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Mirrors > Home > HSE Home > Th. List > hisubcomi | Structured version Visualization version GIF version |
Description: Two vector subtractions simultaneously commute in an inner product. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hisubcom.1 | ⊢ 𝐴 ∈ ℋ |
hisubcom.2 | ⊢ 𝐵 ∈ ℋ |
hisubcom.3 | ⊢ 𝐶 ∈ ℋ |
hisubcom.4 | ⊢ 𝐷 ∈ ℋ |
Ref | Expression |
---|---|
hisubcomi | ⊢ ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hisubcom.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
2 | hisubcom.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
3 | 1, 2 | hvnegdii 27919 | . . 3 ⊢ (-1 ·ℎ (𝐵 −ℎ 𝐴)) = (𝐴 −ℎ 𝐵) |
4 | hisubcom.4 | . . . 4 ⊢ 𝐷 ∈ ℋ | |
5 | hisubcom.3 | . . . 4 ⊢ 𝐶 ∈ ℋ | |
6 | 4, 5 | hvnegdii 27919 | . . 3 ⊢ (-1 ·ℎ (𝐷 −ℎ 𝐶)) = (𝐶 −ℎ 𝐷) |
7 | 3, 6 | oveq12i 6662 | . 2 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) |
8 | neg1cn 11124 | . . . 4 ⊢ -1 ∈ ℂ | |
9 | 1, 2 | hvsubcli 27878 | . . . 4 ⊢ (𝐵 −ℎ 𝐴) ∈ ℋ |
10 | 4, 5 | hvsubcli 27878 | . . . 4 ⊢ (𝐷 −ℎ 𝐶) ∈ ℋ |
11 | 8, 8, 9, 10 | his35i 27946 | . . 3 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((-1 · (∗‘-1)) · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) |
12 | neg1rr 11125 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
13 | cjre 13879 | . . . . . . 7 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ (∗‘-1) = -1 |
15 | 14 | oveq2i 6661 | . . . . 5 ⊢ (-1 · (∗‘-1)) = (-1 · -1) |
16 | ax-1cn 9994 | . . . . . 6 ⊢ 1 ∈ ℂ | |
17 | 16, 16 | mul2negi 10478 | . . . . 5 ⊢ (-1 · -1) = (1 · 1) |
18 | 1t1e1 11175 | . . . . 5 ⊢ (1 · 1) = 1 | |
19 | 15, 17, 18 | 3eqtri 2648 | . . . 4 ⊢ (-1 · (∗‘-1)) = 1 |
20 | 19 | oveq1i 6660 | . . 3 ⊢ ((-1 · (∗‘-1)) · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) = (1 · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) |
21 | 9, 10 | hicli 27938 | . . . 4 ⊢ ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) ∈ ℂ |
22 | 21 | mulid2i 10043 | . . 3 ⊢ (1 · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
23 | 11, 20, 22 | 3eqtri 2648 | . 2 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
24 | 7, 23 | eqtr3i 2646 | 1 ⊢ ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 1c1 9937 · cmul 9941 -cneg 10267 ∗ccj 13836 ℋchil 27776 ·ℎ csm 27778 ·ih csp 27779 −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-pre-mulgt0 10013 ax-hfvadd 27857 ax-hvcom 27858 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hfi 27936 ax-his1 27939 ax-his3 27941 |
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-rmo 2920 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-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-cj 13839 df-re 13840 df-im 13841 df-hvsub 27828 |
This theorem is referenced by: lnophmlem2 28876 |
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