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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 28841 | . . 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 28841 | . . 3 ⊢ (-1 ·ℎ (𝐷 −ℎ 𝐶)) = (𝐶 −ℎ 𝐷) |
7 | 3, 6 | oveq12i 7170 | . 2 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) |
8 | neg1cn 11754 | . . . 4 ⊢ -1 ∈ ℂ | |
9 | 1, 2 | hvsubcli 28800 | . . . 4 ⊢ (𝐵 −ℎ 𝐴) ∈ ℋ |
10 | 4, 5 | hvsubcli 28800 | . . . 4 ⊢ (𝐷 −ℎ 𝐶) ∈ ℋ |
11 | 8, 8, 9, 10 | his35i 28868 | . . 3 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((-1 · (∗‘-1)) · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) |
12 | neg1rr 11755 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
13 | cjre 14500 | . . . . . . 7 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ (∗‘-1) = -1 |
15 | 14 | oveq2i 7169 | . . . . 5 ⊢ (-1 · (∗‘-1)) = (-1 · -1) |
16 | ax-1cn 10597 | . . . . . 6 ⊢ 1 ∈ ℂ | |
17 | 16, 16 | mul2negi 11090 | . . . . 5 ⊢ (-1 · -1) = (1 · 1) |
18 | 1t1e1 11802 | . . . . 5 ⊢ (1 · 1) = 1 | |
19 | 15, 17, 18 | 3eqtri 2850 | . . . 4 ⊢ (-1 · (∗‘-1)) = 1 |
20 | 19 | oveq1i 7168 | . . 3 ⊢ ((-1 · (∗‘-1)) · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) = (1 · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) |
21 | 9, 10 | hicli 28860 | . . . 4 ⊢ ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) ∈ ℂ |
22 | 21 | mulid2i 10648 | . . 3 ⊢ (1 · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
23 | 11, 20, 22 | 3eqtri 2850 | . 2 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
24 | 7, 23 | eqtr3i 2848 | 1 ⊢ ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 1c1 10540 · cmul 10544 -cneg 10873 ∗ccj 14457 ℋchba 28698 ·ℎ csm 28700 ·ih csp 28701 −ℎ cmv 28704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-hfvadd 28779 ax-hvcom 28780 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvmulass 28786 ax-hvdistr1 28787 ax-hfi 28858 ax-his1 28861 ax-his3 28863 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-cj 14460 df-re 14461 df-im 14462 df-hvsub 28750 |
This theorem is referenced by: lnophmlem2 29796 |
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