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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem12N | Structured version Visualization version GIF version |
Description: Lemma for lcfr 37739. (Contributed by NM, 23-Feb-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lcf1o.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcf1o.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcf1o.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcf1o.v | ⊢ 𝑉 = (Base‘𝑈) |
lcf1o.a | ⊢ + = (+g‘𝑈) |
lcf1o.t | ⊢ · = ( ·𝑠 ‘𝑈) |
lcf1o.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lcf1o.r | ⊢ 𝑅 = (Base‘𝑆) |
lcf1o.z | ⊢ 0 = (0g‘𝑈) |
lcf1o.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcf1o.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcf1o.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcf1o.q | ⊢ 𝑄 = (0g‘𝐷) |
lcf1o.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcf1o.j | ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) |
lcflo.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfrlem10.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lcfrlem12.b | ⊢ 𝐵 = (0g‘𝑆) |
lcfrlem12.y | ⊢ (𝜑 → 𝑌 ∈ ( ⊥ ‘{𝑋})) |
Ref | Expression |
---|---|
lcfrlem12N | ⊢ (𝜑 → ((𝐽‘𝑋)‘𝑌) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcf1o.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcf1o.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | lcflo.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlmod 37264 | . 2 ⊢ (𝜑 → 𝑈 ∈ LMod) |
5 | lcf1o.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
6 | lcf1o.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
7 | lcf1o.a | . . 3 ⊢ + = (+g‘𝑈) | |
8 | lcf1o.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
9 | lcf1o.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑈) | |
10 | lcf1o.r | . . 3 ⊢ 𝑅 = (Base‘𝑆) | |
11 | lcf1o.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
12 | lcf1o.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
13 | lcf1o.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
14 | lcf1o.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
15 | lcf1o.q | . . 3 ⊢ 𝑄 = (0g‘𝐷) | |
16 | lcf1o.c | . . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
17 | lcf1o.j | . . 3 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) | |
18 | lcfrlem10.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
19 | 1, 5, 2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 3, 18 | lcfrlem10 37706 | . 2 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐹) |
20 | lcfrlem12.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ ( ⊥ ‘{𝑋})) | |
21 | 1, 5, 2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 3, 18 | lcfrlem11 37707 | . . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘{𝑋})) |
22 | 20, 21 | eleqtrrd 2862 | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝐿‘(𝐽‘𝑋))) |
23 | lcfrlem12.b | . . 3 ⊢ 𝐵 = (0g‘𝑆) | |
24 | 9, 23, 12, 13 | lkrf0 35247 | . 2 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑋) ∈ 𝐹 ∧ 𝑌 ∈ (𝐿‘(𝐽‘𝑋))) → ((𝐽‘𝑋)‘𝑌) = 𝐵) |
25 | 4, 19, 22, 24 | syl3anc 1439 | 1 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝑌) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 {crab 3094 ∖ cdif 3789 {csn 4398 ↦ cmpt 4965 ‘cfv 6135 ℩crio 6882 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 Scalarcsca 16341 ·𝑠 cvsca 16342 0gc0g 16486 LModclmod 19255 LFnlclfn 35211 LKerclk 35239 LDualcld 35277 HLchlt 35504 LHypclh 36138 DVecHcdvh 37232 ocHcoch 37501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35107 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35130 df-lshyp 35131 df-lfl 35212 df-lkr 35240 df-oposet 35330 df-ol 35332 df-oml 35333 df-covers 35420 df-ats 35421 df-atl 35452 df-cvlat 35476 df-hlat 35505 df-llines 35652 df-lplanes 35653 df-lvols 35654 df-lines 35655 df-psubsp 35657 df-pmap 35658 df-padd 35950 df-lhyp 36142 df-laut 36143 df-ldil 36258 df-ltrn 36259 df-trl 36313 df-tgrp 36897 df-tendo 36909 df-edring 36911 df-dveca 37157 df-disoa 37183 df-dvech 37233 df-dib 37293 df-dic 37327 df-dih 37383 df-doch 37502 df-djh 37549 |
This theorem is referenced by: (None) |
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