	12	sneqd 3056	. . . 4 |- (w = W -> {(0v` w)} = {Z})
14		xpeq2 4017	. . . 4 |- ({(0v` w)} = {Z} -> (X X. {(0v` w)}) = (X X. {Z}))
15	13, 14	syl 12	. . 3 |- (w = W -> (X X. {(0v` w)}) = (X X. {Z}))
16		df-0o 9747	. . 3 |- 0op = {<.<.u, w>., o>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ o = ((BaseSet` u) X. {(0v` w)}))}
17	5, 9, 15, 16	oprabval2 4957	. 2 |- ((U e. NrmCVec /\ W e. NrmCVec) -> (U 0op W) = (X X. {Z}))
18		0oval.0	. 2 |- O = (U 0op W)
19	17, 18	syl5eq 1940	1 |- ((U e. NrmCVec /\ W e. NrmCVec) -> O = (X X. {Z}))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044   X. cxp 3984  ` cfv 3998  (class class class)co 4884  NrmCVeccnv 9535  BaseSetcba 9537  0vcn0v 9539   0op c0o 9743

This theorem is referenced by:  0oval 9788  0oo 9789  lnon0 9798  blocni 9805  hh0oi 11466

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-0o 9747

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