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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 40688 | . . . 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 40437 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
11 | lcfl9a.g | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
12 | 4, 8, 9, 10, 11 | lkrssv 38422 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
13 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝐿‘𝐺) ⊆ 𝑉) |
14 | sneq 4630 | . . . . . . . . 9 ⊢ (𝑋 = (0g‘𝑈) → {𝑋} = {(0g‘𝑈)}) | |
15 | 14 | fveq2d 6885 | . . . . . . . 8 ⊢ (𝑋 = (0g‘𝑈) → ( ⊥ ‘{𝑋}) = ( ⊥ ‘{(0g‘𝑈)})) |
16 | eqid 2724 | . . . . . . . . . 10 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
17 | 1, 2, 3, 4, 16 | doch0 40685 | . . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0g‘𝑈)}) = 𝑉) |
18 | 5, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(0g‘𝑈)}) = 𝑉) |
19 | 15, 18 | sylan9eqr 2786 | . . . . . . 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 | eqsstrrd 4013 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → 𝑉 ⊆ (𝐿‘𝐺)) |
23 | 13, 22 | eqssd 3991 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝐿‘𝐺) = 𝑉) |
24 | 23 | fveq2d 6885 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉)) |
25 | 24 | fveq2d 6885 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉))) |
26 | 7, 25, 23 | 3eqtr4d 2774 | . 2 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
27 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉) |
28 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) = 𝑉) | |
29 | 28 | fveq2d 6885 | . . . 4 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉)) |
30 | 29 | fveq2d 6885 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉))) |
31 | 27, 30, 28 | 3eqtr4d 2774 | . 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
32 | lcfl9a.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
33 | 32 | snssd 4804 | . . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
34 | eqid 2724 | . . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
35 | 1, 34, 2, 4, 3 | dochcl 40680 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
36 | 5, 33, 35 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
37 | 1, 34, 3 | dochoc 40694 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
38 | 5, 36, 37 | syl2anc 583 | . . . 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 2724 | . . . . . . 7 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈) | |
42 | 1, 2, 5 | dvhlvec 40436 | . . . . . . . 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 768 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ≠ (0g‘𝑈)) | |
47 | eldifsn 4782 | . . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ {(0g‘𝑈)}) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ (0g‘𝑈))) | |
48 | 45, 46, 47 | sylanbrc 582 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
49 | 1, 3, 2, 4, 16, 41, 44, 48 | dochsnshp 40780 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) ∈ (LSHyp‘𝑈)) |
50 | simprr 770 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ≠ 𝑉) | |
51 | 4, 41, 8, 9, 42, 11 | lkrshp4 38434 | . . . . . . . . 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 231 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) |
54 | 41, 43, 49, 53 | lshpcmp 38314 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺) ↔ ( ⊥ ‘{𝑋}) = (𝐿‘𝐺))) |
55 | 40, 54 | mpbid 231 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) = (𝐿‘𝐺)) |
56 | 55 | fveq2d 6885 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘𝐺))) |
57 | 56 | fveq2d 6885 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
58 | 39, 57, 55 | 3eqtr3d 2772 | . 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
59 | 26, 31, 58 | pm2.61da2ne 3022 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∖ cdif 3937 ⊆ wss 3940 {csn 4620 ran crn 5667 ‘cfv 6533 Basecbs 17140 0gc0g 17381 LVecclvec 20935 LSHypclsh 38301 LFnlclfn 38383 LKerclk 38411 HLchlt 38676 LHypclh 39311 DVecHcdvh 40405 DIsoHcdih 40555 ocHcoch 40674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-riotaBAD 38279 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-undef 8253 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-proset 18247 df-poset 18265 df-plt 18282 df-lub 18298 df-glb 18299 df-join 18300 df-meet 18301 df-p0 18377 df-p1 18378 df-lat 18384 df-clat 18451 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-submnd 18701 df-grp 18853 df-minusg 18854 df-sbg 18855 df-subg 19035 df-cntz 19218 df-lsm 19541 df-cmn 19687 df-abl 19688 df-mgp 20025 df-rng 20043 df-ur 20072 df-ring 20125 df-oppr 20221 df-dvdsr 20244 df-unit 20245 df-invr 20275 df-dvr 20288 df-drng 20574 df-lmod 20693 df-lss 20764 df-lsp 20804 df-lvec 20936 df-lsatoms 38302 df-lshyp 38303 df-lfl 38384 df-lkr 38412 df-oposet 38502 df-ol 38504 df-oml 38505 df-covers 38592 df-ats 38593 df-atl 38624 df-cvlat 38648 df-hlat 38677 df-llines 38825 df-lplanes 38826 df-lvols 38827 df-lines 38828 df-psubsp 38830 df-pmap 38831 df-padd 39123 df-lhyp 39315 df-laut 39316 df-ldil 39431 df-ltrn 39432 df-trl 39486 df-tgrp 40070 df-tendo 40082 df-edring 40084 df-dveca 40330 df-disoa 40356 df-dvech 40406 df-dib 40466 df-dic 40500 df-dih 40556 df-doch 40675 df-djh 40722 |
This theorem is referenced by: mapdsn 40968 |
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