Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochsnkr2cl | Structured version Visualization version GIF version |
Description: The 𝑋 determining functional 𝐺 belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.) |
Ref | Expression |
---|---|
dochsnkr2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochsnkr2.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochsnkr2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochsnkr2.v | ⊢ 𝑉 = (Base‘𝑈) |
dochsnkr2.z | ⊢ 0 = (0g‘𝑈) |
dochsnkr2.a | ⊢ + = (+g‘𝑈) |
dochsnkr2.t | ⊢ · = ( ·𝑠 ‘𝑈) |
dochsnkr2.l | ⊢ 𝐿 = (LKer‘𝑈) |
dochsnkr2.d | ⊢ 𝐷 = (Scalar‘𝑈) |
dochsnkr2.r | ⊢ 𝑅 = (Base‘𝐷) |
dochsnkr2.g | ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) |
dochsnkr2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochsnkr2.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
Ref | Expression |
---|---|
dochsnkr2cl | ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochsnkr2.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dochsnkr2.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | dochsnkr2.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlmod 38261 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
5 | dochsnkr2.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
6 | 5 | eldifad 3948 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
7 | dochsnkr2.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
8 | eqid 2821 | . . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
9 | 7, 8 | lspsnid 19765 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ ((LSpan‘𝑈)‘{𝑋})) |
10 | 4, 6, 9 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((LSpan‘𝑈)‘{𝑋})) |
11 | dochsnkr2.o | . . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
12 | dochsnkr2.z | . . . . . . 7 ⊢ 0 = (0g‘𝑈) | |
13 | dochsnkr2.a | . . . . . . 7 ⊢ + = (+g‘𝑈) | |
14 | dochsnkr2.t | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑈) | |
15 | dochsnkr2.l | . . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈) | |
16 | dochsnkr2.d | . . . . . . 7 ⊢ 𝐷 = (Scalar‘𝑈) | |
17 | dochsnkr2.r | . . . . . . 7 ⊢ 𝑅 = (Base‘𝐷) | |
18 | dochsnkr2.g | . . . . . . 7 ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) | |
19 | 1, 11, 2, 7, 12, 13, 14, 15, 16, 17, 18, 3, 5 | dochsnkr2 38624 | . . . . . 6 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋})) |
20 | 6 | snssd 4742 | . . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
21 | 1, 2, 11, 7, 8, 3, 20 | dochocsp 38530 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})) = ( ⊥ ‘{𝑋})) |
22 | 19, 21 | eqtr4d 2859 | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))) |
23 | 22 | fveq2d 6674 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋})))) |
24 | eqid 2821 | . . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
25 | 1, 2, 7, 8, 24 | dihlsprn 38482 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
26 | 3, 6, 25 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
27 | 1, 24, 11 | dochoc 38518 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))) = ((LSpan‘𝑈)‘{𝑋})) |
28 | 3, 26, 27 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))) = ((LSpan‘𝑈)‘{𝑋})) |
29 | 23, 28 | eqtr2d 2857 | . . 3 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑋}) = ( ⊥ ‘(𝐿‘𝐺))) |
30 | 10, 29 | eleqtrd 2915 | . 2 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺))) |
31 | eldifsni 4722 | . . 3 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
32 | 5, 31 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
33 | eldifsn 4719 | . 2 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) ↔ (𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ 𝑋 ≠ 0 )) | |
34 | 30, 32, 33 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ∖ cdif 3933 {csn 4567 ↦ cmpt 5146 ran crn 5556 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 LModclmod 19634 LSpanclspn 19743 LKerclk 36236 HLchlt 36501 LHypclh 37135 DVecHcdvh 38229 DIsoHcdih 38379 ocHcoch 38498 |
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 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 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-iin 4922 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-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 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-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 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-dvr 19433 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-lsatoms 36127 df-lshyp 36128 df-lfl 36209 df-lkr 36237 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tgrp 37894 df-tendo 37906 df-edring 37908 df-dveca 38154 df-disoa 38180 df-dvech 38230 df-dib 38290 df-dic 38324 df-dih 38380 df-doch 38499 df-djh 38546 |
This theorem is referenced by: lcfl7lem 38650 lcfrlem9 38701 |
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