Theorem 0ofval 8447

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

Hypotheses

Ref	Expression
0oval.1	|- X = (Base` 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 = (Base` U)
2		fvex 3732	. . . . 5 |- (Base` U) e. V
3	1, 2	eqeltr 1544	. . . 4 |- X e. V
4		snex 2750	. . . 4 |- {Z} e. V
5	3, 4	xpex 3260	. . 3 |- (X X. {Z}) e. V
6		fveq2 3724	. . . . 5 |- (u = U -> (Base` u) = (Base` U))
7	6, 1	syl6eqr 1525	. . . 4 |- (u = U -> (Base` u) = X)
8		xpeq1 3200	. . . 4 |- ((Base` u) = X -> ((Base` u) X. {(0v` w)}) = (X X. {(0v` w)}))
9	7, 8	syl 10	. . 3 |- (u = U -> ((Base` u) X. {(0v` w)}) = (X X. {(0v` w)}))
10		fveq2 3724	. . . . . 6 |- (w = W -> (0v` w) = (0v` W))
11		0oval.6	. . . . . 6 |- Z = (0v` W)
12	10, 11	syl6eqr 1525	. . . . 5 |- (w = W -> (0v` w) = Z)
13	12	sneqd 2419	. . . 4 |- (w = W -> {(0v` w)} = {Z})
14		xpeq2 3201	. . . 4 |- ({(0v` w)} = {Z} -> (X X. {(0v` w)}) = (X X. {Z}))
15	13, 14	syl 10	. . 3 |- (w = W -> (X X. {(0v` w)}) = (X X. {Z}))
16		df-0o 8408	. . 3 |- 0op = {<.<.u, w>., o>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ o = ((Base` u) X. {(0v` w)}))}
17	5, 9, 15, 16	oprabval2 4028	. 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 1519	1 |- ((U e. NrmCVec /\ W e. NrmCVec) -> O = (X X. {Z}))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  {csn 2409   X. cxp 3168  ` cfv 3182  (class class class)co 3963  NrmCVeccnv 8203  Basecba 8205  0vcn0v 8207   0op c0o 8404

This theorem is referenced by:  0oval 8448  0oo 8449  lnon0 8458  blocni 8465  hh0o 9829

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opr 3965  df-oprab 3966  df-0o 8408

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