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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2 | Structured version Visualization version GIF version |
Description: The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 39530 through lclkrlem2y 39554 are used for the proof. Here we express lclkrlem2y 39554 in terms of membership in the set 𝐶 of functionals with closed kernels. (Contributed by NM, 18-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lclkrlem2.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lclkrlem2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lclkrlem2.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lclkrlem2.l | ⊢ 𝐿 = (LKer‘𝑈) |
lclkrlem2.d | ⊢ 𝐷 = (LDual‘𝑈) |
lclkrlem2.p | ⊢ + = (+g‘𝐷) |
lclkrlem2.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lclkrlem2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lclkrlem2.e | ⊢ (𝜑 → 𝐸 ∈ 𝐶) |
lclkrlem2.g | ⊢ (𝜑 → 𝐺 ∈ 𝐶) |
Ref | Expression |
---|---|
lclkrlem2 | ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrlem2.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
2 | lclkrlem2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | lclkrlem2.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | lclkrlem2.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | lclkrlem2.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
6 | lclkrlem2.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
7 | lclkrlem2.p | . . 3 ⊢ + = (+g‘𝐷) | |
8 | lclkrlem2.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | lclkrlem2.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐶) | |
10 | lclkrlem2.c | . . . . . 6 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
11 | 10 | lcfl1lem 39514 | . . . . 5 ⊢ (𝐸 ∈ 𝐶 ↔ (𝐸 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸))) |
12 | 11 | simplbi 498 | . . . 4 ⊢ (𝐸 ∈ 𝐶 → 𝐸 ∈ 𝐹) |
13 | 9, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
14 | lclkrlem2.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐶) | |
15 | 10 | lcfl1lem 39514 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
16 | 15 | simplbi 498 | . . . 4 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐹) |
17 | 14, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
18 | 10, 13 | lcfl1 39515 | . . . 4 ⊢ (𝜑 → (𝐸 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸))) |
19 | 9, 18 | mpbid 231 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸)) |
20 | 10, 17 | lcfl1 39515 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
21 | 14, 20 | mpbid 231 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 13, 17, 19, 21 | lclkrlem2y 39554 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
23 | 2, 4, 8 | dvhlmod 39133 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
24 | 5, 6, 7, 23, 13, 17 | ldualvaddcl 37153 | . . 3 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹) |
25 | 10, 24 | lcfl1 39515 | . 2 ⊢ (𝜑 → ((𝐸 + 𝐺) ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))) |
26 | 22, 25 | mpbird 256 | 1 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 {crab 3070 ‘cfv 6432 (class class class)co 7272 +gcplusg 16973 LFnlclfn 37080 LKerclk 37108 LDualcld 37146 HLchlt 37373 LHypclh 38007 DVecHcdvh 39101 ocHcoch 39370 |
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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-riotaBAD 36976 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-of 7528 df-om 7708 df-1st 7825 df-2nd 7826 df-tpos 8034 df-undef 8081 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-er 8490 df-map 8609 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-n0 12245 df-z 12331 df-uz 12594 df-fz 13251 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-sca 16989 df-vsca 16990 df-0g 17163 df-mre 17306 df-mrc 17307 df-acs 17309 df-proset 18024 df-poset 18042 df-plt 18059 df-lub 18075 df-glb 18076 df-join 18077 df-meet 18078 df-p0 18154 df-p1 18155 df-lat 18161 df-clat 18228 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 df-grp 18591 df-minusg 18592 df-sbg 18593 df-subg 18763 df-cntz 18934 df-oppg 18961 df-lsm 19252 df-cmn 19399 df-abl 19400 df-mgp 19732 df-ur 19749 df-ring 19796 df-oppr 19873 df-dvdsr 19894 df-unit 19895 df-invr 19925 df-dvr 19936 df-drng 20004 df-lmod 20136 df-lss 20205 df-lsp 20245 df-lvec 20376 df-lsatoms 36999 df-lshyp 37000 df-lcv 37042 df-lfl 37081 df-lkr 37109 df-ldual 37147 df-oposet 37199 df-ol 37201 df-oml 37202 df-covers 37289 df-ats 37290 df-atl 37321 df-cvlat 37345 df-hlat 37374 df-llines 37521 df-lplanes 37522 df-lvols 37523 df-lines 37524 df-psubsp 37526 df-pmap 37527 df-padd 37819 df-lhyp 38011 df-laut 38012 df-ldil 38127 df-ltrn 38128 df-trl 38182 df-tgrp 38766 df-tendo 38778 df-edring 38780 df-dveca 39026 df-disoa 39052 df-dvech 39102 df-dib 39162 df-dic 39196 df-dih 39252 df-doch 39371 df-djh 39418 |
This theorem is referenced by: lclkr 39556 lclkrslem2 39561 |
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