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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem37 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 37191. (Contributed by NM, 8-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))) |
lcfrlem37.g | ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷)) |
lcfrlem37.gs | ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) |
lcfrlem37.e | ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) |
lcfrlem37.xe | ⊢ (𝜑 → 𝑋 ∈ 𝐸) |
lcfrlem37.ye | ⊢ (𝜑 → 𝑌 ∈ 𝐸) |
Ref | Expression |
---|---|
lcfrlem37 | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem30.c | . . . . 5 ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) | |
2 | lcfrlem25.d | . . . . . 6 ⊢ 𝐷 = (LDual‘𝑈) | |
3 | lcfrlem30.m | . . . . . 6 ⊢ − = (-g‘𝐷) | |
4 | eqid 2651 | . . . . . 6 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷) | |
5 | lcfrlem17.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | lcfrlem17.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | lcfrlem17.k | . . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | 5, 6, 7 | dvhlmod 36716 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
9 | lcfrlem37.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷)) | |
10 | lcfrlem17.o | . . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
11 | lcfrlem17.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
12 | lcfrlem17.p | . . . . . . 7 ⊢ + = (+g‘𝑈) | |
13 | lcfrlem24.t | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑈) | |
14 | lcfrlem24.s | . . . . . . 7 ⊢ 𝑆 = (Scalar‘𝑈) | |
15 | lcfrlem24.r | . . . . . . 7 ⊢ 𝑅 = (Base‘𝑆) | |
16 | lcfrlem17.z | . . . . . . 7 ⊢ 0 = (0g‘𝑈) | |
17 | eqid 2651 | . . . . . . 7 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
18 | lcfrlem24.l | . . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈) | |
19 | eqid 2651 | . . . . . . 7 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
20 | eqid 2651 | . . . . . . 7 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
21 | lcfrlem24.j | . . . . . . 7 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) | |
22 | lcfrlem37.gs | . . . . . . 7 ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) | |
23 | lcfrlem37.e | . . . . . . 7 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) | |
24 | lcfrlem37.xe | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
25 | lcfrlem17.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
26 | eldifsni 4353 | . . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
27 | 25, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
28 | eldifsn 4350 | . . . . . . . 8 ⊢ (𝑋 ∈ (𝐸 ∖ { 0 }) ↔ (𝑋 ∈ 𝐸 ∧ 𝑋 ≠ 0 )) | |
29 | 24, 27, 28 | sylanbrc 699 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐸 ∖ { 0 })) |
30 | 5, 10, 6, 11, 12, 13, 14, 15, 16, 17, 18, 2, 19, 20, 21, 7, 4, 9, 22, 23, 29 | lcfrlem16 37164 | . . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺) |
31 | eqid 2651 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
32 | lcfrlem17.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈) | |
33 | lcfrlem17.a | . . . . . . . 8 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
34 | lcfrlem17.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
35 | lcfrlem17.ne | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
36 | lcfrlem22.b | . . . . . . . 8 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
37 | lcfrlem24.q | . . . . . . . 8 ⊢ 𝑄 = (0g‘𝑆) | |
38 | lcfrlem24.ib | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
39 | lcfrlem28.jn | . . . . . . . 8 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) | |
40 | lcfrlem29.i | . . . . . . . 8 ⊢ 𝐹 = (invr‘𝑆) | |
41 | 5, 10, 6, 11, 12, 16, 32, 33, 7, 25, 34, 35, 36, 13, 14, 37, 15, 21, 38, 18, 2, 39, 40 | lcfrlem29 37177 | . . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅) |
42 | lcfrlem37.ye | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐸) | |
43 | eldifsni 4353 | . . . . . . . . . 10 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
44 | 34, 43 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ 0 ) |
45 | eldifsn 4350 | . . . . . . . . 9 ⊢ (𝑌 ∈ (𝐸 ∖ { 0 }) ↔ (𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 )) | |
46 | 42, 44, 45 | sylanbrc 699 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝐸 ∖ { 0 })) |
47 | 5, 10, 6, 11, 12, 13, 14, 15, 16, 17, 18, 2, 19, 20, 21, 7, 4, 9, 22, 23, 46 | lcfrlem16 37164 | . . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ 𝐺) |
48 | 14, 15, 2, 31, 4, 8, 9, 41, 47 | ldualssvscl 34763 | . . . . . 6 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)) ∈ 𝐺) |
49 | 2, 3, 4, 8, 9, 30, 48 | ldualssvsubcl 34764 | . . . . 5 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) ∈ 𝐺) |
50 | 1, 49 | syl5eqel 2734 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐺) |
51 | 5, 10, 6, 11, 12, 16, 32, 33, 7, 25, 34, 35, 36, 13, 14, 37, 15, 21, 38, 18, 2, 39, 40, 3, 1 | lcfrlem36 37184 | . . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) |
52 | fveq2 6229 | . . . . . . 7 ⊢ (𝑔 = 𝐶 → (𝐿‘𝑔) = (𝐿‘𝐶)) | |
53 | 52 | fveq2d 6233 | . . . . . 6 ⊢ (𝑔 = 𝐶 → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘𝐶))) |
54 | 53 | eleq2d 2716 | . . . . 5 ⊢ (𝑔 = 𝐶 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))) |
55 | 54 | rspcev 3340 | . . . 4 ⊢ ((𝐶 ∈ 𝐺 ∧ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
56 | 50, 51, 55 | syl2anc 694 | . . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
57 | eliun 4556 | . . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) | |
58 | 56, 57 | sylibr 224 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))) |
59 | 58, 23 | syl6eleqr 2741 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∃wrex 2942 {crab 2945 ∖ cdif 3604 ∩ cin 3606 ⊆ wss 3607 {csn 4210 {cpr 4212 ∪ ciun 4552 ↦ cmpt 4762 ‘cfv 5926 ℩crio 6650 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 .rcmulr 15989 Scalarcsca 15991 ·𝑠 cvsca 15992 0gc0g 16147 -gcsg 17471 invrcinvr 18717 LSubSpclss 18980 LSpanclspn 19019 LSAtomsclsa 34579 LFnlclfn 34662 LKerclk 34690 LDualcld 34728 HLchlt 34955 LHypclh 35588 DVecHcdvh 36684 ocHcoch 36953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-riotaBAD 34557 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-undef 7444 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-0g 16149 df-mre 16293 df-mrc 16294 df-acs 16296 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-cntz 17796 df-oppg 17822 df-lsm 18097 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-drng 18797 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lvec 19151 df-lsatoms 34581 df-lshyp 34582 df-lcv 34624 df-lfl 34663 df-lkr 34691 df-ldual 34729 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-lplanes 35103 df-lvols 35104 df-lines 35105 df-psubsp 35107 df-pmap 35108 df-padd 35400 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 df-trl 35764 df-tgrp 36348 df-tendo 36360 df-edring 36362 df-dveca 36608 df-disoa 36635 df-dvech 36685 df-dib 36745 df-dic 36779 df-dih 36835 df-doch 36954 df-djh 37001 |
This theorem is referenced by: lcfrlem38 37186 |
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