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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrsc | Structured version Visualization version GIF version | ||
| Description: The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 9-Oct-2014.) | 
| Ref | Expression | 
|---|---|
| lkrsc.v | ⊢ 𝑉 = (Base‘𝑊) | 
| lkrsc.d | ⊢ 𝐷 = (Scalar‘𝑊) | 
| lkrsc.k | ⊢ 𝐾 = (Base‘𝐷) | 
| lkrsc.t | ⊢ · = (.r‘𝐷) | 
| lkrsc.f | ⊢ 𝐹 = (LFnl‘𝑊) | 
| lkrsc.l | ⊢ 𝐿 = (LKer‘𝑊) | 
| lkrsc.w | ⊢ (𝜑 → 𝑊 ∈ LVec) | 
| lkrsc.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) | 
| lkrsc.r | ⊢ (𝜑 → 𝑅 ∈ 𝐾) | 
| lkrsc.o | ⊢ 0 = (0g‘𝐷) | 
| lkrsc.e | ⊢ (𝜑 → 𝑅 ≠ 0 ) | 
| Ref | Expression | 
|---|---|
| lkrsc | ⊢ (𝜑 → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lkrsc.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊) | |
| 2 | 1 | fvexi 6920 | . . . . . . . 8 ⊢ 𝑉 ∈ V | 
| 3 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ V) | 
| 4 | lkrsc.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝐾) | |
| 5 | lkrsc.w | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
| 6 | lkrsc.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
| 7 | lkrsc.d | . . . . . . . . . 10 ⊢ 𝐷 = (Scalar‘𝑊) | |
| 8 | lkrsc.k | . . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝐷) | |
| 9 | lkrsc.f | . . . . . . . . . 10 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 10 | 7, 8, 1, 9 | lflf 39064 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) | 
| 11 | 5, 6, 10 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) | 
| 12 | 11 | ffnd 6737 | . . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝑉) | 
| 13 | eqidd 2738 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) = (𝐺‘𝑣)) | |
| 14 | 3, 4, 12, 13 | ofc2 7726 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = ((𝐺‘𝑣) · 𝑅)) | 
| 15 | 14 | eqeq1d 2739 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ↔ ((𝐺‘𝑣) · 𝑅) = 0 )) | 
| 16 | lkrsc.o | . . . . . 6 ⊢ 0 = (0g‘𝐷) | |
| 17 | lkrsc.t | . . . . . 6 ⊢ · = (.r‘𝐷) | |
| 18 | 7 | lvecdrng 21104 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝐷 ∈ DivRing) | 
| 19 | 5, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ DivRing) | 
| 20 | 19 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐷 ∈ DivRing) | 
| 21 | 5 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ LVec) | 
| 22 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐺 ∈ 𝐹) | 
| 23 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
| 24 | 7, 8, 1, 9 | lflcl 39065 | . . . . . . 7 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾) | 
| 25 | 21, 22, 23, 24 | syl3anc 1373 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾) | 
| 26 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑅 ∈ 𝐾) | 
| 27 | lkrsc.e | . . . . . . 7 ⊢ (𝜑 → 𝑅 ≠ 0 ) | |
| 28 | 27 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑅 ≠ 0 ) | 
| 29 | 8, 16, 17, 20, 25, 26, 28 | drngmuleq0 20763 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺‘𝑣) · 𝑅) = 0 ↔ (𝐺‘𝑣) = 0 )) | 
| 30 | 15, 29 | bitrd 279 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ↔ (𝐺‘𝑣) = 0 )) | 
| 31 | 30 | pm5.32da 579 | . . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ∧ ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) | 
| 32 | lveclmod 21105 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
| 33 | 5, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | 
| 34 | 1, 7, 8, 17, 9, 33, 6, 4 | lflvscl 39078 | . . . 4 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {𝑅})) ∈ 𝐹) | 
| 35 | lkrsc.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑊) | |
| 36 | 1, 7, 16, 9, 35 | ellkr 39090 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∘f · (𝑉 × {𝑅})) ∈ 𝐹) → (𝑣 ∈ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) ↔ (𝑣 ∈ 𝑉 ∧ ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ))) | 
| 37 | 5, 34, 36 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) ↔ (𝑣 ∈ 𝑉 ∧ ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ))) | 
| 38 | 1, 7, 16, 9, 35 | ellkr 39090 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) | 
| 39 | 5, 6, 38 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) | 
| 40 | 31, 37, 39 | 3bitr4d 311 | . 2 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) ↔ 𝑣 ∈ (𝐿‘𝐺))) | 
| 41 | 40 | eqrdv 2735 | 1 ⊢ (𝜑 → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 {csn 4626 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 Basecbs 17247 .rcmulr 17298 Scalarcsca 17300 0gc0g 17484 DivRingcdr 20729 LModclmod 20858 LVecclvec 21101 LFnlclfn 39058 LKerclk 39086 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-nzr 20513 df-rlreg 20694 df-domn 20695 df-drng 20731 df-lmod 20860 df-lvec 21102 df-lfl 39059 df-lkr 39087 | 
| This theorem is referenced by: lkrscss 39099 ldualkrsc 39168 | 
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