![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elimnv | Structured version Visualization version GIF version |
Description: Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elimnv.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
elimnv.5 | ⊢ 𝑍 = (0vec‘𝑈) |
elimnv.9 | ⊢ 𝑈 ∈ NrmCVec |
Ref | Expression |
---|---|
elimnv | ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elimnv.9 | . . 3 ⊢ 𝑈 ∈ NrmCVec | |
2 | elimnv.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | elimnv.5 | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
4 | 2, 3 | nvzcl 29814 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ 𝑍 ∈ 𝑋 |
6 | 5 | elimel 4592 | 1 ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ifcif 4523 ‘cfv 6533 NrmCVeccnv 29764 BaseSetcba 29766 0veccn0v 29768 |
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 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-un 7709 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-1st 7959 df-2nd 7960 df-grpo 29673 df-gid 29674 df-ablo 29725 df-vc 29739 df-nv 29772 df-va 29775 df-ba 29776 df-sm 29777 df-0v 29778 df-nmcv 29780 |
This theorem is referenced by: elimph 30000 |
Copyright terms: Public domain | W3C validator |