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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldual1dim | Structured version Visualization version GIF version |
Description: Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.) |
Ref | Expression |
---|---|
ldual1dim.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldual1dim.l | ⊢ 𝐿 = (LKer‘𝑊) |
ldual1dim.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldual1dim.n | ⊢ 𝑁 = (LSpan‘𝐷) |
ldual1dim.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
ldual1dim.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
ldual1dim | ⊢ (𝜑 → (𝑁‘{𝐺}) = {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . . . . 8 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2818 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | ldual1dim.d | . . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑊) | |
4 | eqid 2818 | . . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
5 | eqid 2818 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷)) | |
6 | ldual1dim.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | 1, 2, 3, 4, 5, 6 | ldualsbase 36149 | . . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑊))) |
8 | 7 | eleq2d 2895 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ (Base‘(Scalar‘𝐷)) ↔ 𝑘 ∈ (Base‘(Scalar‘𝑊)))) |
9 | 8 | anbi1d 629 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺)) ↔ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺)))) |
10 | ldual1dim.f | . . . . . . . 8 ⊢ 𝐹 = (LFnl‘𝑊) | |
11 | eqid 2818 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
12 | eqid 2818 | . . . . . . . 8 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊)) | |
13 | eqid 2818 | . . . . . . . 8 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
14 | 6 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝑊 ∈ LVec) |
15 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝑘 ∈ (Base‘(Scalar‘𝑊))) | |
16 | ldual1dim.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
17 | 16 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝐺 ∈ 𝐹) |
18 | 10, 11, 1, 2, 12, 3, 13, 14, 15, 17 | ldualvs 36153 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑘( ·𝑠 ‘𝐷)𝐺) = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘}))) |
19 | 18 | eqeq2d 2829 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺) ↔ 𝑔 = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘})))) |
20 | 19 | pm5.32da 579 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺)) ↔ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑔 = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘}))))) |
21 | 9, 20 | bitrd 280 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺)) ↔ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑔 = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘}))))) |
22 | 21 | rexbidv2 3292 | . . 3 ⊢ (𝜑 → (∃𝑘 ∈ (Base‘(Scalar‘𝐷))𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺) ↔ ∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑔 = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘})))) |
23 | 22 | abbidv 2882 | . 2 ⊢ (𝜑 → {𝑔 ∣ ∃𝑘 ∈ (Base‘(Scalar‘𝐷))𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺)} = {𝑔 ∣ ∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑔 = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘}))}) |
24 | lveclmod 19807 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
25 | 3, 24 | lduallmod 36169 | . . . 4 ⊢ (𝑊 ∈ LVec → 𝐷 ∈ LMod) |
26 | 6, 25 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) |
27 | eqid 2818 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
28 | 10, 3, 27, 6, 16 | ldualelvbase 36143 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷)) |
29 | ldual1dim.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝐷) | |
30 | 4, 5, 27, 13, 29 | lspsn 19703 | . . 3 ⊢ ((𝐷 ∈ LMod ∧ 𝐺 ∈ (Base‘𝐷)) → (𝑁‘{𝐺}) = {𝑔 ∣ ∃𝑘 ∈ (Base‘(Scalar‘𝐷))𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺)}) |
31 | 26, 28, 30 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑁‘{𝐺}) = {𝑔 ∣ ∃𝑘 ∈ (Base‘(Scalar‘𝐷))𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺)}) |
32 | ldual1dim.l | . . 3 ⊢ 𝐿 = (LKer‘𝑊) | |
33 | 11, 1, 10, 32, 2, 12, 6, 16 | lfl1dim 36137 | . 2 ⊢ (𝜑 → {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)} = {𝑔 ∣ ∃𝑘 ∈ (Base‘(Scalar‘𝑊))𝑔 = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘}))}) |
34 | 23, 31, 33 | 3eqtr4d 2863 | 1 ⊢ (𝜑 → (𝑁‘{𝐺}) = {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 ∃wrex 3136 {crab 3139 ⊆ wss 3933 {csn 4557 × cxp 5546 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 Basecbs 16471 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 LModclmod 19563 LSpanclspn 19672 LVecclvec 19803 LFnlclfn 36073 LKerclk 36101 LDualcld 36139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lshyp 35993 df-lfl 36074 df-lkr 36102 df-ldual 36140 |
This theorem is referenced by: mapdsn3 38659 |
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