dochexmidlem7 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem7		Structured version Visualization version GIF version

Theorem dochexmidlem7 41446

Description: Lemma for dochexmid 41448. Contradict dochexmidlem6 41445. (Contributed by NM, 15-Jan-2015.)

**Hypotheses**
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)
dochexmidlem6.pp	⊢ (𝜑 → 𝑝 ∈ 𝐴)
dochexmidlem6.z	⊢ 0 = (0_g‘𝑈)
dochexmidlem6.m	⊢ 𝑀 = (𝑋 ⊕ 𝑝)
dochexmidlem6.xn	⊢ (𝜑 → 𝑋 ≠ { 0 })
dochexmidlem6.c	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
dochexmidlem6.pl	⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))

**Assertion**
Ref	Expression
dochexmidlem7	⊢ (𝜑 → 𝑀 ≠ 𝑋)

**Proof of Theorem dochexmidlem7**
Step	Hyp	Ref	Expression
1		dochexmidlem1.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
2		dochexmidlem1.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		dochexmidlem1.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 41090	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		dochexmidlem1.s	. . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈)
6	5	lsssssubg 20948	. . . . . 6 ⊢ (𝑈 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑈))
7	4, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑈))
8		dochexmidlem1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
9	7, 8	sseldd 3983	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘𝑈))
10		dochexmidlem1.a	. . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑈)
11		dochexmidlem6.pp	. . . . . 6 ⊢ (𝜑 → 𝑝 ∈ 𝐴)
12	5, 10, 4, 11	lsatlssel 38976	. . . . 5 ⊢ (𝜑 → 𝑝 ∈ 𝑆)
13	7, 12	sseldd 3983	. . . 4 ⊢ (𝜑 → 𝑝 ∈ (SubGrp‘𝑈))
14		dochexmidlem1.p	. . . . 5 ⊢ ⊕ = (LSSum‘𝑈)
15	14	lsmub2 19672	. . . 4 ⊢ ((𝑋 ∈ (SubGrp‘𝑈) ∧ 𝑝 ∈ (SubGrp‘𝑈)) → 𝑝 ⊆ (𝑋 ⊕ 𝑝))
16	9, 13, 15	syl2anc 584	. . 3 ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ 𝑝))
17		dochexmidlem6.m	. . 3 ⊢ 𝑀 = (𝑋 ⊕ 𝑝)
18	16, 17	sseqtrrdi 4024	. 2 ⊢ (𝜑 → 𝑝 ⊆ 𝑀)
19		dochexmidlem6.pl	. . 3 ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
20		dochexmidlem1.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
21	20, 5	lssss 20926	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉)
22	8, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
23		dochexmidlem1.o	. . . . . . . 8 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
24	1, 2, 20, 5, 23	dochlss 41334	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆)
25	3, 22, 24	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆)
26	7, 25	sseldd 3983	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ (SubGrp‘𝑈))
27	14	lsmub1 19671	. . . . 5 ⊢ ((𝑋 ∈ (SubGrp‘𝑈) ∧ ( ⊥ ‘𝑋) ∈ (SubGrp‘𝑈)) → 𝑋 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
28	9, 26, 27	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
29		sstr2 3989	. . . 4 ⊢ (𝑝 ⊆ 𝑋 → (𝑋 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)) → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))))
30	28, 29	syl5com 31	. . 3 ⊢ (𝜑 → (𝑝 ⊆ 𝑋 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))))
31	19, 30	mtod 198	. 2 ⊢ (𝜑 → ¬ 𝑝 ⊆ 𝑋)
32		sseq2 4009	. . . 4 ⊢ (𝑀 = 𝑋 → (𝑝 ⊆ 𝑀 ↔ 𝑝 ⊆ 𝑋))
33	32	biimpcd 249	. . 3 ⊢ (𝑝 ⊆ 𝑀 → (𝑀 = 𝑋 → 𝑝 ⊆ 𝑋))
34	33	necon3bd 2953	. 2 ⊢ (𝑝 ⊆ 𝑀 → (¬ 𝑝 ⊆ 𝑋 → 𝑀 ≠ 𝑋))
35	18, 31, 34	sylc 65	1 ⊢ (𝜑 → 𝑀 ≠ 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 ⊆ wss 3950 {csn 4624 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 0_gc0g 17480 SubGrpcsubg 19134 LSSumclsm 19648 LModclmod 20850 LSubSpclss 20921 LSpanclspn 20961 LSAtomsclsa 38953 HLchlt 39329 LHypclh 39964 DVecHcdvh 41058 ocHcoch 41327

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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-riotaBAD 38932

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-tpos 8247 df-undef 8294 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-map 8864 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-n0 12523 df-z 12610 df-uz 12875 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-sca 17309 df-vsca 17310 df-0g 17482 df-proset 18336 df-poset 18355 df-plt 18371 df-lub 18387 df-glb 18388 df-join 18389 df-meet 18390 df-p0 18466 df-p1 18467 df-lat 18473 df-clat 18540 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-submnd 18793 df-grp 18950 df-minusg 18951 df-sbg 18952 df-subg 19137 df-cntz 19331 df-lsm 19650 df-cmn 19796 df-abl 19797 df-mgp 20134 df-rng 20146 df-ur 20175 df-ring 20228 df-oppr 20326 df-dvdsr 20349 df-unit 20350 df-invr 20380 df-dvr 20393 df-drng 20723 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lvec 21094 df-lsatoms 38955 df-oposet 39155 df-ol 39157 df-oml 39158 df-covers 39245 df-ats 39246 df-atl 39277 df-cvlat 39301 df-hlat 39330 df-llines 39478 df-lplanes 39479 df-lvols 39480 df-lines 39481 df-psubsp 39483 df-pmap 39484 df-padd 39776 df-lhyp 39968 df-laut 39969 df-ldil 40084 df-ltrn 40085 df-trl 40139 df-tendo 40735 df-edring 40737 df-disoa 41009 df-dvech 41059 df-dib 41119 df-dic 41153 df-dih 41209 df-doch 41328

This theorem is referenced by: dochexmidlem8 41447

W3C validator