| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > nvm1 | Structured version Visualization version GIF version | ||
| Description: The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nvs.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
| nvs.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
| nvs.6 | ⊢ 𝑁 = (normCV‘𝑈) |
| Ref | Expression |
|---|---|
| nvm1 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(-1𝑆𝐴)) = (𝑁‘𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neg1cn 12139 | . . 3 ⊢ -1 ∈ ℂ | |
| 2 | nvs.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 3 | nvs.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
| 4 | nvs.6 | . . . 4 ⊢ 𝑁 = (normCV‘𝑈) | |
| 5 | 2, 3, 4 | nvs 30754 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑁‘(-1𝑆𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
| 6 | 1, 5 | mp3an2 1458 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(-1𝑆𝐴)) = ((abs‘-1) · (𝑁‘𝐴))) |
| 7 | ax-1cn 11092 | . . . . . 6 ⊢ 1 ∈ ℂ | |
| 8 | 7 | absnegi 15358 | . . . . 5 ⊢ (abs‘-1) = (abs‘1) |
| 9 | abs1 15254 | . . . . 5 ⊢ (abs‘1) = 1 | |
| 10 | 8, 9 | eqtri 2764 | . . . 4 ⊢ (abs‘-1) = 1 |
| 11 | 10 | oveq1i 7369 | . . 3 ⊢ ((abs‘-1) · (𝑁‘𝐴)) = (1 · (𝑁‘𝐴)) |
| 12 | 2, 4 | nvcl 30752 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
| 13 | 12 | recnd 11169 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℂ) |
| 14 | 13 | mullidd 11159 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1 · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
| 15 | 11, 14 | eqtrid 2788 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((abs‘-1) · (𝑁‘𝐴)) = (𝑁‘𝐴)) |
| 16 | 6, 15 | eqtrd 2776 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(-1𝑆𝐴)) = (𝑁‘𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ‘cfv 6488 (class class class)co 7359 ℂcc 11032 1c1 11035 · cmul 11039 -cneg 11374 abscabs 15191 NrmCVeccnv 30675 BaseSetcba 30677 ·𝑠OLD cns 30678 normCVcnmcv 30681 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-pre-sup 11112 |
| This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-om 7810 df-1st 7933 df-2nd 7934 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-sup 9349 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-z 12520 df-uz 12784 df-rp 12938 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-vc 30650 df-nv 30683 df-va 30686 df-ba 30687 df-sm 30688 df-0v 30689 df-nmcv 30691 |
| This theorem is referenced by: nvdif 30757 nvge0 30764 |
| Copyright terms: Public domain | W3C validator |