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Mirrors > Home > HSE Home > Th. List > hvaddcl | Structured version Visualization version GIF version |
Description: Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvaddcl | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hfvadd 29350 | . 2 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
2 | 1 | fovcl 7394 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2110 (class class class)co 7269 ℋchba 29269 +ℎ cva 29270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-hfvadd 29350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-fv 6439 df-ov 7272 |
This theorem is referenced by: hvsubf 29365 hvsubcl 29367 hvaddcli 29368 hvadd4 29386 hvsub4 29387 hvpncan 29389 hvaddsubass 29391 hvsubass 29394 hv2times 29411 hvaddsub4 29428 his7 29440 normpyc 29496 hhph 29528 hlimadd 29543 helch 29593 ocsh 29633 spanunsni 29929 3oalem1 30012 pjcompi 30022 mayete3i 30078 hoscl 30095 hoaddcl 30108 unoplin 30270 hmoplin 30292 braadd 30295 0lnfn 30335 lnopmi 30350 lnophsi 30351 lnopcoi 30353 lnopeq0i 30357 nlelshi 30410 cnlnadjlem2 30418 cnlnadjlem6 30422 adjlnop 30436 superpos 30704 cdj3lem2b 30787 cdj3i 30791 |
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