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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 2736 | . . . . . . . 8 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 2 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 3 | ldual1dim.d | . . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑊) | |
| 4 | eqid 2736 | . . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
| 5 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷)) | |
| 6 | ldual1dim.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
| 7 | 1, 2, 3, 4, 5, 6 | ldualsbase 39135 | . . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑊))) | 
| 8 | 7 | eleq2d 2826 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ (Base‘(Scalar‘𝐷)) ↔ 𝑘 ∈ (Base‘(Scalar‘𝑊)))) | 
| 9 | 8 | anbi1d 631 | . . . . 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 2736 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 12 | eqid 2736 | . . . . . . . 8 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊)) | |
| 13 | eqid 2736 | . . . . . . . 8 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
| 14 | 6 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝑊 ∈ LVec) | 
| 15 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝑘 ∈ (Base‘(Scalar‘𝑊))) | |
| 16 | ldual1dim.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
| 17 | 16 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝐺 ∈ 𝐹) | 
| 18 | 10, 11, 1, 2, 12, 3, 13, 14, 15, 17 | ldualvs 39139 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑘( ·𝑠 ‘𝐷)𝐺) = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘}))) | 
| 19 | 18 | eqeq2d 2747 | . . . . . 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 279 | . . . 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 3174 | . . 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 2807 | . 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 21106 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
| 25 | 3, 24 | lduallmod 39155 | . . . 4 ⊢ (𝑊 ∈ LVec → 𝐷 ∈ LMod) | 
| 26 | 6, 25 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) | 
| 27 | eqid 2736 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
| 28 | 10, 3, 27, 6, 16 | ldualelvbase 39129 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷)) | 
| 29 | ldual1dim.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝐷) | |
| 30 | 4, 5, 27, 13, 29 | lspsn 21001 | . . 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 39123 | . 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 2786 | 1 ⊢ (𝜑 → (𝑁‘{𝐺}) = {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 {crab 3435 ⊆ wss 3950 {csn 4625 × cxp 5682 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 Basecbs 17248 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 LModclmod 20859 LSpanclspn 20970 LVecclvec 21102 LFnlclfn 39059 LKerclk 39087 LDualcld 39125 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-cntz 19336 df-lsm 19655 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-nzr 20514 df-rlreg 20695 df-domn 20696 df-drng 20732 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lvec 21103 df-lshyp 38979 df-lfl 39060 df-lkr 39088 df-ldual 39126 | 
| This theorem is referenced by: mapdsn3 41646 | 
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