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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 28433 | . 2 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
2 | 1 | fovcl 7044 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 (class class class)co 6924 ℋchba 28352 +ℎ cva 28353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-hfvadd 28433 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-fv 6145 df-ov 6927 |
This theorem is referenced by: hvsubf 28448 hvsubcl 28450 hvaddcli 28451 hvadd4 28469 hvsub4 28470 hvpncan 28472 hvaddsubass 28474 hvsubass 28477 hv2times 28494 hvaddsub4 28511 his7 28523 normpyc 28579 hhph 28611 hlimadd 28626 helch 28676 ocsh 28718 spanunsni 29014 3oalem1 29097 pjcompi 29107 mayete3i 29163 hoscl 29180 hoaddcl 29193 unoplin 29355 hmoplin 29377 braadd 29380 0lnfn 29420 lnopmi 29435 lnophsi 29436 lnopcoi 29438 lnopeq0i 29442 nlelshi 29495 cnlnadjlem2 29503 cnlnadjlem6 29507 adjlnop 29521 superpos 29789 cdj3lem2b 29872 cdj3i 29876 |
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