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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2b | Structured version Visualization version GIF version |
Description: Lemma for lclkr 38673. (Contributed by NM, 17-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2a.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lclkrlem2a.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lclkrlem2a.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lclkrlem2a.v | ⊢ 𝑉 = (Base‘𝑈) |
lclkrlem2a.z | ⊢ 0 = (0g‘𝑈) |
lclkrlem2a.p | ⊢ ⊕ = (LSSum‘𝑈) |
lclkrlem2a.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lclkrlem2a.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
lclkrlem2a.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lclkrlem2a.b | ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2a.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2a.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2a.e | ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌})) |
lclkrlem2b.da | ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) |
Ref | Expression |
---|---|
lclkrlem2b | ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrlem2a.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lclkrlem2a.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | lclkrlem2a.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lclkrlem2a.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | lclkrlem2a.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
6 | lclkrlem2a.p | . . 3 ⊢ ⊕ = (LSSum‘𝑈) | |
7 | lclkrlem2a.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
8 | lclkrlem2a.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
9 | lclkrlem2a.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 9 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
11 | lclkrlem2a.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 })) | |
12 | 11 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → 𝐵 ∈ (𝑉 ∖ { 0 })) |
13 | lclkrlem2a.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
14 | 13 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
15 | lclkrlem2a.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
16 | 15 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
17 | lclkrlem2a.e | . . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌})) | |
18 | 17 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌})) |
19 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) | |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 19 | lclkrlem2a 38647 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴) |
21 | 1, 3, 9 | dvhlmod 38250 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
22 | lmodabl 19684 | . . . . . . 7 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel) | |
23 | 21, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Abel) |
24 | eqid 2824 | . . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
25 | 24 | lsssssubg 19733 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
26 | 21, 25 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
27 | 13 | eldifad 3951 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
28 | 4, 24, 7 | lspsncl 19752 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
29 | 21, 27, 28 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
30 | 26, 29 | sseldd 3971 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑈)) |
31 | 15 | eldifad 3951 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
32 | 4, 24, 7 | lspsncl 19752 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
33 | 21, 31, 32 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
34 | 26, 33 | sseldd 3971 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑈)) |
35 | 6 | lsmcom 18981 | . . . . . 6 ⊢ ((𝑈 ∈ Abel ∧ (𝑁‘{𝑋}) ∈ (SubGrp‘𝑈) ∧ (𝑁‘{𝑌}) ∈ (SubGrp‘𝑈)) → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = ((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋}))) |
36 | 23, 30, 34, 35 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = ((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋}))) |
37 | 36 | ineq1d 4191 | . . . 4 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})) ∩ ( ⊥ ‘{𝐵}))) |
38 | 37 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})) ∩ ( ⊥ ‘{𝐵}))) |
39 | 9 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
40 | 11 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → 𝐵 ∈ (𝑉 ∖ { 0 })) |
41 | 15 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
42 | 13 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
43 | 17 | necomd 3074 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ≠ ( ⊥ ‘{𝑋})) |
44 | 43 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → ( ⊥ ‘{𝑌}) ≠ ( ⊥ ‘{𝑋})) |
45 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) | |
46 | 1, 2, 3, 4, 5, 6, 7, 8, 39, 40, 41, 42, 44, 45 | lclkrlem2a 38647 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴) |
47 | 38, 46 | eqeltrd 2916 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴) |
48 | lclkrlem2b.da | . 2 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) | |
49 | 20, 47, 48 | mpjaodan 955 | 1 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∖ cdif 3936 ∩ cin 3938 ⊆ wss 3939 {csn 4570 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 0gc0g 16716 SubGrpcsubg 18276 LSSumclsm 18762 Abelcabl 18910 LModclmod 19637 LSubSpclss 19706 LSpanclspn 19746 LSAtomsclsa 36114 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-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-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: lclkrlem2c 38649 |
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