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Mirrors > Home > HSE Home > Th. List > hvsubeq0i | Structured version Visualization version GIF version |
Description: If the difference between two vectors is zero, they are equal. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvnegdi.1 | ⊢ 𝐴 ∈ ℋ |
hvnegdi.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
hvsubeq0i | ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvnegdi.1 | . . . . . 6 ⊢ 𝐴 ∈ ℋ | |
2 | hvnegdi.2 | . . . . . 6 ⊢ 𝐵 ∈ ℋ | |
3 | 1, 2 | hvsubvali 27877 | . . . . 5 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | eqeq1i 2627 | . . . 4 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ) |
5 | oveq1 6657 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵)) | |
6 | 4, 5 | sylbi 207 | . . 3 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵)) |
7 | neg1cn 11124 | . . . . . 6 ⊢ -1 ∈ ℂ | |
8 | 7, 2 | hvmulcli 27871 | . . . . 5 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
9 | 1, 8, 2 | hvadd32i 27911 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) |
10 | 1, 2, 8 | hvassi 27910 | . . . . 5 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) |
11 | 2 | hvnegidi 27887 | . . . . . . 7 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ |
12 | 11 | oveq2i 6661 | . . . . . 6 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ 0ℎ) |
13 | ax-hvaddid 27861 | . . . . . . 7 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
14 | 1, 13 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 +ℎ 0ℎ) = 𝐴 |
15 | 12, 14 | eqtri 2644 | . . . . 5 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = 𝐴 |
16 | 10, 15 | eqtri 2644 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = 𝐴 |
17 | 9, 16 | eqtri 2644 | . . 3 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ 𝐵) = 𝐴 |
18 | 2 | hvaddid2i 27886 | . . 3 ⊢ (0ℎ +ℎ 𝐵) = 𝐵 |
19 | 6, 17, 18 | 3eqtr3g 2679 | . 2 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ → 𝐴 = 𝐵) |
20 | oveq1 6657 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 −ℎ 𝐵) = (𝐵 −ℎ 𝐵)) | |
21 | hvsubid 27883 | . . . 4 ⊢ (𝐵 ∈ ℋ → (𝐵 −ℎ 𝐵) = 0ℎ) | |
22 | 2, 21 | ax-mp 5 | . . 3 ⊢ (𝐵 −ℎ 𝐵) = 0ℎ |
23 | 20, 22 | syl6eq 2672 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 −ℎ 𝐵) = 0ℎ) |
24 | 19, 23 | impbii 199 | 1 ⊢ ((𝐴 −ℎ 𝐵) = 0ℎ ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 1c1 9937 -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-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 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-iun 4522 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: hvsubeq0 27925 bcseqi 27977 normsub0i 27992 pjss2i 28539 |
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