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Mirrors > Home > MPE Home > Th. List > nvmf | Structured version Visualization version GIF version |
Description: Mapping for the vector subtraction operation. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvmf.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvmf.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
Ref | Expression |
---|---|
nvmf | ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑈 ∈ NrmCVec) | |
2 | simprl 794 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋) | |
3 | neg1cn 11124 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
4 | nvmf.1 | . . . . . . . 8 ⊢ 𝑋 = (BaseSet‘𝑈) | |
5 | eqid 2622 | . . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
6 | 4, 5 | nvscl 27481 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝑦 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝑦) ∈ 𝑋) |
7 | 3, 6 | mp3an2 1412 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝑦) ∈ 𝑋) |
8 | 7 | adantrl 752 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (-1( ·𝑠OLD ‘𝑈)𝑦) ∈ 𝑋) |
9 | eqid 2622 | . . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
10 | 4, 9 | nvgcl 27475 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ (-1( ·𝑠OLD ‘𝑈)𝑦) ∈ 𝑋) → (𝑥( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋) |
11 | 1, 2, 8, 10 | syl3anc 1326 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋) |
12 | 11 | ralrimivva 2971 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋) |
13 | eqid 2622 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑦))) | |
14 | 13 | fmpt2 7237 | . . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑦)) ∈ 𝑋 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑦))):(𝑋 × 𝑋)⟶𝑋) |
15 | 12, 14 | sylib 208 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑦))):(𝑋 × 𝑋)⟶𝑋) |
16 | nvmf.3 | . . . 4 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
17 | 4, 9, 5, 16 | nvmfval 27499 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑦)))) |
18 | 17 | feq1d 6030 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑀:(𝑋 × 𝑋)⟶𝑋 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥( +𝑣 ‘𝑈)(-1( ·𝑠OLD ‘𝑈)𝑦))):(𝑋 × 𝑋)⟶𝑋)) |
19 | 15, 18 | mpbird 247 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 × cxp 5112 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ℂcc 9934 1c1 9937 -cneg 10267 NrmCVeccnv 27439 +𝑣 cpv 27440 BaseSetcba 27441 ·𝑠OLD cns 27442 −𝑣 cnsb 27444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 |
This theorem is referenced by: nvmcl 27501 imsdval 27541 imsdf 27544 sspm 27589 hhssvsf 28130 |
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