Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem1 | Structured version Visualization version GIF version |
Description: Lemma for dochexmid 39478. 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 2740 | . . . . 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 39120 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
7 | dochexmidlem1.rr | . . . . 5 ⊢ (𝜑 → 𝑟 ∈ 𝐴) | |
8 | 1, 2, 6, 7 | lsatn0 37009 | . . . 4 ⊢ (𝜑 → 𝑟 ≠ {(0g‘𝑈)}) |
9 | dochexmidlem1.s | . . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈) | |
10 | 9, 2, 6, 7 | lsatlssel 37007 | . . . . . 6 ⊢ (𝜑 → 𝑟 ∈ 𝑆) |
11 | 1, 9 | lssle0 20209 | . . . . . 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 2983 | . . . 4 ⊢ (𝜑 → (¬ 𝑟 ⊆ {(0g‘𝑈)} ↔ 𝑟 ≠ {(0g‘𝑈)})) |
14 | 8, 13 | mpbird 256 | . . 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 39401 | . . . . 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 3958 | . . 3 ⊢ (𝜑 → (𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋)) ↔ 𝑟 ⊆ {(0g‘𝑈)})) |
20 | 14, 19 | mtbird 325 | . 2 ⊢ (𝜑 → ¬ 𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋))) |
21 | dochexmidlem1.ql | . . . . . 6 ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋)) | |
22 | sseq1 3951 | . . . . . 6 ⊢ (𝑞 = 𝑟 → (𝑞 ⊆ ( ⊥ ‘𝑋) ↔ 𝑟 ⊆ ( ⊥ ‘𝑋))) | |
23 | 21, 22 | syl5ibcom 244 | . . . . 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 4170 | . . . 4 ⊢ ((𝑟 ⊆ 𝑋 ∧ 𝑟 ⊆ ( ⊥ ‘𝑋)) ↔ 𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋))) | |
27 | 25, 26 | syl6ib 250 | . . 3 ⊢ (𝜑 → (𝑞 = 𝑟 → 𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋)))) |
28 | 27 | necon3bd 2959 | . 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 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∩ cin 3891 ⊆ wss 3892 {csn 4567 ‘cfv 6432 Basecbs 16910 0gc0g 17148 LSSumclsm 19237 LModclmod 20121 LSubSpclss 20191 LSpanclspn 20231 LSAtomsclsa 36984 HLchlt 37360 LHypclh 37994 DVecHcdvh 39088 ocHcoch 39357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-riotaBAD 36963 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-tpos 8033 df-undef 8080 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-0g 17150 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 df-lsatoms 36986 df-oposet 37186 df-ol 37188 df-oml 37189 df-covers 37276 df-ats 37277 df-atl 37308 df-cvlat 37332 df-hlat 37361 df-llines 37508 df-lplanes 37509 df-lvols 37510 df-lines 37511 df-psubsp 37513 df-pmap 37514 df-padd 37806 df-lhyp 37998 df-laut 37999 df-ldil 38114 df-ltrn 38115 df-trl 38169 df-tendo 38765 df-edring 38767 df-disoa 39039 df-dvech 39089 df-dib 39149 df-dic 39183 df-dih 39239 df-doch 39358 |
This theorem is referenced by: dochexmidlem3 39472 |
Copyright terms: Public domain | W3C validator |