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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqlkr4 | Structured version Visualization version GIF version |
Description: Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 4-Feb-2015.) |
Ref | Expression |
---|---|
eqlkr4.s | ⊢ 𝑆 = (Scalar‘𝑊) |
eqlkr4.r | ⊢ 𝑅 = (Base‘𝑆) |
eqlkr4.f | ⊢ 𝐹 = (LFnl‘𝑊) |
eqlkr4.k | ⊢ 𝐾 = (LKer‘𝑊) |
eqlkr4.d | ⊢ 𝐷 = (LDual‘𝑊) |
eqlkr4.t | ⊢ · = ( ·𝑠 ‘𝐷) |
eqlkr4.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
eqlkr4.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
eqlkr4.h | ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
eqlkr4.e | ⊢ (𝜑 → (𝐾‘𝐺) = (𝐾‘𝐻)) |
Ref | Expression |
---|---|
eqlkr4 | ⊢ (𝜑 → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqlkr4.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | eqlkr4.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
3 | eqlkr4.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
4 | eqlkr4.e | . . 3 ⊢ (𝜑 → (𝐾‘𝐺) = (𝐾‘𝐻)) | |
5 | eqlkr4.s | . . . 4 ⊢ 𝑆 = (Scalar‘𝑊) | |
6 | eqlkr4.r | . . . 4 ⊢ 𝑅 = (Base‘𝑆) | |
7 | eqid 2821 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
8 | eqid 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
9 | eqlkr4.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
10 | eqlkr4.k | . . . 4 ⊢ 𝐾 = (LKer‘𝑊) | |
11 | 5, 6, 7, 8, 9, 10 | eqlkr2 36251 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝐺 ∘f (.r‘𝑆)((Base‘𝑊) × {𝑟}))) |
12 | 1, 2, 3, 4, 11 | syl121anc 1371 | . 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑅 𝐻 = (𝐺 ∘f (.r‘𝑆)((Base‘𝑊) × {𝑟}))) |
13 | eqlkr4.d | . . . . 5 ⊢ 𝐷 = (LDual‘𝑊) | |
14 | eqlkr4.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐷) | |
15 | 1 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑊 ∈ LVec) |
16 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ 𝑅) | |
17 | 2 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐺 ∈ 𝐹) |
18 | 9, 8, 5, 6, 7, 13, 14, 15, 16, 17 | ldualvs 36288 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝑟 · 𝐺) = (𝐺 ∘f (.r‘𝑆)((Base‘𝑊) × {𝑟}))) |
19 | 18 | eqeq2d 2832 | . . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐻 = (𝑟 · 𝐺) ↔ 𝐻 = (𝐺 ∘f (.r‘𝑆)((Base‘𝑊) × {𝑟})))) |
20 | 19 | rexbidva 3296 | . 2 ⊢ (𝜑 → (∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺) ↔ ∃𝑟 ∈ 𝑅 𝐻 = (𝐺 ∘f (.r‘𝑆)((Base‘𝑊) × {𝑟})))) |
21 | 12, 20 | mpbird 259 | 1 ⊢ (𝜑 → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 {csn 4567 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 Basecbs 16483 .rcmulr 16566 Scalarcsca 16568 ·𝑠 cvsca 16569 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-grp 18106 df-minusg 18107 df-sbg 18108 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-lvec 19875 df-lfl 36209 df-lkr 36237 df-ldual 36275 |
This theorem is referenced by: lkrss2N 36320 lcfrlem16 38709 mapdrvallem2 38796 |
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