Theorem 0ofval 31120

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 6883	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3		0oval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	2, 3	eqtr4di 2816	. . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5	4	xpeq1d 5692	. . 3 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec‘𝑤)}) = (𝑋 × {(0vec‘𝑤)}))
6		fveq2 6883	. . . . . 6 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = (0vec‘𝑊))
7		0oval.6	. . . . . 6 ⊢ 𝑍 = (0vec‘𝑊)
8	6, 7	eqtr4di 2816	. . . . 5 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = 𝑍)
9	8	sneqd 4602	. . . 4 ⊢ (𝑤 = 𝑊 → {(0vec‘𝑤)} = {𝑍})
10	9	xpeq2d 5693	. . 3 ⊢ (𝑤 = 𝑊 → (𝑋 × {(0vec‘𝑤)}) = (𝑋 × {𝑍}))
11		df-0o 31080	. . 3 ⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)}))
12	3	fvexi 6897	. . . 4 ⊢ 𝑋 ∈ V
13		snex 5412	. . . 4 ⊢ {𝑍} ∈ V
14	12, 13	xpex 7753	. . 3 ⊢ (𝑋 × {𝑍}) ∈ V
15	5, 10, 11, 14	ovmpo 7572	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍}))
16	1, 15	eqtrid 2810	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {csn 4590   × cxp 5661  ‘cfv 6538  (class class class)co 7412  NrmCVeccnv 30917  BaseSetcba 30919  0_veccn0v 30921   0_op c0o 31076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-0o 31080

This theorem is referenced by:  0oval  31121  0oo  31122  lnon0  31131  blocni  31138  hh0oi  32236