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Theorem 0oval 30993

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 = (0_vec‘𝑊)
0oval.0	⊢ 𝑂 = (𝑈 0_op 𝑊)

**Assertion**
Ref	Expression
0oval	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = 𝑍)

**Proof of Theorem 0oval**
Step	Hyp	Ref	Expression
1		0oval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
2		0oval.6	. . . . 5 ⊢ 𝑍 = (0_vec‘𝑊)
3		0oval.0	. . . . 5 ⊢ 𝑂 = (𝑈 0_op 𝑊)
4	1, 2, 3	0ofval 30992	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
5	4	fveq1d 6871	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑂‘𝐴) = ((𝑋 × {𝑍})‘𝐴))
6	5	3adant3 1146	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = ((𝑋 × {𝑍})‘𝐴))
7	2	fvexi 6883	. . . 4 ⊢ 𝑍 ∈ V
8	7	fvconst2 7190	. . 3 ⊢ (𝐴 ∈ 𝑋 → ((𝑋 × {𝑍})‘𝐴) = 𝑍)
9	8	3ad2ant3 1149	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑍})‘𝐴) = 𝑍)
10	6, 9	eqtrd 2799	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1099 = wceq 1562 ∈ wcel 2144 {csn 4584 × cxp 5647 ‘cfv 6523 (class class class)co 7398 NrmCVeccnv 30789 BaseSetcba 30791 0_veccn0v 30793 0_op c0o 30948

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3079 df-rex 3089 df-rab 3417 df-v 3458 df-sbc 3747 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5544 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-fv 6531 df-ov 7401 df-oprab 7402 df-mpo 7403 df-0o 30952

This theorem is referenced by: 0lno 30995 nmoo0 30996 nmlno0lem 30998

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