Theorem 0ofval 30731

Description: The zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
0oval.1	⊢ 𝑋 = (BaseSet‘𝑈)
0oval.6	⊢ 𝑍 = (0vec‘𝑊)
0oval.0	⊢ 𝑂 = (𝑈 0op 𝑊)

Assertion

Ref	Expression
0ofval	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))

Proof of Theorem 0ofval

Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0oval.0	. 2 ⊢ 𝑂 = (𝑈 0op 𝑊)
2		fveq2 6822	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3		0oval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	2, 3	eqtr4di 2782	. . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5	4	xpeq1d 5648	. . 3 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec‘𝑤)}) = (𝑋 × {(0vec‘𝑤)}))
6		fveq2 6822	. . . . . 6 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = (0vec‘𝑊))
7		0oval.6	. . . . . 6 ⊢ 𝑍 = (0vec‘𝑊)
8	6, 7	eqtr4di 2782	. . . . 5 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = 𝑍)
9	8	sneqd 4589	. . . 4 ⊢ (𝑤 = 𝑊 → {(0vec‘𝑤)} = {𝑍})
10	9	xpeq2d 5649	. . 3 ⊢ (𝑤 = 𝑊 → (𝑋 × {(0vec‘𝑤)}) = (𝑋 × {𝑍}))
11		df-0o 30691	. . 3 ⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)}))
12	3	fvexi 6836	. . . 4 ⊢ 𝑋 ∈ V
13		snex 5375	. . . 4 ⊢ {𝑍} ∈ V
14	12, 13	xpex 7689	. . 3 ⊢ (𝑋 × {𝑍}) ∈ V
15	5, 10, 11, 14	ovmpo 7509	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍}))
16	1, 15	eqtrid 2776	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4577   × cxp 5617  ‘cfv 6482  (class class class)co 7349  NrmCVeccnv 30528  BaseSetcba 30530  0_veccn0v 30532   0_op c0o 30687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-0o 30691

This theorem is referenced by:  0oval  30732  0oo  30733  lnon0  30742  blocni  30749  hh0oi  31847