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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem1 | Structured version Visualization version GIF version |
Description: Lemma for dochexmid 38484. Holland's proof implicitly requires 𝑞 ≠ 𝑟, which we prove here. (Contributed by NM, 14-Jan-2015.) |
Ref | Expression |
---|---|
dochexmidlem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochexmidlem1.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochexmidlem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochexmidlem1.v | ⊢ 𝑉 = (Base‘𝑈) |
dochexmidlem1.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
dochexmidlem1.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dochexmidlem1.p | ⊢ ⊕ = (LSSum‘𝑈) |
dochexmidlem1.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
dochexmidlem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochexmidlem1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
dochexmidlem1.pp | ⊢ (𝜑 → 𝑝 ∈ 𝐴) |
dochexmidlem1.qq | ⊢ (𝜑 → 𝑞 ∈ 𝐴) |
dochexmidlem1.rr | ⊢ (𝜑 → 𝑟 ∈ 𝐴) |
dochexmidlem1.ql | ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋)) |
dochexmidlem1.rl | ⊢ (𝜑 → 𝑟 ⊆ 𝑋) |
Ref | Expression |
---|---|
dochexmidlem1 | ⊢ (𝜑 → 𝑞 ≠ 𝑟) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . 5 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
2 | dochexmidlem1.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
3 | dochexmidlem1.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dochexmidlem1.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | dochexmidlem1.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 3, 4, 5 | dvhlmod 38126 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
7 | dochexmidlem1.rr | . . . . 5 ⊢ (𝜑 → 𝑟 ∈ 𝐴) | |
8 | 1, 2, 6, 7 | lsatn0 36015 | . . . 4 ⊢ (𝜑 → 𝑟 ≠ {(0g‘𝑈)}) |
9 | dochexmidlem1.s | . . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈) | |
10 | 9, 2, 6, 7 | lsatlssel 36013 | . . . . . 6 ⊢ (𝜑 → 𝑟 ∈ 𝑆) |
11 | 1, 9 | lssle0 19650 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑟 ∈ 𝑆) → (𝑟 ⊆ {(0g‘𝑈)} ↔ 𝑟 = {(0g‘𝑈)})) |
12 | 6, 10, 11 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑟 ⊆ {(0g‘𝑈)} ↔ 𝑟 = {(0g‘𝑈)})) |
13 | 12 | necon3bbid 3050 | . . . 4 ⊢ (𝜑 → (¬ 𝑟 ⊆ {(0g‘𝑈)} ↔ 𝑟 ≠ {(0g‘𝑈)})) |
14 | 8, 13 | mpbird 258 | . . 3 ⊢ (𝜑 → ¬ 𝑟 ⊆ {(0g‘𝑈)}) |
15 | dochexmidlem1.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
16 | dochexmidlem1.o | . . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
17 | 3, 4, 9, 1, 16 | dochnoncon 38407 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) = {(0g‘𝑈)}) |
18 | 5, 15, 17 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑋 ∩ ( ⊥ ‘𝑋)) = {(0g‘𝑈)}) |
19 | 18 | sseq2d 3996 | . . 3 ⊢ (𝜑 → (𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋)) ↔ 𝑟 ⊆ {(0g‘𝑈)})) |
20 | 14, 19 | mtbird 326 | . 2 ⊢ (𝜑 → ¬ 𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋))) |
21 | dochexmidlem1.ql | . . . . . 6 ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋)) | |
22 | sseq1 3989 | . . . . . 6 ⊢ (𝑞 = 𝑟 → (𝑞 ⊆ ( ⊥ ‘𝑋) ↔ 𝑟 ⊆ ( ⊥ ‘𝑋))) | |
23 | 21, 22 | syl5ibcom 246 | . . . . 5 ⊢ (𝜑 → (𝑞 = 𝑟 → 𝑟 ⊆ ( ⊥ ‘𝑋))) |
24 | dochexmidlem1.rl | . . . . 5 ⊢ (𝜑 → 𝑟 ⊆ 𝑋) | |
25 | 23, 24 | jctild 526 | . . . 4 ⊢ (𝜑 → (𝑞 = 𝑟 → (𝑟 ⊆ 𝑋 ∧ 𝑟 ⊆ ( ⊥ ‘𝑋)))) |
26 | ssin 4204 | . . . 4 ⊢ ((𝑟 ⊆ 𝑋 ∧ 𝑟 ⊆ ( ⊥ ‘𝑋)) ↔ 𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋))) | |
27 | 25, 26 | syl6ib 252 | . . 3 ⊢ (𝜑 → (𝑞 = 𝑟 → 𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋)))) |
28 | 27 | necon3bd 3027 | . 2 ⊢ (𝜑 → (¬ 𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋)) → 𝑞 ≠ 𝑟)) |
29 | 20, 28 | mpd 15 | 1 ⊢ (𝜑 → 𝑞 ≠ 𝑟) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∩ cin 3932 ⊆ wss 3933 {csn 4557 ‘cfv 6348 Basecbs 16471 0gc0g 16701 LSSumclsm 18688 LModclmod 19563 LSubSpclss 19632 LSpanclspn 19672 LSAtomsclsa 35990 HLchlt 36366 LHypclh 37000 DVecHcdvh 38094 ocHcoch 38363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lsatoms 35992 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tendo 37771 df-edring 37773 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 df-dih 38245 df-doch 38364 |
This theorem is referenced by: dochexmidlem3 38478 |
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