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| Mirrors > Home > MPE Home > Th. List > nvmval2 | Structured version Visualization version GIF version | ||
| Description: Value of vector subtraction on a normed complex vector space. (Contributed by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nvmval.1 | β’ π = (BaseSetβπ) |
| nvmval.2 | β’ πΊ = ( +π£ βπ) |
| nvmval.4 | β’ π = ( Β·π OLD βπ) |
| nvmval.3 | β’ π = ( βπ£ βπ) |
| Ref | Expression |
|---|---|
| nvmval2 | β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) = ((-1ππ΅)πΊπ΄)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvmval.1 | . . 3 β’ π = (BaseSetβπ) | |
| 2 | nvmval.2 | . . 3 β’ πΊ = ( +π£ βπ) | |
| 3 | nvmval.4 | . . 3 β’ π = ( Β·π OLD βπ) | |
| 4 | nvmval.3 | . . 3 β’ π = ( βπ£ βπ) | |
| 5 | 1, 2, 3, 4 | nvmval 30472 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) = (π΄πΊ(-1ππ΅))) |
| 6 | neg1cn 12364 | . . . . 5 β’ -1 β β | |
| 7 | 1, 3 | nvscl 30456 | . . . . 5 β’ ((π β NrmCVec β§ -1 β β β§ π΅ β π) β (-1ππ΅) β π) |
| 8 | 6, 7 | mp3an2 1445 | . . . 4 β’ ((π β NrmCVec β§ π΅ β π) β (-1ππ΅) β π) |
| 9 | 8 | 3adant2 1128 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (-1ππ΅) β π) |
| 10 | 1, 2 | nvcom 30451 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ (-1ππ΅) β π) β (π΄πΊ(-1ππ΅)) = ((-1ππ΅)πΊπ΄)) |
| 11 | 9, 10 | syld3an3 1406 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄πΊ(-1ππ΅)) = ((-1ππ΅)πΊπ΄)) |
| 12 | 5, 11 | eqtrd 2768 | 1 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) = ((-1ππ΅)πΊπ΄)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 βcc 11144 1c1 11147 -cneg 11483 NrmCVeccnv 30414 +π£ cpv 30415 BaseSetcba 30416 Β·π OLD cns 30417 βπ£ cnsb 30419 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-ltxr 11291 df-sub 11484 df-neg 11485 df-grpo 30323 df-gid 30324 df-ginv 30325 df-gdiv 30326 df-ablo 30375 df-vc 30389 df-nv 30422 df-va 30425 df-ba 30426 df-sm 30427 df-0v 30428 df-vs 30429 df-nmcv 30430 |
| This theorem is referenced by: lnosub 30589 |
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