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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 2821 | . . . . . . . 8 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | ldual1dim.d | . . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑊) | |
4 | eqid 2821 | . . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
5 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷)) | |
6 | ldual1dim.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | 1, 2, 3, 4, 5, 6 | ldualsbase 36284 | . . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑊))) |
8 | 7 | eleq2d 2898 | . . . . . 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 2821 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
12 | eqid 2821 | . . . . . . . 8 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊)) | |
13 | eqid 2821 | . . . . . . . 8 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
14 | 6 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝑊 ∈ LVec) |
15 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝑘 ∈ (Base‘(Scalar‘𝑊))) | |
16 | ldual1dim.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
17 | 16 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → 𝐺 ∈ 𝐹) |
18 | 10, 11, 1, 2, 12, 3, 13, 14, 15, 17 | ldualvs 36288 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑘( ·𝑠 ‘𝐷)𝐺) = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘}))) |
19 | 18 | eqeq2d 2832 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊))) → (𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺) ↔ 𝑔 = (𝐺 ∘f (.r‘(Scalar‘𝑊))((Base‘𝑊) × {𝑘})))) |
20 | 19 | pm5.32da 581 | . . . . 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 281 | . . . 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 3295 | . . 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 2885 | . 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 19878 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
25 | 3, 24 | lduallmod 36304 | . . . 4 ⊢ (𝑊 ∈ LVec → 𝐷 ∈ LMod) |
26 | 6, 25 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) |
27 | eqid 2821 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
28 | 10, 3, 27, 6, 16 | ldualelvbase 36278 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷)) |
29 | ldual1dim.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝐷) | |
30 | 4, 5, 27, 13, 29 | lspsn 19774 | . . 3 ⊢ ((𝐷 ∈ LMod ∧ 𝐺 ∈ (Base‘𝐷)) → (𝑁‘{𝐺}) = {𝑔 ∣ ∃𝑘 ∈ (Base‘(Scalar‘𝐷))𝑔 = (𝑘( ·𝑠 ‘𝐷)𝐺)}) |
31 | 26, 28, 30 | syl2anc 586 | . 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 36272 | . 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 2866 | 1 ⊢ (𝜑 → (𝑁‘{𝐺}) = {𝑔 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑔)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2799 ∃wrex 3139 {crab 3142 ⊆ wss 3936 {csn 4567 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 Basecbs 16483 .rcmulr 16566 Scalarcsca 16568 ·𝑠 cvsca 16569 LModclmod 19634 LSpanclspn 19743 LVecclvec 19874 LFnlclfn 36208 LKerclk 36236 LDualcld 36274 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-lshyp 36128 df-lfl 36209 df-lkr 36237 df-ldual 36275 |
This theorem is referenced by: mapdsn3 38794 |
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