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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 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvmval.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvmval.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nvmval.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
Ref | Expression |
---|---|
nvmval2 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = ((-1𝑆𝐵)𝐺𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvmval.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | nvmval.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
3 | nvmval.4 | . . 3 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
4 | nvmval.3 | . . 3 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
5 | 1, 2, 3, 4 | nvmval 30163 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵))) |
6 | neg1cn 12331 | . . . . 5 ⊢ -1 ∈ ℂ | |
7 | 1, 3 | nvscl 30147 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) ∈ 𝑋) |
8 | 6, 7 | mp3an2 1448 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) ∈ 𝑋) |
9 | 8 | 3adant2 1130 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) ∈ 𝑋) |
10 | 1, 2 | nvcom 30142 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐵)) = ((-1𝑆𝐵)𝐺𝐴)) |
11 | 9, 10 | syld3an3 1408 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐵)) = ((-1𝑆𝐵)𝐺𝐴)) |
12 | 5, 11 | eqtrd 2771 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = ((-1𝑆𝐵)𝐺𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 ℂcc 11112 1c1 11115 -cneg 11450 NrmCVeccnv 30105 +𝑣 cpv 30106 BaseSetcba 30107 ·𝑠OLD cns 30108 −𝑣 cnsb 30110 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-ltxr 11258 df-sub 11451 df-neg 11452 df-grpo 30014 df-gid 30015 df-ginv 30016 df-gdiv 30017 df-ablo 30066 df-vc 30080 df-nv 30113 df-va 30116 df-ba 30117 df-sm 30118 df-0v 30119 df-vs 30120 df-nmcv 30121 |
This theorem is referenced by: lnosub 30280 |
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