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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 30511 | . 2 ⊢ 0ℎ ∈ ℋ | |
2 | 1 | elimel 4597 | 1 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ifcif 4528 ℋchba 30427 0ℎc0v 30432 |
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-ext 2703 ax-hv0cl 30511 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-if 4529 |
This theorem is referenced by: hvsubsub4 30568 hvnegdi 30575 hvsubeq0 30576 hvaddcan 30578 hvsubadd 30585 normlem9at 30629 normsq 30642 normsub0 30644 norm-ii 30646 norm-iii 30648 normsub 30651 normpyth 30653 norm3dif 30658 norm3lemt 30660 norm3adifi 30661 normpar 30663 polid 30667 bcs 30689 pjoc1 30942 pjoc2 30947 h1de2ci 31064 spansn 31067 elspansn 31074 elspansn2 31075 h1datom 31090 spansnj 31155 spansncv 31161 pjch1 31178 pjadji 31193 pjaddi 31194 pjinormi 31195 pjsubi 31196 pjmuli 31197 pjcjt2 31200 pjch 31202 pjopyth 31228 pjnorm 31232 pjpyth 31233 pjnel 31234 eigre 31343 eigorth 31346 lnopeq0lem2 31514 lnopunii 31520 lnophmi 31526 pjss2coi 31672 pjssmi 31673 pjssge0i 31674 pjdifnormi 31675 |
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