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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 27303 | . . 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 27303 | . . 3 ⊢ (-1 ·ℎ (𝐷 −ℎ 𝐶)) = (𝐶 −ℎ 𝐷) |
7 | 3, 6 | oveq12i 6561 | . 2 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) |
8 | neg1cn 11001 | . . . 4 ⊢ -1 ∈ ℂ | |
9 | 1, 2 | hvsubcli 27262 | . . . 4 ⊢ (𝐵 −ℎ 𝐴) ∈ ℋ |
10 | 4, 5 | hvsubcli 27262 | . . . 4 ⊢ (𝐷 −ℎ 𝐶) ∈ ℋ |
11 | 8, 8, 9, 10 | his35i 27330 | . . 3 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((-1 · (∗‘-1)) · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) |
12 | neg1rr 11002 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
13 | cjre 13727 | . . . . . . 7 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ (∗‘-1) = -1 |
15 | 14 | oveq2i 6560 | . . . . 5 ⊢ (-1 · (∗‘-1)) = (-1 · -1) |
16 | ax-1cn 9873 | . . . . . 6 ⊢ 1 ∈ ℂ | |
17 | 16, 16 | mul2negi 10357 | . . . . 5 ⊢ (-1 · -1) = (1 · 1) |
18 | 1t1e1 11052 | . . . . 5 ⊢ (1 · 1) = 1 | |
19 | 15, 17, 18 | 3eqtri 2636 | . . . 4 ⊢ (-1 · (∗‘-1)) = 1 |
20 | 19 | oveq1i 6559 | . . 3 ⊢ ((-1 · (∗‘-1)) · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) = (1 · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) |
21 | 9, 10 | hicli 27322 | . . . 4 ⊢ ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) ∈ ℂ |
22 | 21 | mulid2i 9922 | . . 3 ⊢ (1 · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
23 | 11, 20, 22 | 3eqtri 2636 | . 2 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
24 | 7, 23 | eqtr3i 2634 | 1 ⊢ ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 1c1 9816 · cmul 9820 -cneg 10146 ∗ccj 13684 ℋchil 27160 ·ℎ csm 27162 ·ih csp 27163 −ℎ cmv 27166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-hfvadd 27241 ax-hvcom 27242 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hfi 27320 ax-his1 27323 ax-his3 27325 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-cj 13687 df-re 13688 df-im 13689 df-hvsub 27212 |
This theorem is referenced by: lnophmlem2 28260 |
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