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Mirrors > Home > HSE Home > Th. List > hvsubcl | Structured version Visualization version GIF version |
Description: Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvsubcl | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvsubval 30819 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) | |
2 | neg1cn 12350 | . . . 4 ⊢ -1 ∈ ℂ | |
3 | hvmulcl 30816 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ 𝐵) ∈ ℋ) | |
4 | 2, 3 | mpan 689 | . . 3 ⊢ (𝐵 ∈ ℋ → (-1 ·ℎ 𝐵) ∈ ℋ) |
5 | hvaddcl 30815 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ (-1 ·ℎ 𝐵) ∈ ℋ) → (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ ℋ) | |
6 | 4, 5 | sylan2 592 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ ℋ) |
7 | 1, 6 | eqeltrd 2829 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 (class class class)co 7414 ℂcc 11130 1c1 11133 -cneg 11469 ℋchba 30722 +ℎ cva 30723 ·ℎ csm 30724 −ℎ cmv 30728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-hfvadd 30803 ax-hfvmul 30808 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-ltxr 11277 df-sub 11470 df-neg 11471 df-hvsub 30774 |
This theorem is referenced by: hvsubcli 30824 hvmulcan 30875 hvsubcan2 30878 hvaddsub4 30881 his2sub2 30896 hi2eq 30908 hial2eq 30909 hhph 30981 pjhthlem1 31194 pjhthlem2 31195 chscllem2 31441 5oalem2 31458 5oalem3 31459 5oalem5 31461 3oalem2 31466 hodcl 31550 hosubcli 31572 unopf1o 31719 lnopeq0i 31810 lnconi 31836 riesz3i 31865 riesz4i 31866 hmopidmpji 31955 pjclem4 32002 pj3si 32010 cdj1i 32236 |
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