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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2b | Structured version Visualization version GIF version |
Description: Lemma for lclkr 38671. (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 38645 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴) |
21 | 1, 3, 9 | dvhlmod 38248 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
22 | lmodabl 19683 | . . . . . . 7 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel) | |
23 | 21, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Abel) |
24 | eqid 2823 | . . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
25 | 24 | lsssssubg 19732 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
26 | 21, 25 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) |
27 | 13 | eldifad 3950 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
28 | 4, 24, 7 | lspsncl 19751 | . . . . . . . 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 3970 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑈)) |
31 | 15 | eldifad 3950 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
32 | 4, 24, 7 | lspsncl 19751 | . . . . . . . 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 3970 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑈)) |
35 | 6 | lsmcom 18980 | . . . . . 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 4190 | . . . 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 3073 | . . . . 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 38645 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})) → (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑋})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴) |
47 | 38, 46 | eqeltrd 2915 | . 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 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 {csn 4569 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 0gc0g 16715 SubGrpcsubg 18275 LSSumclsm 18761 Abelcabl 18909 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 LSAtomsclsa 36112 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 ocHcoch 38485 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 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 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-mre 16859 df-mrc 16860 df-acs 16862 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-oppg 18476 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 df-lshyp 36115 df-lcv 36157 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tgrp 37881 df-tendo 37893 df-edring 37895 df-dveca 38141 df-disoa 38167 df-dvech 38217 df-dib 38277 df-dic 38311 df-dih 38367 df-doch 38486 df-djh 38533 |
This theorem is referenced by: lclkrlem2c 38647 |
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