![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0oval | Structured version Visualization version GIF version |
Description: Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0oval.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
0oval.6 | ⊢ 𝑍 = (0vec‘𝑊) |
0oval.0 | ⊢ 𝑂 = (𝑈 0op 𝑊) |
Ref | Expression |
---|---|
0oval | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0oval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | 0oval.6 | . . . . 5 ⊢ 𝑍 = (0vec‘𝑊) | |
3 | 0oval.0 | . . . . 5 ⊢ 𝑂 = (𝑈 0op 𝑊) | |
4 | 1, 2, 3 | 0ofval 29627 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍})) |
5 | 4 | fveq1d 6842 | . . 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 6854 | . . . 4 ⊢ 𝑍 ∈ V |
8 | 7 | fvconst2 7150 | . . 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 2776 | 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 4585 × cxp 5630 ‘cfv 6494 (class class class)co 7354 NrmCVeccnv 29424 BaseSetcba 29426 0veccn0v 29428 0op c0o 29583 |
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 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-fv 6502 df-ov 7357 df-oprab 7358 df-mpo 7359 df-0o 29587 |
This theorem is referenced by: 0lno 29630 nmoo0 29631 nmlno0lem 29633 |
Copyright terms: Public domain | W3C validator |