Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11083 | . . . . . 6 ⊢ (𝑦 ∈ ℂ → (-1 · 𝑦) = -𝑦) | |
2 | 1 | adantl 484 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (-1 · 𝑦) = -𝑦) |
3 | 2 | oveq2d 7174 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + (-1 · 𝑦)) = (𝑥 + -𝑦)) |
4 | negsub 10936 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + -𝑦) = (𝑥 − 𝑦)) | |
5 | 3, 4 | eqtr2d 2859 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) = (𝑥 + (-1 · 𝑦))) |
6 | 5 | mpoeq3ia 7234 | . 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + (-1 · 𝑦))) |
7 | subf 10890 | . . . 4 ⊢ − :(ℂ × ℂ)⟶ℂ | |
8 | ffn 6516 | . . . 4 ⊢ ( − :(ℂ × ℂ)⟶ℂ → − Fn (ℂ × ℂ)) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ − Fn (ℂ × ℂ) |
10 | fnov 7284 | . . 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 28456 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
14 | 12 | cnnvba 28458 | . . . 4 ⊢ ℂ = (BaseSet‘𝑈) |
15 | 12 | cnnvg 28457 | . . . 4 ⊢ + = ( +𝑣 ‘𝑈) |
16 | 12 | cnnvs 28459 | . . . 4 ⊢ · = ( ·𝑠OLD ‘𝑈) |
17 | eqid 2823 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
18 | 14, 15, 16, 17 | nvmfval 28423 | . . 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 2856 | 1 ⊢ − = ( −𝑣 ‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4575 × cxp 5555 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ℂcc 10537 1c1 10540 + caddc 10542 · cmul 10544 − cmin 10872 -cneg 10873 abscabs 14595 NrmCVeccnv 28363 −𝑣 cnsb 28368 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 |
This theorem is referenced by: cnims 28472 |
Copyright terms: Public domain | W3C validator |