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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem1 | Structured version Visualization version GIF version |
Description: Lemma for dochexmid 37543. 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 2825 | . . . . 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 37185 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
7 | dochexmidlem1.rr | . . . . 5 ⊢ (𝜑 → 𝑟 ∈ 𝐴) | |
8 | 1, 2, 6, 7 | lsatn0 35074 | . . . 4 ⊢ (𝜑 → 𝑟 ≠ {(0g‘𝑈)}) |
9 | dochexmidlem1.s | . . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈) | |
10 | 9, 2, 6, 7 | lsatlssel 35072 | . . . . . 6 ⊢ (𝜑 → 𝑟 ∈ 𝑆) |
11 | 1, 9 | lssle0 19306 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑟 ∈ 𝑆) → (𝑟 ⊆ {(0g‘𝑈)} ↔ 𝑟 = {(0g‘𝑈)})) |
12 | 6, 10, 11 | syl2anc 581 | . . . . 5 ⊢ (𝜑 → (𝑟 ⊆ {(0g‘𝑈)} ↔ 𝑟 = {(0g‘𝑈)})) |
13 | 12 | necon3bbid 3036 | . . . 4 ⊢ (𝜑 → (¬ 𝑟 ⊆ {(0g‘𝑈)} ↔ 𝑟 ≠ {(0g‘𝑈)})) |
14 | 8, 13 | mpbird 249 | . . 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 37466 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) = {(0g‘𝑈)}) |
18 | 5, 15, 17 | syl2anc 581 | . . . 4 ⊢ (𝜑 → (𝑋 ∩ ( ⊥ ‘𝑋)) = {(0g‘𝑈)}) |
19 | 18 | sseq2d 3858 | . . 3 ⊢ (𝜑 → (𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋)) ↔ 𝑟 ⊆ {(0g‘𝑈)})) |
20 | 14, 19 | mtbird 317 | . 2 ⊢ (𝜑 → ¬ 𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋))) |
21 | dochexmidlem1.ql | . . . . . 6 ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋)) | |
22 | sseq1 3851 | . . . . . 6 ⊢ (𝑞 = 𝑟 → (𝑞 ⊆ ( ⊥ ‘𝑋) ↔ 𝑟 ⊆ ( ⊥ ‘𝑋))) | |
23 | 21, 22 | syl5ibcom 237 | . . . . 5 ⊢ (𝜑 → (𝑞 = 𝑟 → 𝑟 ⊆ ( ⊥ ‘𝑋))) |
24 | dochexmidlem1.rl | . . . . 5 ⊢ (𝜑 → 𝑟 ⊆ 𝑋) | |
25 | 23, 24 | jctild 523 | . . . 4 ⊢ (𝜑 → (𝑞 = 𝑟 → (𝑟 ⊆ 𝑋 ∧ 𝑟 ⊆ ( ⊥ ‘𝑋)))) |
26 | ssin 4059 | . . . 4 ⊢ ((𝑟 ⊆ 𝑋 ∧ 𝑟 ⊆ ( ⊥ ‘𝑋)) ↔ 𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋))) | |
27 | 25, 26 | syl6ib 243 | . . 3 ⊢ (𝜑 → (𝑞 = 𝑟 → 𝑟 ⊆ (𝑋 ∩ ( ⊥ ‘𝑋)))) |
28 | 27 | necon3bd 3013 | . 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 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∩ cin 3797 ⊆ wss 3798 {csn 4397 ‘cfv 6123 Basecbs 16222 0gc0g 16453 LSSumclsm 18400 LModclmod 19219 LSubSpclss 19288 LSpanclspn 19330 LSAtomsclsa 35049 HLchlt 35425 LHypclh 36059 DVecHcdvh 37153 ocHcoch 37422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-riotaBAD 35028 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-undef 7664 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-0g 16455 df-proset 17281 df-poset 17299 df-plt 17311 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-p1 17393 df-lat 17399 df-clat 17461 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 df-lsm 18402 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-oppr 18977 df-dvdsr 18995 df-unit 18996 df-invr 19026 df-dvr 19037 df-drng 19105 df-lmod 19221 df-lss 19289 df-lsp 19331 df-lvec 19462 df-lsatoms 35051 df-oposet 35251 df-ol 35253 df-oml 35254 df-covers 35341 df-ats 35342 df-atl 35373 df-cvlat 35397 df-hlat 35426 df-llines 35573 df-lplanes 35574 df-lvols 35575 df-lines 35576 df-psubsp 35578 df-pmap 35579 df-padd 35871 df-lhyp 36063 df-laut 36064 df-ldil 36179 df-ltrn 36180 df-trl 36234 df-tendo 36830 df-edring 36832 df-disoa 37104 df-dvech 37154 df-dib 37214 df-dic 37248 df-dih 37304 df-doch 37423 |
This theorem is referenced by: dochexmidlem3 37537 |
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