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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl9a | Structured version Visualization version GIF version |
Description: Property implying that a functional has a closed kernel. (Contributed by NM, 16-Feb-2015.) |
Ref | Expression |
---|---|
lcfl9a.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfl9a.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfl9a.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfl9a.v | ⊢ 𝑉 = (Base‘𝑈) |
lcfl9a.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcfl9a.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfl9a.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfl9a.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lcfl9a.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lcfl9a.s | ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺)) |
Ref | Expression |
---|---|
lcfl9a | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfl9a.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcfl9a.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | lcfl9a.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | lcfl9a.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
5 | lcfl9a.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 1, 2, 3, 4, 5 | dochoc1 35668 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉) |
8 | lcfl9a.f | . . . . . . . 8 ⊢ 𝐹 = (LFnl‘𝑈) | |
9 | lcfl9a.l | . . . . . . . 8 ⊢ 𝐿 = (LKer‘𝑈) | |
10 | 1, 2, 5 | dvhlmod 35417 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
11 | lcfl9a.g | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
12 | 4, 8, 9, 10, 11 | lkrssv 33401 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
13 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝐿‘𝐺) ⊆ 𝑉) |
14 | sneq 4135 | . . . . . . . . 9 ⊢ (𝑋 = (0g‘𝑈) → {𝑋} = {(0g‘𝑈)}) | |
15 | 14 | fveq2d 6107 | . . . . . . . 8 ⊢ (𝑋 = (0g‘𝑈) → ( ⊥ ‘{𝑋}) = ( ⊥ ‘{(0g‘𝑈)})) |
16 | eqid 2610 | . . . . . . . . . 10 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
17 | 1, 2, 3, 4, 16 | doch0 35665 | . . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0g‘𝑈)}) = 𝑉) |
18 | 5, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(0g‘𝑈)}) = 𝑉) |
19 | 15, 18 | sylan9eqr 2666 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → ( ⊥ ‘{𝑋}) = 𝑉) |
20 | lcfl9a.s | . . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺)) | |
21 | 20 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺)) |
22 | 19, 21 | eqsstr3d 3603 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → 𝑉 ⊆ (𝐿‘𝐺)) |
23 | 13, 22 | eqssd 3585 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝐿‘𝐺) = 𝑉) |
24 | 23 | fveq2d 6107 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉)) |
25 | 24 | fveq2d 6107 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉))) |
26 | 7, 25, 23 | 3eqtr4d 2654 | . 2 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
27 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉) |
28 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) = 𝑉) | |
29 | 28 | fveq2d 6107 | . . . 4 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉)) |
30 | 29 | fveq2d 6107 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉))) |
31 | 27, 30, 28 | 3eqtr4d 2654 | . 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
32 | lcfl9a.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
33 | 32 | snssd 4281 | . . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
34 | eqid 2610 | . . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
35 | 1, 34, 2, 4, 3 | dochcl 35660 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
36 | 5, 33, 35 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
37 | 1, 34, 3 | dochoc 35674 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
38 | 5, 36, 37 | syl2anc 691 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
39 | 38 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
40 | 20 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺)) |
41 | eqid 2610 | . . . . . . 7 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈) | |
42 | 1, 2, 5 | dvhlvec 35416 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec) |
43 | 42 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑈 ∈ LVec) |
44 | 5 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
45 | 32 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ∈ 𝑉) |
46 | simprl 790 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ≠ (0g‘𝑈)) | |
47 | eldifsn 4260 | . . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ {(0g‘𝑈)}) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ (0g‘𝑈))) | |
48 | 45, 46, 47 | sylanbrc 695 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
49 | 1, 3, 2, 4, 16, 41, 44, 48 | dochsnshp 35760 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) ∈ (LSHyp‘𝑈)) |
50 | simprr 792 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ≠ 𝑉) | |
51 | 4, 41, 8, 9, 42, 11 | lkrshp4 33413 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ (𝐿‘𝐺) ∈ (LSHyp‘𝑈))) |
52 | 51 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ((𝐿‘𝐺) ≠ 𝑉 ↔ (𝐿‘𝐺) ∈ (LSHyp‘𝑈))) |
53 | 50, 52 | mpbid 221 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) |
54 | 41, 43, 49, 53 | lshpcmp 33293 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺) ↔ ( ⊥ ‘{𝑋}) = (𝐿‘𝐺))) |
55 | 40, 54 | mpbid 221 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) = (𝐿‘𝐺)) |
56 | 55 | fveq2d 6107 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘𝐺))) |
57 | 56 | fveq2d 6107 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
58 | 39, 57, 55 | 3eqtr3d 2652 | . 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
59 | 26, 31, 58 | pm2.61da2ne 2870 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 ran crn 5039 ‘cfv 5804 Basecbs 15695 0gc0g 15923 LVecclvec 18923 LSHypclsh 33280 LFnlclfn 33362 LKerclk 33390 HLchlt 33655 LHypclh 34288 DVecHcdvh 35385 DIsoHcdih 35535 ocHcoch 35654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-lshyp 33282 df-lfl 33363 df-lkr 33391 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tgrp 35049 df-tendo 35061 df-edring 35063 df-dveca 35309 df-disoa 35336 df-dvech 35386 df-dib 35446 df-dic 35480 df-dih 35536 df-doch 35655 df-djh 35702 |
This theorem is referenced by: mapdsn 35948 |
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