Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > hvsubid | Structured version Visualization version GIF version |
Description: Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvsubid | ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hvmulid 28710 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | |
2 | 1 | oveq1d 7160 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴)) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) |
3 | ax-1cn 10583 | . . . . 5 ⊢ 1 ∈ ℂ | |
4 | neg1cn 11739 | . . . . 5 ⊢ -1 ∈ ℂ | |
5 | ax-hvdistr2 28713 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 + -1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴))) | |
6 | 3, 4, 5 | mp3an12 1442 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((1 + -1) ·ℎ 𝐴) = ((1 ·ℎ 𝐴) +ℎ (-1 ·ℎ 𝐴))) |
7 | hvsubval 28720 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 −ℎ 𝐴) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) | |
8 | 7 | anidms 567 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = (𝐴 +ℎ (-1 ·ℎ 𝐴))) |
9 | 2, 6, 8 | 3eqtr4rd 2864 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = ((1 + -1) ·ℎ 𝐴)) |
10 | 1pneg1e0 11744 | . . . 4 ⊢ (1 + -1) = 0 | |
11 | 10 | oveq1i 7155 | . . 3 ⊢ ((1 + -1) ·ℎ 𝐴) = (0 ·ℎ 𝐴) |
12 | 9, 11 | syl6eq 2869 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = (0 ·ℎ 𝐴)) |
13 | ax-hvmul0 28714 | . 2 ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) | |
14 | 12, 13 | eqtrd 2853 | 1 ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 𝐴) = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 -cneg 10859 ℋchba 28623 +ℎ cva 28624 ·ℎ csm 28625 0ℎc0v 28628 −ℎ cmv 28629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-hvmulid 28710 ax-hvdistr2 28713 ax-hvmul0 28714 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 df-neg 10861 df-hvsub 28675 |
This theorem is referenced by: hvnegid 28731 hvsubeq0i 28767 hvaddsub4 28782 norm3difi 28851 5oalem1 29358 5oalem2 29359 5oalem3 29360 5oalem5 29362 3oalem2 29367 pjsslem 29383 ho0val 29454 lnop0 29670 0cnop 29683 pjclem4 29903 pj3si 29911 |
Copyright terms: Public domain | W3C validator |