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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem37 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 38725. (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 2824 | . . . . . 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 38250 | . . . . . 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 2824 | . . . . . . 7 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
18 | lcfrlem24.l | . . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈) | |
19 | eqid 2824 | . . . . . . 7 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
20 | eqid 2824 | . . . . . . 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 4725 | . . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
27 | 25, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
28 | eldifsn 4722 | . . . . . . . 8 ⊢ (𝑋 ∈ (𝐸 ∖ { 0 }) ↔ (𝑋 ∈ 𝐸 ∧ 𝑋 ≠ 0 )) | |
29 | 24, 27, 28 | sylanbrc 585 | . . . . . . 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 38698 | . . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺) |
31 | eqid 2824 | . . . . . . 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 38711 | . . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅) |
42 | lcfrlem37.ye | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐸) | |
43 | eldifsni 4725 | . . . . . . . . . 10 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
44 | 34, 43 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ 0 ) |
45 | eldifsn 4722 | . . . . . . . . 9 ⊢ (𝑌 ∈ (𝐸 ∖ { 0 }) ↔ (𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 )) | |
46 | 42, 44, 45 | sylanbrc 585 | . . . . . . . 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 38698 | . . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ 𝐺) |
48 | 14, 15, 2, 31, 4, 8, 9, 41, 47 | ldualssvscl 36298 | . . . . . 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 36299 | . . . . 5 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) ∈ 𝐺) |
50 | 1, 49 | eqeltrid 2920 | . . . 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 38718 | . . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) |
52 | 2fveq3 6678 | . . . . . 6 ⊢ (𝑔 = 𝐶 → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘𝐶))) | |
53 | 52 | eleq2d 2901 | . . . . 5 ⊢ (𝑔 = 𝐶 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))) |
54 | 53 | rspcev 3626 | . . . 4 ⊢ ((𝐶 ∈ 𝐺 ∧ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
55 | 50, 51, 54 | syl2anc 586 | . . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
56 | eliun 4926 | . . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) | |
57 | 55, 56 | sylibr 236 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))) |
58 | 57, 23 | eleqtrrdi 2927 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∃wrex 3142 {crab 3145 ∖ cdif 3936 ∩ cin 3938 ⊆ wss 3939 {csn 4570 {cpr 4572 ∪ ciun 4922 ↦ cmpt 5149 ‘cfv 6358 ℩crio 7116 (class class class)co 7159 Basecbs 16486 +gcplusg 16568 .rcmulr 16569 Scalarcsca 16571 ·𝑠 cvsca 16572 0gc0g 16716 -gcsg 18108 invrcinvr 19424 LSubSpclss 19706 LSpanclspn 19746 LSAtomsclsa 36114 LFnlclfn 36197 LKerclk 36225 LDualcld 36263 HLchlt 36490 LHypclh 37124 DVecHcdvh 38218 ocHcoch 38487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-riotaBAD 36093 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-undef 7942 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-0g 16718 df-mre 16860 df-mrc 16861 df-acs 16863 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-p1 17653 df-lat 17659 df-clat 17721 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-cntz 18450 df-oppg 18477 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-drng 19507 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878 df-lsatoms 36116 df-lshyp 36117 df-lcv 36159 df-lfl 36198 df-lkr 36226 df-ldual 36264 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-llines 36638 df-lplanes 36639 df-lvols 36640 df-lines 36641 df-psubsp 36643 df-pmap 36644 df-padd 36936 df-lhyp 37128 df-laut 37129 df-ldil 37244 df-ltrn 37245 df-trl 37299 df-tgrp 37883 df-tendo 37895 df-edring 37897 df-dveca 38143 df-disoa 38169 df-dvech 38219 df-dib 38279 df-dic 38313 df-dih 38369 df-doch 38488 df-djh 38535 |
This theorem is referenced by: lcfrlem38 38720 |
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