dochexmidlem6 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem6		Structured version Visualization version GIF version

Theorem dochexmidlem6 41964

Description: Lemma for dochexmid 41967. (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 41963	. . . . . 6 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 })
17	16	fveq2d 6838	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = ( ⊥ ‘{ 0 }))
18	1, 3, 2, 4, 12	doch0 41857	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = 𝑉)
19	9, 18	syl 17	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{ 0 }) = 𝑉)
20	17, 19	eqtrd 2775	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = 𝑉)
21	20	ineq1d 4155	. . 3 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = (𝑉 ∩ 𝑀))
22		eqid 2740	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
23		dochexmidlem6.c	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
24	4, 5	lssss 20933	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉)
25	10, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
26	1, 3, 4, 2	dochssv 41854	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉)
27	9, 25, 26	syl2anc 590	. . . . . . . . 9 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ 𝑉)
28	1, 22, 3, 4, 2	dochcl 41852	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
29	9, 27, 28	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
30	23, 29	eqeltrrd 2841	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ran ((DIsoH‘𝐾)‘𝑊))
31	1, 22, 3, 7, 8, 9, 30, 11	dihsmatrn 41935	. . . . . 6 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	13, 31	eqeltrid 2844	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	1, 3, 22, 5	dihrnlss 41776	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑀 ∈ 𝑆)
34	9, 32, 33	syl2anc 590	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑆)
35	1, 3, 9	dvhlmod 41609	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
36	5, 8, 35, 11	lsatlssel 39496	. . . . . . . . 9 ⊢ (𝜑 → 𝑝 ∈ 𝑆)
37	5, 7	lsmcl 21080	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑝 ∈ 𝑆) → (𝑋 ⊕ 𝑝) ∈ 𝑆)
38	35, 10, 36, 37	syl3anc 1379	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ 𝑆)
39	4, 5	lssss 20933	. . . . . . . 8 ⊢ ((𝑋 ⊕ 𝑝) ∈ 𝑆 → (𝑋 ⊕ 𝑝) ⊆ 𝑉)
40	38, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ⊆ 𝑉)
41	13, 40	eqsstrid 3960	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ 𝑉)
42	1, 22, 3, 4, 2, 9, 41	dochoccl 41868	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀))
43	32, 42	mpbid 233	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀)
44	5	lsssssubg 20955	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑈))
45	35, 44	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑈))
46	45, 10	sseldd 3923	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘𝑈))
47	45, 36	sseldd 3923	. . . . . 6 ⊢ (𝜑 → 𝑝 ∈ (SubGrp‘𝑈))
48	7	lsmub1 19630	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑈) ∧ 𝑝 ∈ (SubGrp‘𝑈)) → 𝑋 ⊆ (𝑋 ⊕ 𝑝))
49	46, 47, 48	syl2anc 590	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ (𝑋 ⊕ 𝑝))
50	49, 13	sseqtrrdi 3963	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑀)
51	1, 3, 5, 2, 9, 10, 34, 43, 50	dihoml4 41876	. . 3 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = ( ⊥ ‘( ⊥ ‘𝑋)))
52		sseqin2 4159	. . . 4 ⊢ (𝑀 ⊆ 𝑉 ↔ (𝑉 ∩ 𝑀) = 𝑀)
53	41, 52	sylib 219	. . 3 ⊢ (𝜑 → (𝑉 ∩ 𝑀) = 𝑀)
54	21, 51, 53	3eqtr3rd 2784	. 2 ⊢ (𝜑 → 𝑀 = ( ⊥ ‘( ⊥ ‘𝑋)))
55	54, 23	eqtrd 2775	1 ⊢ (𝜑 → 𝑀 = 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∩ cin 3889 ⊆ wss 3890 {csn 4562 ran crn 5626 ‘cfv 6492 (class class class)co 7363 Basecbs 17177 0_gc0g 17400 SubGrpcsubg 19094 LSSumclsm 19607 LModclmod 20857 LSubSpclss 20928 LSpanclspn 20968 LSAtomsclsa 39473 HLchlt 39849 LHypclh 40483 DVecHcdvh 41577 DIsoHcdih 41727 ocHcoch 41846

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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-riotaBAD 39452

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-tpos 8173 df-undef 8220 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-0g 17402 df-mre 17546 df-mrc 17547 df-acs 17549 df-proset 18258 df-poset 18277 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18396 df-clat 18463 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-cntz 19290 df-oppg 19319 df-lsm 19609 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-drng 20710 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lvec 21100 df-lsatoms 39475 df-lcv 39518 df-oposet 39675 df-ol 39677 df-oml 39678 df-covers 39765 df-ats 39766 df-atl 39797 df-cvlat 39821 df-hlat 39850 df-llines 39997 df-lplanes 39998 df-lvols 39999 df-lines 40000 df-psubsp 40002 df-pmap 40003 df-padd 40295 df-lhyp 40487 df-laut 40488 df-ldil 40603 df-ltrn 40604 df-trl 40658 df-tgrp 41242 df-tendo 41254 df-edring 41256 df-dveca 41502 df-disoa 41528 df-dvech 41578 df-dib 41638 df-dic 41672 df-dih 41728 df-doch 41847 df-djh 41894

This theorem is referenced by: dochexmidlem8 41966

W3C validator