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Theorem 0oval 29793

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 29792	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
5	4	fveq1d 6849	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑂‘𝐴) = ((𝑋 × {𝑍})‘𝐴))
6	5	3adant3 1132	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = ((𝑋 × {𝑍})‘𝐴))
7	2	fvexi 6861	. . . 4 ⊢ 𝑍 ∈ V
8	7	fvconst2 7158	. . 3 ⊢ (𝐴 ∈ 𝑋 → ((𝑋 × {𝑍})‘𝐴) = 𝑍)
9	8	3ad2ant3 1135	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑍})‘𝐴) = 𝑍)
10	6, 9	eqtrd 2771	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {csn 4591 × cxp 5636 ‘cfv 6501 (class class class)co 7362 NrmCVeccnv 29589 BaseSetcba 29591 0_veccn0v 29593 0_op c0o 29748

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3406 df-v 3448 df-sbc 3743 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-0o 29752

This theorem is referenced by: 0lno 29795 nmoo0 29796 nmlno0lem 29798

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