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Mirrors > Home > MPE Home > Th. List > cnnvm | Structured version Visualization version GIF version |
Description: The vector subtraction operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnnvm.6 | β’ π = β¨β¨ + , Β· β©, absβ© |
Ref | Expression |
---|---|
cnnvm | β’ β = ( βπ£ βπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1 11686 | . . . . . 6 β’ (π¦ β β β (-1 Β· π¦) = -π¦) | |
2 | 1 | adantl 481 | . . . . 5 β’ ((π₯ β β β§ π¦ β β) β (-1 Β· π¦) = -π¦) |
3 | 2 | oveq2d 7436 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯ + (-1 Β· π¦)) = (π₯ + -π¦)) |
4 | negsub 11539 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯ + -π¦) = (π₯ β π¦)) | |
5 | 3, 4 | eqtr2d 2769 | . . 3 β’ ((π₯ β β β§ π¦ β β) β (π₯ β π¦) = (π₯ + (-1 Β· π¦))) |
6 | 5 | mpoeq3ia 7498 | . 2 β’ (π₯ β β, π¦ β β β¦ (π₯ β π¦)) = (π₯ β β, π¦ β β β¦ (π₯ + (-1 Β· π¦))) |
7 | subf 11493 | . . . 4 β’ β :(β Γ β)βΆβ | |
8 | ffn 6722 | . . . 4 β’ ( β :(β Γ β)βΆβ β β Fn (β Γ β)) | |
9 | 7, 8 | ax-mp 5 | . . 3 β’ β Fn (β Γ β) |
10 | fnov 7552 | . . 3 β’ ( β Fn (β Γ β) β β = (π₯ β β, π¦ β β β¦ (π₯ β π¦))) | |
11 | 9, 10 | mpbi 229 | . 2 β’ β = (π₯ β β, π¦ β β β¦ (π₯ β π¦)) |
12 | cnnvm.6 | . . . 4 β’ π = β¨β¨ + , Β· β©, absβ© | |
13 | 12 | cnnv 30500 | . . 3 β’ π β NrmCVec |
14 | 12 | cnnvba 30502 | . . . 4 β’ β = (BaseSetβπ) |
15 | 12 | cnnvg 30501 | . . . 4 β’ + = ( +π£ βπ) |
16 | 12 | cnnvs 30503 | . . . 4 β’ Β· = ( Β·π OLD βπ) |
17 | eqid 2728 | . . . 4 β’ ( βπ£ βπ) = ( βπ£ βπ) | |
18 | 14, 15, 16, 17 | nvmfval 30467 | . . 3 β’ (π β NrmCVec β ( βπ£ βπ) = (π₯ β β, π¦ β β β¦ (π₯ + (-1 Β· π¦)))) |
19 | 13, 18 | ax-mp 5 | . 2 β’ ( βπ£ βπ) = (π₯ β β, π¦ β β β¦ (π₯ + (-1 Β· π¦))) |
20 | 6, 11, 19 | 3eqtr4i 2766 | 1 β’ β = ( βπ£ βπ) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1534 β wcel 2099 β¨cop 4635 Γ cxp 5676 Fn wfn 6543 βΆwf 6544 βcfv 6548 (class class class)co 7420 β cmpo 7422 βcc 11137 1c1 11140 + caddc 11142 Β· cmul 11144 β cmin 11475 -cneg 11476 abscabs 15214 NrmCVeccnv 30407 βπ£ cnsb 30412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-grpo 30316 df-gid 30317 df-ginv 30318 df-gdiv 30319 df-ablo 30368 df-vc 30382 df-nv 30415 df-va 30418 df-ba 30419 df-sm 30420 df-0v 30421 df-vs 30422 df-nmcv 30423 |
This theorem is referenced by: cnims 30516 |
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