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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 31091 | . . 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 31091 | . . 3 ⊢ (-1 ·ℎ (𝐷 −ℎ 𝐶)) = (𝐶 −ℎ 𝐷) |
7 | 3, 6 | oveq12i 7443 | . 2 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) |
8 | neg1cn 12378 | . . . 4 ⊢ -1 ∈ ℂ | |
9 | 1, 2 | hvsubcli 31050 | . . . 4 ⊢ (𝐵 −ℎ 𝐴) ∈ ℋ |
10 | 4, 5 | hvsubcli 31050 | . . . 4 ⊢ (𝐷 −ℎ 𝐶) ∈ ℋ |
11 | 8, 8, 9, 10 | his35i 31118 | . . 3 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((-1 · (∗‘-1)) · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) |
12 | neg1rr 12379 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
13 | cjre 15175 | . . . . . . 7 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ (∗‘-1) = -1 |
15 | 14 | oveq2i 7442 | . . . . 5 ⊢ (-1 · (∗‘-1)) = (-1 · -1) |
16 | ax-1cn 11211 | . . . . . 6 ⊢ 1 ∈ ℂ | |
17 | 16, 16 | mul2negi 11709 | . . . . 5 ⊢ (-1 · -1) = (1 · 1) |
18 | 1t1e1 12426 | . . . . 5 ⊢ (1 · 1) = 1 | |
19 | 15, 17, 18 | 3eqtri 2767 | . . . 4 ⊢ (-1 · (∗‘-1)) = 1 |
20 | 19 | oveq1i 7441 | . . 3 ⊢ ((-1 · (∗‘-1)) · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) = (1 · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) |
21 | 9, 10 | hicli 31110 | . . . 4 ⊢ ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) ∈ ℂ |
22 | 21 | mullidi 11264 | . . 3 ⊢ (1 · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
23 | 11, 20, 22 | 3eqtri 2767 | . 2 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
24 | 7, 23 | eqtr3i 2765 | 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 2106 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 1c1 11154 · cmul 11158 -cneg 11491 ∗ccj 15132 ℋchba 30948 ·ℎ csm 30950 ·ih csp 30951 −ℎ cmv 30954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-hfvadd 31029 ax-hvcom 31030 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvmulass 31036 ax-hvdistr1 31037 ax-hfi 31108 ax-his1 31111 ax-his3 31113 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-cj 15135 df-re 15136 df-im 15137 df-hvsub 31000 |
This theorem is referenced by: lnophmlem2 32046 |
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