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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 28435 | . . 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 28435 | . . 3 ⊢ (-1 ·ℎ (𝐷 −ℎ 𝐶)) = (𝐶 −ℎ 𝐷) |
7 | 3, 6 | oveq12i 6888 | . 2 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) |
8 | neg1cn 11430 | . . . 4 ⊢ -1 ∈ ℂ | |
9 | 1, 2 | hvsubcli 28394 | . . . 4 ⊢ (𝐵 −ℎ 𝐴) ∈ ℋ |
10 | 4, 5 | hvsubcli 28394 | . . . 4 ⊢ (𝐷 −ℎ 𝐶) ∈ ℋ |
11 | 8, 8, 9, 10 | his35i 28462 | . . 3 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((-1 · (∗‘-1)) · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) |
12 | neg1rr 11431 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
13 | cjre 14216 | . . . . . . 7 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ (∗‘-1) = -1 |
15 | 14 | oveq2i 6887 | . . . . 5 ⊢ (-1 · (∗‘-1)) = (-1 · -1) |
16 | ax-1cn 10280 | . . . . . 6 ⊢ 1 ∈ ℂ | |
17 | 16, 16 | mul2negi 10768 | . . . . 5 ⊢ (-1 · -1) = (1 · 1) |
18 | 1t1e1 11478 | . . . . 5 ⊢ (1 · 1) = 1 | |
19 | 15, 17, 18 | 3eqtri 2823 | . . . 4 ⊢ (-1 · (∗‘-1)) = 1 |
20 | 19 | oveq1i 6886 | . . 3 ⊢ ((-1 · (∗‘-1)) · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) = (1 · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) |
21 | 9, 10 | hicli 28454 | . . . 4 ⊢ ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) ∈ ℂ |
22 | 21 | mulid2i 10332 | . . 3 ⊢ (1 · ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
23 | 11, 20, 22 | 3eqtri 2823 | . 2 ⊢ ((-1 ·ℎ (𝐵 −ℎ 𝐴)) ·ih (-1 ·ℎ (𝐷 −ℎ 𝐶))) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
24 | 7, 23 | eqtr3i 2821 | 1 ⊢ ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 ‘cfv 6099 (class class class)co 6876 ℝcr 10221 1c1 10223 · cmul 10227 -cneg 10555 ∗ccj 14173 ℋchba 28292 ·ℎ csm 28294 ·ih csp 28295 −ℎ cmv 28298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-hfvadd 28373 ax-hvcom 28374 ax-hfvmul 28378 ax-hvmulid 28379 ax-hvmulass 28380 ax-hvdistr1 28381 ax-hfi 28452 ax-his1 28455 ax-his3 28457 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-po 5231 df-so 5232 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-2 11372 df-cj 14176 df-re 14177 df-im 14178 df-hvsub 28344 |
This theorem is referenced by: lnophmlem2 29392 |
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