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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem37 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 39853. (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 2736 | . . . . . 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 39378 | . . . . . 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 2736 | . . . . . . 7 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
18 | lcfrlem24.l | . . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈) | |
19 | eqid 2736 | . . . . . . 7 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
20 | eqid 2736 | . . . . . . 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 4737 | . . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
27 | 25, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
28 | eldifsn 4734 | . . . . . . . 8 ⊢ (𝑋 ∈ (𝐸 ∖ { 0 }) ↔ (𝑋 ∈ 𝐸 ∧ 𝑋 ≠ 0 )) | |
29 | 24, 27, 28 | sylanbrc 583 | . . . . . . 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 39826 | . . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺) |
31 | eqid 2736 | . . . . . . 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 39839 | . . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅) |
42 | lcfrlem37.ye | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐸) | |
43 | eldifsni 4737 | . . . . . . . . . 10 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
44 | 34, 43 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ 0 ) |
45 | eldifsn 4734 | . . . . . . . . 9 ⊢ (𝑌 ∈ (𝐸 ∖ { 0 }) ↔ (𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 )) | |
46 | 42, 44, 45 | sylanbrc 583 | . . . . . . . 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 39826 | . . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ 𝐺) |
48 | 14, 15, 2, 31, 4, 8, 9, 41, 47 | ldualssvscl 37425 | . . . . . 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 37426 | . . . . 5 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) ∈ 𝐺) |
50 | 1, 49 | eqeltrid 2841 | . . . 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 39846 | . . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) |
52 | 2fveq3 6830 | . . . . . 6 ⊢ (𝑔 = 𝐶 → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘𝐶))) | |
53 | 52 | eleq2d 2822 | . . . . 5 ⊢ (𝑔 = 𝐶 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))) |
54 | 53 | rspcev 3570 | . . . 4 ⊢ ((𝐶 ∈ 𝐺 ∧ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
55 | 50, 51, 54 | syl2anc 584 | . . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
56 | eliun 4945 | . . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) | |
57 | 55, 56 | sylibr 233 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))) |
58 | 57, 23 | eleqtrrdi 2848 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 ∃wrex 3070 {crab 3403 ∖ cdif 3895 ∩ cin 3897 ⊆ wss 3898 {csn 4573 {cpr 4575 ∪ ciun 4941 ↦ cmpt 5175 ‘cfv 6479 ℩crio 7292 (class class class)co 7337 Basecbs 17009 +gcplusg 17059 .rcmulr 17060 Scalarcsca 17062 ·𝑠 cvsca 17063 0gc0g 17247 -gcsg 18675 invrcinvr 20008 LSubSpclss 20299 LSpanclspn 20339 LSAtomsclsa 37241 LFnlclfn 37324 LKerclk 37352 LDualcld 37390 HLchlt 37617 LHypclh 38252 DVecHcdvh 39346 ocHcoch 39615 |
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 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-riotaBAD 37220 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-of 7595 df-om 7781 df-1st 7899 df-2nd 7900 df-tpos 8112 df-undef 8159 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-n0 12335 df-z 12421 df-uz 12684 df-fz 13341 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-sca 17075 df-vsca 17076 df-0g 17249 df-mre 17392 df-mrc 17393 df-acs 17395 df-proset 18110 df-poset 18128 df-plt 18145 df-lub 18161 df-glb 18162 df-join 18163 df-meet 18164 df-p0 18240 df-p1 18241 df-lat 18247 df-clat 18314 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-submnd 18528 df-grp 18676 df-minusg 18677 df-sbg 18678 df-subg 18848 df-cntz 19019 df-oppg 19046 df-lsm 19337 df-cmn 19483 df-abl 19484 df-mgp 19816 df-ur 19833 df-ring 19880 df-oppr 19957 df-dvdsr 19978 df-unit 19979 df-invr 20009 df-dvr 20020 df-drng 20095 df-lmod 20231 df-lss 20300 df-lsp 20340 df-lvec 20471 df-lsatoms 37243 df-lshyp 37244 df-lcv 37286 df-lfl 37325 df-lkr 37353 df-ldual 37391 df-oposet 37443 df-ol 37445 df-oml 37446 df-covers 37533 df-ats 37534 df-atl 37565 df-cvlat 37589 df-hlat 37618 df-llines 37766 df-lplanes 37767 df-lvols 37768 df-lines 37769 df-psubsp 37771 df-pmap 37772 df-padd 38064 df-lhyp 38256 df-laut 38257 df-ldil 38372 df-ltrn 38373 df-trl 38427 df-tgrp 39011 df-tendo 39023 df-edring 39025 df-dveca 39271 df-disoa 39297 df-dvech 39347 df-dib 39407 df-dic 39441 df-dih 39497 df-doch 39616 df-djh 39663 |
This theorem is referenced by: lcfrlem38 39848 |
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