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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl6lem | Structured version Visualization version GIF version |
Description: Lemma for lcfl6 40837. A functional 𝐺 (whose kernel is closed by dochsnkr 40809) is comletely determined by a vector 𝑋 in the orthocomplement in its kernel at which the functional value is 1. Note that the ∖ { 0 } in the 𝑋 hypothesis is redundant by the last hypothesis but allows easier use of other theorems. (Contributed by NM, 3-Jan-2015.) |
Ref | Expression |
---|---|
lcfl6lem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfl6lem.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfl6lem.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfl6lem.v | ⊢ 𝑉 = (Base‘𝑈) |
lcfl6lem.a | ⊢ + = (+g‘𝑈) |
lcfl6lem.t | ⊢ · = ( ·𝑠 ‘𝑈) |
lcfl6lem.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lcfl6lem.i | ⊢ 1 = (1r‘𝑆) |
lcfl6lem.r | ⊢ 𝑅 = (Base‘𝑆) |
lcfl6lem.z | ⊢ 0 = (0g‘𝑈) |
lcfl6lem.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcfl6lem.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfl6lem.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfl6lem.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lcfl6lem.x | ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) |
lcfl6lem.y | ⊢ (𝜑 → (𝐺‘𝑋) = 1 ) |
Ref | Expression |
---|---|
lcfl6lem | ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfl6lem.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
2 | lcfl6lem.s | . 2 ⊢ 𝑆 = (Scalar‘𝑈) | |
3 | lcfl6lem.r | . 2 ⊢ 𝑅 = (Base‘𝑆) | |
4 | eqid 2731 | . 2 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
5 | lcfl6lem.f | . 2 ⊢ 𝐹 = (LFnl‘𝑈) | |
6 | lcfl6lem.l | . 2 ⊢ 𝐿 = (LKer‘𝑈) | |
7 | lcfl6lem.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | lcfl6lem.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
9 | lcfl6lem.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 7, 8, 9 | dvhlvec 40446 | . 2 ⊢ (𝜑 → 𝑈 ∈ LVec) |
11 | 7, 8, 9 | dvhlmod 40447 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | lcfl6lem.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
13 | 1, 5, 6, 11, 12 | lkrssv 38432 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
14 | lcfl6lem.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
15 | 7, 8, 1, 14 | dochssv 40692 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
16 | 9, 13, 15 | syl2anc 583 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
17 | lcfl6lem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) | |
18 | 17 | eldifad 3960 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺))) |
19 | 16, 18 | sseldd 3983 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
20 | lcfl6lem.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
21 | lcfl6lem.a | . . 3 ⊢ + = (+g‘𝑈) | |
22 | lcfl6lem.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
23 | eqid 2731 | . . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) | |
24 | eldifsni 4793 | . . . . 5 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) → 𝑋 ≠ 0 ) | |
25 | 17, 24 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
26 | eldifsn 4790 | . . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
27 | 19, 25, 26 | sylanbrc 582 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
28 | 7, 14, 8, 1, 20, 21, 22, 5, 2, 3, 23, 9, 27 | dochflcl 40812 | . 2 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) ∈ 𝐹) |
29 | 7, 14, 8, 1, 20, 5, 6, 9, 12, 17 | dochsnkr 40809 | . . 3 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋})) |
30 | 7, 14, 8, 1, 20, 21, 22, 6, 2, 3, 23, 9, 27 | dochsnkr2 40810 | . . 3 ⊢ (𝜑 → (𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) = ( ⊥ ‘{𝑋})) |
31 | 29, 30 | eqtr4d 2774 | . 2 ⊢ (𝜑 → (𝐿‘𝐺) = (𝐿‘(𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))) |
32 | lcfl6lem.y | . . 3 ⊢ (𝜑 → (𝐺‘𝑋) = 1 ) | |
33 | lcfl6lem.i | . . . 4 ⊢ 1 = (1r‘𝑆) | |
34 | 7, 14, 8, 1, 21, 22, 20, 2, 3, 33, 9, 27, 23 | dochfl1 40813 | . . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))‘𝑋) = 1 ) |
35 | 32, 34 | eqtr4d 2774 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) = ((𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))‘𝑋)) |
36 | 7, 14, 8, 1, 2, 4, 20, 5, 6, 9, 12, 17 | dochfln0 40814 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) ≠ (0g‘𝑆)) |
37 | 1, 2, 3, 4, 5, 6, 10, 19, 12, 28, 31, 35, 36 | eqlkr3 38437 | 1 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∃wrex 3069 ∖ cdif 3945 ⊆ wss 3948 {csn 4628 ↦ cmpt 5231 ‘cfv 6543 ℩crio 7367 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 Scalarcsca 17207 ·𝑠 cvsca 17208 0gc0g 17392 1rcur 20082 LFnlclfn 38393 LKerclk 38421 HLchlt 38686 LHypclh 39321 DVecHcdvh 40415 ocHcoch 40684 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38289 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cntz 19229 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20947 df-lsatoms 38312 df-lshyp 38313 df-lfl 38394 df-lkr 38422 df-oposet 38512 df-ol 38514 df-oml 38515 df-covers 38602 df-ats 38603 df-atl 38634 df-cvlat 38658 df-hlat 38687 df-llines 38835 df-lplanes 38836 df-lvols 38837 df-lines 38838 df-psubsp 38840 df-pmap 38841 df-padd 39133 df-lhyp 39325 df-laut 39326 df-ldil 39441 df-ltrn 39442 df-trl 39496 df-tgrp 40080 df-tendo 40092 df-edring 40094 df-dveca 40340 df-disoa 40366 df-dvech 40416 df-dib 40476 df-dic 40510 df-dih 40566 df-doch 40685 df-djh 40732 |
This theorem is referenced by: lcfl6 40837 |
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