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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem30 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 40919. (Contributed by NM, 6-Mar-2015.) |
Ref | Expression |
---|---|
lcfrlem17.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfrlem17.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfrlem17.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfrlem17.v | ⊢ 𝑉 = (Base‘𝑈) |
lcfrlem17.p | ⊢ + = (+g‘𝑈) |
lcfrlem17.z | ⊢ 0 = (0g‘𝑈) |
lcfrlem17.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lcfrlem17.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
lcfrlem17.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfrlem17.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lcfrlem17.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
lcfrlem17.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
lcfrlem22.b | ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) |
lcfrlem24.t | ⊢ · = ( ·𝑠 ‘𝑈) |
lcfrlem24.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lcfrlem24.q | ⊢ 𝑄 = (0g‘𝑆) |
lcfrlem24.r | ⊢ 𝑅 = (Base‘𝑆) |
lcfrlem24.j | ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) |
lcfrlem24.ib | ⊢ (𝜑 → 𝐼 ∈ 𝐵) |
lcfrlem24.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfrlem25.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcfrlem28.jn | ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) |
lcfrlem29.i | ⊢ 𝐹 = (invr‘𝑆) |
lcfrlem30.m | ⊢ − = (-g‘𝐷) |
lcfrlem30.c | ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) |
Ref | Expression |
---|---|
lcfrlem30 | ⊢ (𝜑 → 𝐶 ∈ (LFnl‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem30.c | . 2 ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) | |
2 | eqid 2731 | . . 3 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
3 | lcfrlem25.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
4 | lcfrlem30.m | . . 3 ⊢ − = (-g‘𝐷) | |
5 | lcfrlem17.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | lcfrlem17.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | lcfrlem17.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | 5, 6, 7 | dvhlmod 40444 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
9 | lcfrlem17.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
10 | lcfrlem17.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
11 | lcfrlem17.p | . . . 4 ⊢ + = (+g‘𝑈) | |
12 | lcfrlem24.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑈) | |
13 | lcfrlem24.s | . . . 4 ⊢ 𝑆 = (Scalar‘𝑈) | |
14 | lcfrlem24.r | . . . 4 ⊢ 𝑅 = (Base‘𝑆) | |
15 | lcfrlem17.z | . . . 4 ⊢ 0 = (0g‘𝑈) | |
16 | lcfrlem24.l | . . . 4 ⊢ 𝐿 = (LKer‘𝑈) | |
17 | eqid 2731 | . . . 4 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
18 | eqid 2731 | . . . 4 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
19 | lcfrlem24.j | . . . 4 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) | |
20 | lcfrlem17.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
21 | 5, 9, 6, 10, 11, 12, 13, 14, 15, 2, 16, 3, 17, 18, 19, 7, 20 | lcfrlem10 40886 | . . 3 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LFnl‘𝑈)) |
22 | eqid 2731 | . . . 4 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
23 | lcfrlem17.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
24 | lcfrlem17.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
25 | lcfrlem17.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
26 | lcfrlem17.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
27 | lcfrlem22.b | . . . . 5 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
28 | lcfrlem24.q | . . . . 5 ⊢ 𝑄 = (0g‘𝑆) | |
29 | lcfrlem24.ib | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
30 | lcfrlem28.jn | . . . . 5 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) | |
31 | lcfrlem29.i | . . . . 5 ⊢ 𝐹 = (invr‘𝑆) | |
32 | 5, 9, 6, 10, 11, 15, 23, 24, 7, 20, 25, 26, 27, 12, 13, 28, 14, 19, 29, 16, 3, 30, 31 | lcfrlem29 40905 | . . . 4 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅) |
33 | 5, 9, 6, 10, 11, 12, 13, 14, 15, 2, 16, 3, 17, 18, 19, 7, 25 | lcfrlem10 40886 | . . . 4 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈)) |
34 | 2, 13, 14, 3, 22, 8, 32, 33 | ldualvscl 38472 | . . 3 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)) ∈ (LFnl‘𝑈)) |
35 | 2, 3, 4, 8, 21, 34 | ldualvsubcl 38489 | . 2 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) ∈ (LFnl‘𝑈)) |
36 | 1, 35 | eqeltrid 2836 | 1 ⊢ (𝜑 → 𝐶 ∈ (LFnl‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∃wrex 3069 {crab 3431 ∖ cdif 3945 ∩ cin 3947 {csn 4628 {cpr 4630 ↦ cmpt 5231 ‘cfv 6543 ℩crio 7367 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 .rcmulr 17205 Scalarcsca 17207 ·𝑠 cvsca 17208 0gc0g 17392 -gcsg 18863 invrcinvr 20285 LSpanclspn 20814 LSAtomsclsa 38307 LFnlclfn 38390 LKerclk 38418 LDualcld 38456 HLchlt 38683 LHypclh 39318 DVecHcdvh 40412 ocHcoch 40681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38286 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-mre 17537 df-mrc 17538 df-acs 17540 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cntz 19229 df-oppg 19258 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20946 df-lsatoms 38309 df-lshyp 38310 df-lcv 38352 df-lfl 38391 df-ldual 38457 df-oposet 38509 df-ol 38511 df-oml 38512 df-covers 38599 df-ats 38600 df-atl 38631 df-cvlat 38655 df-hlat 38684 df-llines 38832 df-lplanes 38833 df-lvols 38834 df-lines 38835 df-psubsp 38837 df-pmap 38838 df-padd 39130 df-lhyp 39322 df-laut 39323 df-ldil 39438 df-ltrn 39439 df-trl 39493 df-tgrp 40077 df-tendo 40089 df-edring 40091 df-dveca 40337 df-disoa 40363 df-dvech 40413 df-dib 40473 df-dic 40507 df-dih 40563 df-doch 40682 df-djh 40729 |
This theorem is referenced by: lcfrlem35 40911 lcfrlem36 40912 |
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