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Mirrors > Home > HSE Home > Th. List > hvsubid | Structured version Visualization version GIF version |
Description: Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvsubid | ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hvmulid 27863 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | |
2 | 1 | oveq1d 6665 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴)) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) |
3 | ax-1cn 9994 | . . . . 5 ⊢ 1 ∈ ℂ | |
4 | neg1cn 11124 | . . . . 5 ⊢ -1 ∈ ℂ | |
5 | ax-hvdistr2 27866 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 + -1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴))) | |
6 | 3, 4, 5 | mp3an12 1414 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((1 + -1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴))) |
7 | hvsubval 27873 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 −ℎ 𝐴) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) | |
8 | 7 | anidms 677 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) |
9 | 2, 6, 8 | 3eqtr4rd 2667 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = ((1 + -1) ·ℎ 𝐴)) |
10 | 1pneg1e0 11129 | . . . 4 ⊢ (1 + -1) = 0 | |
11 | 10 | oveq1i 6660 | . . 3 ⊢ ((1 + -1) ·ℎ 𝐴) = (0 ·ℎ 𝐴) |
12 | 9, 11 | syl6eq 2672 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = (0 ·ℎ 𝐴)) |
13 | ax-hvmul0 27867 | . 2 ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) | |
14 | 12, 13 | eqtrd 2656 | 1 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 -cneg 10267 ℋchil 27776 +ℎ cva 27777 ·ℎ csm 27778 0ℎc0v 27781 −ℎ cmv 27782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-hvmulid 27863 ax-hvdistr2 27866 ax-hvmul0 27867 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-hvsub 27828 |
This theorem is referenced by: hvnegid 27884 hvsubeq0i 27920 hvaddsub4 27935 norm3difi 28004 5oalem1 28513 5oalem2 28514 5oalem3 28515 5oalem5 28517 3oalem2 28522 pjsslem 28538 ho0val 28609 lnop0 28825 0cnop 28838 pjclem4 29058 pj3si 29066 |
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