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Theorem 0oval 30867

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 30866	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
5	4	fveq1d 6837	. . 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 6849	. . . 4 ⊢ 𝑍 ∈ V
8	7	fvconst2 7152	. . 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 4581 × cxp 5623 ‘cfv 6493 (class class class)co 7360 NrmCVeccnv 30663 BaseSetcba 30665 0_veccn0v 30667 0_op c0o 30822

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 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682

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 3062 df-rab 3401 df-v 3443 df-sbc 3742 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7363 df-oprab 7364 df-mpo 7365 df-0o 30826

This theorem is referenced by: 0lno 30869 nmoo0 30870 nmlno0lem 30872

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