Theorem 0ofval 28566

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 6672	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3		0oval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	2, 3	syl6eqr 2876	. . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5	4	xpeq1d 5586	. . 3 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec‘𝑤)}) = (𝑋 × {(0vec‘𝑤)}))
6		fveq2 6672	. . . . . 6 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = (0vec‘𝑊))
7		0oval.6	. . . . . 6 ⊢ 𝑍 = (0vec‘𝑊)
8	6, 7	syl6eqr 2876	. . . . 5 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = 𝑍)
9	8	sneqd 4581	. . . 4 ⊢ (𝑤 = 𝑊 → {(0vec‘𝑤)} = {𝑍})
10	9	xpeq2d 5587	. . 3 ⊢ (𝑤 = 𝑊 → (𝑋 × {(0vec‘𝑤)}) = (𝑋 × {𝑍}))
11		df-0o 28526	. . 3 ⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)}))
12	3	fvexi 6686	. . . 4 ⊢ 𝑋 ∈ V
13		snex 5334	. . . 4 ⊢ {𝑍} ∈ V
14	12, 13	xpex 7478	. . 3 ⊢ (𝑋 × {𝑍}) ∈ V
15	5, 10, 11, 14	ovmpo 7312	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍}))
16	1, 15	syl5eq 2870	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {csn 4569   × cxp 5555  ‘cfv 6357  (class class class)co 7158  NrmCVeccnv 28363  BaseSetcba 28365  0_veccn0v 28367   0_op c0o 28522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-0o 28526

This theorem is referenced by:  0oval  28567  0oo  28568  lnon0  28577  blocni  28584  hh0oi  29682