![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > ifhvhv0 | Structured version Visualization version GIF version |
Description: Prove if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ. (Contributed by David A. Wheeler, 7-Dec-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ifhvhv0 | ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 31032 | . 2 ⊢ 0ℎ ∈ ℋ | |
2 | 1 | elimel 4600 | 1 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ifcif 4531 ℋchba 30948 0ℎc0v 30953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-hv0cl 31032 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-if 4532 |
This theorem is referenced by: hvsubsub4 31089 hvnegdi 31096 hvsubeq0 31097 hvaddcan 31099 hvsubadd 31106 normlem9at 31150 normsq 31163 normsub0 31165 norm-ii 31167 norm-iii 31169 normsub 31172 normpyth 31174 norm3dif 31179 norm3lemt 31181 norm3adifi 31182 normpar 31184 polid 31188 bcs 31210 pjoc1 31463 pjoc2 31468 h1de2ci 31585 spansn 31588 elspansn 31595 elspansn2 31596 h1datom 31611 spansnj 31676 spansncv 31682 pjch1 31699 pjadji 31714 pjaddi 31715 pjinormi 31716 pjsubi 31717 pjmuli 31718 pjcjt2 31721 pjch 31723 pjopyth 31749 pjnorm 31753 pjpyth 31754 pjnel 31755 eigre 31864 eigorth 31867 lnopeq0lem2 32035 lnopunii 32041 lnophmi 32047 pjss2coi 32193 pjssmi 32194 pjssge0i 32195 pjdifnormi 32196 |
Copyright terms: Public domain | W3C validator |