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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 | ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 |
Ref | Expression |
---|---|
cnnvm | ⊢ − = ( −𝑣 ‘𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1 10878 | . . . . . 6 ⊢ (𝑦 ∈ ℂ → (-1 · 𝑦) = -𝑦) | |
2 | 1 | adantl 474 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (-1 · 𝑦) = -𝑦) |
3 | 2 | oveq2d 6990 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + (-1 · 𝑦)) = (𝑥 + -𝑦)) |
4 | negsub 10731 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + -𝑦) = (𝑥 − 𝑦)) | |
5 | 3, 4 | eqtr2d 2812 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) = (𝑥 + (-1 · 𝑦))) |
6 | 5 | mpoeq3ia 7048 | . 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + (-1 · 𝑦))) |
7 | subf 10684 | . . . 4 ⊢ − :(ℂ × ℂ)⟶ℂ | |
8 | ffn 6342 | . . . 4 ⊢ ( − :(ℂ × ℂ)⟶ℂ → − Fn (ℂ × ℂ)) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ − Fn (ℂ × ℂ) |
10 | fnov 7096 | . . 3 ⊢ ( − Fn (ℂ × ℂ) ↔ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦))) | |
11 | 9, 10 | mpbi 222 | . 2 ⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) |
12 | cnnvm.6 | . . . 4 ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 | |
13 | 12 | cnnv 28225 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
14 | 12 | cnnvba 28227 | . . . 4 ⊢ ℂ = (BaseSet‘𝑈) |
15 | 12 | cnnvg 28226 | . . . 4 ⊢ + = ( +𝑣 ‘𝑈) |
16 | 12 | cnnvs 28228 | . . . 4 ⊢ · = ( ·𝑠OLD ‘𝑈) |
17 | eqid 2775 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
18 | 14, 15, 16, 17 | nvmfval 28192 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ( −𝑣 ‘𝑈) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + (-1 · 𝑦)))) |
19 | 13, 18 | ax-mp 5 | . 2 ⊢ ( −𝑣 ‘𝑈) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + (-1 · 𝑦))) |
20 | 6, 11, 19 | 3eqtr4i 2809 | 1 ⊢ − = ( −𝑣 ‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1507 ∈ wcel 2048 〈cop 4445 × cxp 5402 Fn wfn 6181 ⟶wf 6182 ‘cfv 6186 (class class class)co 6974 ∈ cmpo 6976 ℂcc 10329 1c1 10332 + caddc 10334 · cmul 10336 − cmin 10666 -cneg 10667 abscabs 14448 NrmCVeccnv 28132 −𝑣 cnsb 28137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2747 ax-rep 5047 ax-sep 5058 ax-nul 5065 ax-pow 5117 ax-pr 5184 ax-un 7277 ax-cnex 10387 ax-resscn 10388 ax-1cn 10389 ax-icn 10390 ax-addcl 10391 ax-addrcl 10392 ax-mulcl 10393 ax-mulrcl 10394 ax-mulcom 10395 ax-addass 10396 ax-mulass 10397 ax-distr 10398 ax-i2m1 10399 ax-1ne0 10400 ax-1rid 10401 ax-rnegex 10402 ax-rrecex 10403 ax-cnre 10404 ax-pre-lttri 10405 ax-pre-lttrn 10406 ax-pre-ltadd 10407 ax-pre-mulgt0 10408 ax-pre-sup 10409 ax-addf 10410 ax-mulf 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-ne 2965 df-nel 3071 df-ral 3090 df-rex 3091 df-reu 3092 df-rmo 3093 df-rab 3094 df-v 3414 df-sbc 3681 df-csb 3786 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-pss 3844 df-nul 4178 df-if 4349 df-pw 4422 df-sn 4440 df-pr 4442 df-tp 4444 df-op 4446 df-uni 4711 df-iun 4792 df-br 4928 df-opab 4990 df-mpt 5007 df-tr 5029 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7498 df-2nd 7499 df-wrecs 7747 df-recs 7809 df-rdg 7847 df-er 8085 df-en 8303 df-dom 8304 df-sdom 8305 df-sup 8697 df-pnf 10472 df-mnf 10473 df-xr 10474 df-ltxr 10475 df-le 10476 df-sub 10668 df-neg 10669 df-div 11095 df-nn 11436 df-2 11500 df-3 11501 df-n0 11705 df-z 11791 df-uz 12056 df-rp 12202 df-seq 13182 df-exp 13242 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-grpo 28041 df-gid 28042 df-ginv 28043 df-gdiv 28044 df-ablo 28093 df-vc 28107 df-nv 28140 df-va 28143 df-ba 28144 df-sm 28145 df-0v 28146 df-vs 28147 df-nmcv 28148 |
This theorem is referenced by: cnims 28241 |
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