dochexmidlem6 - Mathbox for Norm Megill

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Theorem dochexmidlem6 39479

Description: Lemma for dochexmid 39482. (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
dochexmidlem6	⊢ (𝜑 → 𝑀 = 𝑋)

**Proof of Theorem dochexmidlem6**
Step	Hyp	Ref	Expression
1		dochexmidlem1.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
2		dochexmidlem1.o	. . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		dochexmidlem1.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		dochexmidlem1.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
5		dochexmidlem1.s	. . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈)
6		dochexmidlem1.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
7		dochexmidlem1.p	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝑈)
8		dochexmidlem1.a	. . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑈)
9		dochexmidlem1.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		dochexmidlem1.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
11		dochexmidlem6.pp	. . . . . . 7 ⊢ (𝜑 → 𝑝 ∈ 𝐴)
12		dochexmidlem6.z	. . . . . . 7 ⊢ 0 = (0_g‘𝑈)
13		dochexmidlem6.m	. . . . . . 7 ⊢ 𝑀 = (𝑋 ⊕ 𝑝)
14		dochexmidlem6.xn	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ { 0 })
15		dochexmidlem6.pl	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	dochexmidlem5 39478	. . . . . 6 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 })
17	16	fveq2d 6778	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = ( ⊥ ‘{ 0 }))
18	1, 3, 2, 4, 12	doch0 39372	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = 𝑉)
19	9, 18	syl 17	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{ 0 }) = 𝑉)
20	17, 19	eqtrd 2778	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = 𝑉)
21	20	ineq1d 4145	. . 3 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = (𝑉 ∩ 𝑀))
22		eqid 2738	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
23		dochexmidlem6.c	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
24	4, 5	lssss 20198	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉)
25	10, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
26	1, 3, 4, 2	dochssv 39369	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉)
27	9, 25, 26	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ 𝑉)
28	1, 22, 3, 4, 2	dochcl 39367	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
29	9, 27, 28	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
30	23, 29	eqeltrrd 2840	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ran ((DIsoH‘𝐾)‘𝑊))
31	1, 22, 3, 7, 8, 9, 30, 11	dihsmatrn 39450	. . . . . 6 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	13, 31	eqeltrid 2843	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	1, 3, 22, 5	dihrnlss 39291	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑀 ∈ 𝑆)
34	9, 32, 33	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑆)
35	1, 3, 9	dvhlmod 39124	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
36	5, 8, 35, 11	lsatlssel 37011	. . . . . . . . 9 ⊢ (𝜑 → 𝑝 ∈ 𝑆)
37	5, 7	lsmcl 20345	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑝 ∈ 𝑆) → (𝑋 ⊕ 𝑝) ∈ 𝑆)
38	35, 10, 36, 37	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ 𝑆)
39	4, 5	lssss 20198	. . . . . . . 8 ⊢ ((𝑋 ⊕ 𝑝) ∈ 𝑆 → (𝑋 ⊕ 𝑝) ⊆ 𝑉)
40	38, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ⊆ 𝑉)
41	13, 40	eqsstrid 3969	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ 𝑉)
42	1, 22, 3, 4, 2, 9, 41	dochoccl 39383	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀))
43	32, 42	mpbid 231	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀)
44	5	lsssssubg 20220	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑈))
45	35, 44	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑈))
46	45, 10	sseldd 3922	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘𝑈))
47	45, 36	sseldd 3922	. . . . . 6 ⊢ (𝜑 → 𝑝 ∈ (SubGrp‘𝑈))
48	7	lsmub1 19262	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑈) ∧ 𝑝 ∈ (SubGrp‘𝑈)) → 𝑋 ⊆ (𝑋 ⊕ 𝑝))
49	46, 47, 48	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ (𝑋 ⊕ 𝑝))
50	49, 13	sseqtrrdi 3972	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑀)
51	1, 3, 5, 2, 9, 10, 34, 43, 50	dihoml4 39391	. . 3 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = ( ⊥ ‘( ⊥ ‘𝑋)))
52		sseqin2 4149	. . . 4 ⊢ (𝑀 ⊆ 𝑉 ↔ (𝑉 ∩ 𝑀) = 𝑀)
53	41, 52	sylib 217	. . 3 ⊢ (𝜑 → (𝑉 ∩ 𝑀) = 𝑀)
54	21, 51, 53	3eqtr3rd 2787	. 2 ⊢ (𝜑 → 𝑀 = ( ⊥ ‘( ⊥ ‘𝑋)))
55	54, 23	eqtrd 2778	1 ⊢ (𝜑 → 𝑀 = 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∩ cin 3886 ⊆ wss 3887 {csn 4561 ran crn 5590 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 0_gc0g 17150 SubGrpcsubg 18749 LSSumclsm 19239 LModclmod 20123 LSubSpclss 20193 LSpanclspn 20233 LSAtomsclsa 36988 HLchlt 37364 LHypclh 37998 DVecHcdvh 39092 DIsoHcdih 39242 ocHcoch 39361

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 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 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-mre 17295 df-mrc 17296 df-acs 17298 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-oppg 18950 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-lcv 37033 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tgrp 38757 df-tendo 38769 df-edring 38771 df-dveca 39017 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243 df-doch 39362 df-djh 39409

This theorem is referenced by: dochexmidlem8 39481

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