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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 11081 | . . . . . 6 ⊢ (𝑦 ∈ ℂ → (-1 · 𝑦) = -𝑦) | |
2 | 1 | adantl 484 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (-1 · 𝑦) = -𝑦) |
3 | 2 | oveq2d 7172 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + (-1 · 𝑦)) = (𝑥 + -𝑦)) |
4 | negsub 10934 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + -𝑦) = (𝑥 − 𝑦)) | |
5 | 3, 4 | eqtr2d 2857 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) = (𝑥 + (-1 · 𝑦))) |
6 | 5 | mpoeq3ia 7232 | . 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + (-1 · 𝑦))) |
7 | subf 10888 | . . . 4 ⊢ − :(ℂ × ℂ)⟶ℂ | |
8 | ffn 6514 | . . . 4 ⊢ ( − :(ℂ × ℂ)⟶ℂ → − Fn (ℂ × ℂ)) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ − Fn (ℂ × ℂ) |
10 | fnov 7282 | . . 3 ⊢ ( − Fn (ℂ × ℂ) ↔ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦))) | |
11 | 9, 10 | mpbi 232 | . 2 ⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) |
12 | cnnvm.6 | . . . 4 ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 | |
13 | 12 | cnnv 28454 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
14 | 12 | cnnvba 28456 | . . . 4 ⊢ ℂ = (BaseSet‘𝑈) |
15 | 12 | cnnvg 28455 | . . . 4 ⊢ + = ( +𝑣 ‘𝑈) |
16 | 12 | cnnvs 28457 | . . . 4 ⊢ · = ( ·𝑠OLD ‘𝑈) |
17 | eqid 2821 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
18 | 14, 15, 16, 17 | nvmfval 28421 | . . 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 2854 | 1 ⊢ − = ( −𝑣 ‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 × cxp 5553 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ℂcc 10535 1c1 10538 + caddc 10540 · cmul 10542 − cmin 10870 -cneg 10871 abscabs 14593 NrmCVeccnv 28361 −𝑣 cnsb 28366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-grpo 28270 df-gid 28271 df-ginv 28272 df-gdiv 28273 df-ablo 28322 df-vc 28336 df-nv 28369 df-va 28372 df-ba 28373 df-sm 28374 df-0v 28375 df-vs 28376 df-nmcv 28377 |
This theorem is referenced by: cnims 28470 |
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