dochexmidlem7 - Mathbox for Norm Megill

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Theorem dochexmidlem7 41410

Description: Lemma for dochexmid 41412. Contradict dochexmidlem6 41409. (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 41054	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		dochexmidlem1.s	. . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈)
6	5	lsssssubg 20955	. . . . . 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 3996	. . . 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 38940	. . . . 5 ⊢ (𝜑 → 𝑝 ∈ 𝑆)
13	7, 12	sseldd 3996	. . . 4 ⊢ (𝜑 → 𝑝 ∈ (SubGrp‘𝑈))
14		dochexmidlem1.p	. . . . 5 ⊢ ⊕ = (LSSum‘𝑈)
15	14	lsmub2 19676	. . . 4 ⊢ ((𝑋 ∈ (SubGrp‘𝑈) ∧ 𝑝 ∈ (SubGrp‘𝑈)) → 𝑝 ⊆ (𝑋 ⊕ 𝑝))
16	9, 13, 15	syl2anc 583	. . 3 ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ 𝑝))
17		dochexmidlem6.m	. . 3 ⊢ 𝑀 = (𝑋 ⊕ 𝑝)
18	16, 17	sseqtrrdi 4047	. 2 ⊢ (𝜑 → 𝑝 ⊆ 𝑀)
19		dochexmidlem6.pl	. . 3 ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
20		dochexmidlem1.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
21	20, 5	lssss 20933	. . . . . . . 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 41298	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆)
25	3, 22, 24	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆)
26	7, 25	sseldd 3996	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ (SubGrp‘𝑈))
27	14	lsmub1 19675	. . . . 5 ⊢ ((𝑋 ∈ (SubGrp‘𝑈) ∧ ( ⊥ ‘𝑋) ∈ (SubGrp‘𝑈)) → 𝑋 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
28	9, 26, 27	syl2anc 583	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
29		sstr2 4002	. . . 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 4022	. . . 4 ⊢ (𝑀 = 𝑋 → (𝑝 ⊆ 𝑀 ↔ 𝑝 ⊆ 𝑋))
33	32	biimpcd 249	. . 3 ⊢ (𝑝 ⊆ 𝑀 → (𝑀 = 𝑋 → 𝑝 ⊆ 𝑋))
34	33	necon3bd 2950	. 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 1535 ∈ wcel 2104 ≠ wne 2936 ⊆ wss 3963 {csn 4630 ‘cfv 6558 (class class class)co 7425 Basecbs 17234 0_gc0g 17475 SubGrpcsubg 19136 LSSumclsm 19652 LModclmod 20856 LSubSpclss 20928 LSpanclspn 20968 LSAtomsclsa 38917 HLchlt 39293 LHypclh 39928 DVecHcdvh 41022 ocHcoch 41291

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-riotaBAD 38896

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4915 df-int 4954 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-1st 8007 df-2nd 8008 df-tpos 8244 df-undef 8291 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-1o 8499 df-er 8738 df-map 8861 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-n0 12518 df-z 12605 df-uz 12870 df-fz 13538 df-struct 17170 df-sets 17187 df-slot 17205 df-ndx 17217 df-base 17235 df-ress 17264 df-plusg 17300 df-mulr 17301 df-sca 17303 df-vsca 17304 df-0g 17477 df-proset 18341 df-poset 18359 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-p1 18472 df-lat 18478 df-clat 18545 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18795 df-grp 18952 df-minusg 18953 df-sbg 18954 df-subg 19139 df-cntz 19333 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20156 df-ur 20185 df-ring 20238 df-oppr 20336 df-dvdsr 20359 df-unit 20360 df-invr 20390 df-dvr 20403 df-drng 20729 df-lmod 20858 df-lss 20929 df-lsp 20969 df-lvec 21101 df-lsatoms 38919 df-oposet 39119 df-ol 39121 df-oml 39122 df-covers 39209 df-ats 39210 df-atl 39241 df-cvlat 39265 df-hlat 39294 df-llines 39442 df-lplanes 39443 df-lvols 39444 df-lines 39445 df-psubsp 39447 df-pmap 39448 df-padd 39740 df-lhyp 39932 df-laut 39933 df-ldil 40048 df-ltrn 40049 df-trl 40103 df-tendo 40699 df-edring 40701 df-disoa 40973 df-dvech 41023 df-dib 41083 df-dic 41117 df-dih 41173 df-doch 41292

This theorem is referenced by: dochexmidlem8 41411

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