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Mirrors > Home > HSE Home > Th. List > hvsubcli | Structured version Visualization version GIF version |
Description: Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvaddcl.1 | ⊢ 𝐴 ∈ ℋ |
hvaddcl.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
hvsubcli | ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvaddcl.1 | . 2 ⊢ 𝐴 ∈ ℋ | |
2 | hvaddcl.2 | . 2 ⊢ 𝐵 ∈ ℋ | |
3 | hvsubcl 31051 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2108 (class class class)co 7450 ℋchba 30953 −ℎ cmv 30959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-hfvadd 31034 ax-hfvmul 31039 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-ltxr 11331 df-sub 11524 df-neg 11525 df-hvsub 31005 |
This theorem is referenced by: hisubcomi 31138 normlem5 31148 bcseqi 31154 normsub0i 31169 normsubi 31175 norm3difi 31181 norm3adifii 31182 norm3lem 31183 normpari 31188 normpar2i 31190 polidi 31192 h1de2i 31587 pjsslem 31713 pjss2i 31714 pjssmii 31715 pjcji 31718 lnophmlem2 32051 |
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