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Theorem 0oval 28663
 Description: Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
0oval.1 𝑋 = (BaseSet‘𝑈)
0oval.6 𝑍 = (0vec𝑊)
0oval.0 𝑂 = (𝑈 0op 𝑊)
Assertion
Ref Expression
0oval ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑂𝐴) = 𝑍)

Proof of Theorem 0oval
StepHypRef Expression
1 0oval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
2 0oval.6 . . . . 5 𝑍 = (0vec𝑊)
3 0oval.0 . . . . 5 𝑂 = (𝑈 0op 𝑊)
41, 2, 30ofval 28662 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
54fveq1d 6661 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑂𝐴) = ((𝑋 × {𝑍})‘𝐴))
653adant3 1130 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑂𝐴) = ((𝑋 × {𝑍})‘𝐴))
72fvexi 6673 . . . 4 𝑍 ∈ V
87fvconst2 6958 . . 3 (𝐴𝑋 → ((𝑋 × {𝑍})‘𝐴) = 𝑍)
983ad2ant3 1133 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴𝑋) → ((𝑋 × {𝑍})‘𝐴) = 𝑍)
106, 9eqtrd 2794 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑂𝐴) = 𝑍)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  {csn 4523   × cxp 5523  ‘cfv 6336  (class class class)co 7151  NrmCVeccnv 28459  BaseSetcba 28461  0veccn0v 28463   0op c0o 28618 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-0o 28622 This theorem is referenced by:  0lno  28665  nmoo0  28666  nmlno0lem  28668
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