Theorem 0ofval 30874

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 6842	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3		0oval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	2, 3	eqtr4di 2790	. . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5	4	xpeq1d 5661	. . 3 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec‘𝑤)}) = (𝑋 × {(0vec‘𝑤)}))
6		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = (0vec‘𝑊))
7		0oval.6	. . . . . 6 ⊢ 𝑍 = (0vec‘𝑊)
8	6, 7	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = 𝑍)
9	8	sneqd 4594	. . . 4 ⊢ (𝑤 = 𝑊 → {(0vec‘𝑤)} = {𝑍})
10	9	xpeq2d 5662	. . 3 ⊢ (𝑤 = 𝑊 → (𝑋 × {(0vec‘𝑤)}) = (𝑋 × {𝑍}))
11		df-0o 30834	. . 3 ⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)}))
12	3	fvexi 6856	. . . 4 ⊢ 𝑋 ∈ V
13		snex 5385	. . . 4 ⊢ {𝑍} ∈ V
14	12, 13	xpex 7708	. . 3 ⊢ (𝑋 × {𝑍}) ∈ V
15	5, 10, 11, 14	ovmpo 7528	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍}))
16	1, 15	eqtrid 2784	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4582   × cxp 5630  ‘cfv 6500  (class class class)co 7368  NrmCVeccnv 30671  BaseSetcba 30673  0_veccn0v 30675   0_op c0o 30830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-0o 30834

This theorem is referenced by:  0oval  30875  0oo  30876  lnon0  30885  blocni  30892  hh0oi  31990