Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochkrshp2 | Structured version Visualization version GIF version |
Description: Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.) |
Ref | Expression |
---|---|
dochkrshp2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochkrshp2.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochkrshp2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochkrshp2.v | ⊢ 𝑉 = (Base‘𝑈) |
dochkrshp2.y | ⊢ 𝑌 = (LSHyp‘𝑈) |
dochkrshp2.f | ⊢ 𝐹 = (LFnl‘𝑈) |
dochkrshp2.l | ⊢ 𝐿 = (LKer‘𝑈) |
dochkrshp2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochkrshp2.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
dochkrshp2 | ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochkrshp2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dochkrshp2.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | dochkrshp2.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | dochkrshp2.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | dochkrshp2.y | . . 3 ⊢ 𝑌 = (LSHyp‘𝑈) | |
6 | dochkrshp2.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
7 | dochkrshp2.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
8 | dochkrshp2.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | dochkrshp2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | dochkrshp 39055 | . 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)) |
11 | 1, 2, 3, 6, 5, 7, 8, 9 | dochlkr 39054 | . 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))) |
12 | 10, 11 | bitrd 282 | 1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ‘cfv 6349 Basecbs 16598 LSHypclsh 36644 LFnlclfn 36726 LKerclk 36754 HLchlt 37019 LHypclh 37653 DVecHcdvh 38747 ocHcoch 39016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-riotaBAD 36622 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-tpos 7933 df-undef 7980 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-map 8451 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-n0 11989 df-z 12075 df-uz 12337 df-fz 12994 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-sca 16696 df-vsca 16697 df-0g 16830 df-proset 17666 df-poset 17684 df-plt 17696 df-lub 17712 df-glb 17713 df-join 17714 df-meet 17715 df-p0 17777 df-p1 17778 df-lat 17784 df-clat 17846 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-submnd 18085 df-grp 18234 df-minusg 18235 df-sbg 18236 df-subg 18406 df-cntz 18577 df-lsm 18891 df-cmn 19038 df-abl 19039 df-mgp 19371 df-ur 19383 df-ring 19430 df-oppr 19507 df-dvdsr 19525 df-unit 19526 df-invr 19556 df-dvr 19567 df-drng 19635 df-lmod 19767 df-lss 19835 df-lsp 19875 df-lvec 20006 df-lsatoms 36645 df-lshyp 36646 df-lfl 36727 df-lkr 36755 df-oposet 36845 df-ol 36847 df-oml 36848 df-covers 36935 df-ats 36936 df-atl 36967 df-cvlat 36991 df-hlat 37020 df-llines 37167 df-lplanes 37168 df-lvols 37169 df-lines 37170 df-psubsp 37172 df-pmap 37173 df-padd 37465 df-lhyp 37657 df-laut 37658 df-ldil 37773 df-ltrn 37774 df-trl 37828 df-tendo 38424 df-edring 38426 df-disoa 38698 df-dvech 38748 df-dib 38808 df-dic 38842 df-dih 38898 df-doch 39017 |
This theorem is referenced by: dochkrshp3 39057 |
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