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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem26 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 38881. Special case of lcfrlem36 38874 when ((𝐽‘𝑌)‘𝐼) is zero. (Contributed by NM, 11-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) |
lcfrlem25.jz | ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄) |
lcfrlem25.in | ⊢ (𝜑 → 𝐼 ≠ 0 ) |
Ref | Expression |
---|---|
lcfrlem26 | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem17.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcfrlem17.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | lcfrlem17.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | lcfrlem17.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
5 | lcfrlem17.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
6 | lcfrlem17.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | lcfrlem17.p | . . . . . . 7 ⊢ + = (+g‘𝑈) | |
8 | lcfrlem17.z | . . . . . . 7 ⊢ 0 = (0g‘𝑈) | |
9 | lcfrlem17.a | . . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
10 | lcfrlem17.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
11 | lcfrlem17.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
12 | lcfrlem17.ne | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
13 | 1, 3, 2, 4, 7, 8, 5, 9, 6, 10, 11, 12 | lcfrlem17 38855 | . . . . . 6 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 })) |
14 | 13 | eldifad 3893 | . . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
15 | 1, 2, 3, 4, 5, 6, 14 | dochocsn 38677 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)})) |
16 | lcfrlem22.b | . . . . . 6 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
17 | lcfrlem24.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑈) | |
18 | lcfrlem24.s | . . . . . 6 ⊢ 𝑆 = (Scalar‘𝑈) | |
19 | lcfrlem24.q | . . . . . 6 ⊢ 𝑄 = (0g‘𝑆) | |
20 | lcfrlem24.r | . . . . . 6 ⊢ 𝑅 = (Base‘𝑆) | |
21 | lcfrlem24.j | . . . . . 6 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) | |
22 | lcfrlem24.ib | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
23 | lcfrlem24.l | . . . . . 6 ⊢ 𝐿 = (LKer‘𝑈) | |
24 | lcfrlem25.d | . . . . . 6 ⊢ 𝐷 = (LDual‘𝑈) | |
25 | lcfrlem25.jz | . . . . . 6 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄) | |
26 | lcfrlem25.in | . . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 0 ) | |
27 | 1, 3, 2, 4, 7, 8, 5, 9, 6, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 | lcfrlem25 38863 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘(𝐽‘𝑌))) |
28 | 27 | fveq2d 6649 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
29 | 15, 28 | eqtr3d 2835 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
30 | eqimss 3971 | . . 3 ⊢ ((𝑁‘{(𝑋 + 𝑌)}) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) | |
31 | 29, 30 | syl 17 | . 2 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
32 | eqid 2798 | . . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
33 | 1, 2, 6 | dvhlmod 38406 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
34 | eqid 2798 | . . . . 5 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
35 | eqid 2798 | . . . . . 6 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
36 | eqid 2798 | . . . . . 6 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
37 | 1, 3, 2, 4, 7, 17, 18, 20, 8, 34, 23, 24, 35, 36, 21, 6, 11 | lcfrlem10 38848 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈)) |
38 | 4, 34, 23, 33, 37 | lkrssv 36392 | . . . 4 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ⊆ 𝑉) |
39 | 1, 2, 4, 32, 3 | dochlss 38650 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘(𝐽‘𝑌)) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ∈ (LSubSp‘𝑈)) |
40 | 6, 38, 39 | syl2anc 587 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ∈ (LSubSp‘𝑈)) |
41 | 4, 32, 5, 33, 40, 14 | lspsnel5 19760 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ↔ (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))) |
42 | 31, 41 | mpbird 260 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 {crab 3110 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 {csn 4525 {cpr 4527 ↦ cmpt 5110 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 LSubSpclss 19696 LSpanclspn 19736 LSAtomsclsa 36270 LFnlclfn 36353 LKerclk 36381 LDualcld 36419 HLchlt 36646 LHypclh 37280 DVecHcdvh 38374 ocHcoch 38643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-lshyp 36273 df-lcv 36315 df-lfl 36354 df-lkr 36382 df-ldual 36420 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tgrp 38039 df-tendo 38051 df-edring 38053 df-dveca 38299 df-disoa 38325 df-dvech 38375 df-dib 38435 df-dic 38469 df-dih 38525 df-doch 38644 df-djh 38691 |
This theorem is referenced by: lcfrlem27 38865 |
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