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Mathbox for Norm Megill |
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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 2725 | . . . . . . . 8 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2725 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | ldual1dim.d | . . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑊) | |
4 | eqid 2725 | . . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
5 | eqid 2725 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷)) | |
6 | ldual1dim.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | 1, 2, 3, 4, 5, 6 | ldualsbase 38735 | . . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑊))) |
8 | 7 | eleq2d 2811 | . . . . . 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 2725 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
12 | eqid 2725 | . . . . . . . 8 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊)) | |
13 | eqid 2725 | . . . . . . . 8 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
14 | 6 | adantr 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝑊 ∈ LVec) |
15 | simpr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝑘 ∈ (Base‘(Scalar‘𝑊))) | |
16 | ldual1dim.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
17 | 16 | adantr 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝐺 ∈ 𝐹) |
18 | 10, 11, 1, 2, 12, 3, 13, 14, 15, 17 | ldualvs 38739 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑘( ·𝑠 ‘𝐷)𝐺) = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘}))) |
19 | 18 | eqeq2d 2736 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺) ↔ 𝑔 = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘})))) |
20 | 19 | pm5.32da 577 | . . . . 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 278 | . . . 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 3164 | . . 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 2794 | . 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 21003 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
25 | 3, 24 | lduallmod 38755 | . . . 4 ⊢ (𝑊 ∈ LVec → 𝐷 ∈ LMod) |
26 | 6, 25 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) |
27 | eqid 2725 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
28 | 10, 3, 27, 6, 16 | ldualelvbase 38729 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷)) |
29 | ldual1dim.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝐷) | |
30 | 4, 5, 27, 13, 29 | lspsn 20898 | . . 3 ⊢ ((𝐷 ∈ LMod ∧ 𝐺 ∈ (Base‘𝐷)) → (𝑁‘{𝐺}) = {𝑔 ∣ ∃𝑘 ∈ (Base‘(Scalar‘𝐷))𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺)}) |
31 | 26, 28, 30 | syl2anc 582 | . 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 38723 | . 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 2775 | 1 ⊢ (𝜑 → (𝑁‘{𝐺}) = {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {cab 2702 ∃wrex 3059 {crab 3418 ⊆ wss 3944 {csn 4630 × cxp 5676 ‘cfv 6549 (class class class)co 7419 ∘f cof 7683 Basecbs 17183 .rcmulr 17237 Scalarcsca 17239 ·𝑠 cvsca 17240 LModclmod 20755 LSpanclspn 20867 LVecclvec 20999 LFnlclfn 38659 LKerclk 38687 LDualcld 38725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cntz 19280 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lvec 21000 df-lshyp 38579 df-lfl 38660 df-lkr 38688 df-ldual 38726 |
This theorem is referenced by: mapdsn3 41246 |
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