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Mirrors > Home > HSE Home > Th. List > hvsubaddi | Structured version Visualization version GIF version |
Description: Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvnegdi.1 | ⊢ 𝐴 ∈ ℋ |
hvnegdi.2 | ⊢ 𝐵 ∈ ℋ |
hvaddcan.3 | ⊢ 𝐶 ∈ ℋ |
Ref | Expression |
---|---|
hvsubaddi | ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvnegdi.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
2 | hvnegdi.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
3 | 1, 2 | hvsubvali 27877 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | eqeq1i 2627 | . 2 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = 𝐶) |
5 | neg1cn 11124 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
6 | 5, 2 | hvmulcli 27871 | . . . . . 6 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
7 | 2, 1, 6 | hvadd12i 27914 | . . . . 5 ⊢ (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) |
8 | 2 | hvnegidi 27887 | . . . . . 6 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ |
9 | 8 | oveq2i 6661 | . . . . 5 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ 0ℎ) |
10 | ax-hvaddid 27861 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
11 | 1, 10 | ax-mp 5 | . . . . 5 ⊢ (𝐴 +ℎ 0ℎ) = 𝐴 |
12 | 7, 9, 11 | 3eqtri 2648 | . . . 4 ⊢ (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = 𝐴 |
13 | 12 | eqeq1i 2627 | . . 3 ⊢ ((𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (𝐵 +ℎ 𝐶) ↔ 𝐴 = (𝐵 +ℎ 𝐶)) |
14 | 1, 6 | hvaddcli 27875 | . . . 4 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ ℋ |
15 | hvaddcan.3 | . . . 4 ⊢ 𝐶 ∈ ℋ | |
16 | 2, 14, 15 | hvaddcani 27922 | . . 3 ⊢ ((𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (𝐵 +ℎ 𝐶) ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = 𝐶) |
17 | eqcom 2629 | . . 3 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) ↔ (𝐵 +ℎ 𝐶) = 𝐴) | |
18 | 13, 16, 17 | 3bitr3i 290 | . 2 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
19 | 4, 18 | bitri 264 | 1 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
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-hfvadd 27857 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: hvsubadd 27934 omlsilem 28261 |
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