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Theorem 0oval 30882

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 30881	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
5	4	fveq1d 6846	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑂‘𝐴) = ((𝑋 × {𝑍})‘𝐴))
6	5	3adant3 1133	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = ((𝑋 × {𝑍})‘𝐴))
7	2	fvexi 6858	. . . 4 ⊢ 𝑍 ∈ V
8	7	fvconst2 7162	. . 3 ⊢ (𝐴 ∈ 𝑋 → ((𝑋 × {𝑍})‘𝐴) = 𝑍)
9	8	3ad2ant3 1136	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑋 × {𝑍})‘𝐴) = 𝑍)
10	6, 9	eqtrd 2772	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {csn 4582 × cxp 5632 ‘cfv 6502 (class class class)co 7370 NrmCVeccnv 30678 BaseSetcba 30680 0_veccn0v 30682 0_op c0o 30837

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 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692

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-mpt 5182 df-id 5529 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-fv 6510 df-ov 7373 df-oprab 7374 df-mpo 7375 df-0o 30841

This theorem is referenced by: 0lno 30884 nmoo0 30885 nmlno0lem 30887

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