![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > hvaddeq0 | Structured version Visualization version GIF version |
Description: If the sum of two vectors is zero, one is the negative of the other. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvaddeq0 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) = 0ℎ ↔ 𝐴 = (-1 ·ℎ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvaddsubval 28604 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐴 −ℎ (-1 ·ℎ 𝐵))) | |
2 | 1 | eqeq1d 2782 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) = 0ℎ ↔ (𝐴 −ℎ (-1 ·ℎ 𝐵)) = 0ℎ)) |
3 | neg1cn 11567 | . . . 4 ⊢ -1 ∈ ℂ | |
4 | hvmulcl 28584 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ 𝐵) ∈ ℋ) | |
5 | 3, 4 | mpan 678 | . . 3 ⊢ (𝐵 ∈ ℋ → (-1 ·ℎ 𝐵) ∈ ℋ) |
6 | hvsubeq0 28639 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ (-1 ·ℎ 𝐵) ∈ ℋ) → ((𝐴 −ℎ (-1 ·ℎ 𝐵)) = 0ℎ ↔ 𝐴 = (-1 ·ℎ 𝐵))) | |
7 | 5, 6 | sylan2 584 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −ℎ (-1 ·ℎ 𝐵)) = 0ℎ ↔ 𝐴 = (-1 ·ℎ 𝐵))) |
8 | 2, 7 | bitrd 271 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) = 0ℎ ↔ 𝐴 = (-1 ·ℎ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 (class class class)co 6982 ℂcc 10339 1c1 10342 -cneg 10677 ℋchba 28490 +ℎ cva 28491 ·ℎ csm 28492 0ℎc0v 28495 −ℎ cmv 28496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-hvcom 28572 ax-hvass 28573 ax-hv0cl 28574 ax-hvaddid 28575 ax-hfvmul 28576 ax-hvmulid 28577 ax-hvmulass 28578 ax-hvdistr2 28580 ax-hvmul0 28581 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-id 5316 df-po 5330 df-so 5331 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 df-pnf 10482 df-mnf 10483 df-ltxr 10485 df-sub 10678 df-neg 10679 df-hvsub 28542 |
This theorem is referenced by: superpos 29927 |
Copyright terms: Public domain | W3C validator |