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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem26 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 38715. Special case of lcfrlem36 38708 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 38689 | . . . . . 6 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 })) |
14 | 13 | eldifad 3948 | . . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
15 | 1, 2, 3, 4, 5, 6, 14 | dochocsn 38511 | . . . 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 38697 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘(𝐽‘𝑌))) |
28 | 27 | fveq2d 6669 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
29 | 15, 28 | eqtr3d 2858 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
30 | eqimss 4023 | . . 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 2821 | . . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
33 | 1, 2, 6 | dvhlmod 38240 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
34 | eqid 2821 | . . . . 5 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
35 | eqid 2821 | . . . . . 6 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
36 | eqid 2821 | . . . . . 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 38682 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈)) |
38 | 4, 34, 23, 33, 37 | lkrssv 36226 | . . . 4 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ⊆ 𝑉) |
39 | 1, 2, 4, 32, 3 | dochlss 38484 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘(𝐽‘𝑌)) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ∈ (LSubSp‘𝑈)) |
40 | 6, 38, 39 | syl2anc 586 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ∈ (LSubSp‘𝑈)) |
41 | 4, 32, 5, 33, 40, 14 | lspsnel5 19761 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ↔ (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))) |
42 | 31, 41 | mpbird 259 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 {crab 3142 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 {csn 4561 {cpr 4563 ↦ cmpt 5139 ‘cfv 6350 ℩crio 7107 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 LSubSpclss 19697 LSpanclspn 19737 LSAtomsclsa 36104 LFnlclfn 36187 LKerclk 36215 LDualcld 36253 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 ocHcoch 38477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-riotaBAD 36083 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-undef 7933 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-0g 16709 df-mre 16851 df-mrc 16852 df-acs 16854 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-oppg 18468 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-lshyp 36107 df-lcv 36149 df-lfl 36188 df-lkr 36216 df-ldual 36254 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tgrp 37873 df-tendo 37885 df-edring 37887 df-dveca 38133 df-disoa 38159 df-dvech 38209 df-dib 38269 df-dic 38303 df-dih 38359 df-doch 38478 df-djh 38525 |
This theorem is referenced by: lcfrlem27 38699 |
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