dochexmidlem6 - Mathbox for Norm Megill

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Theorem dochexmidlem6 40938

Description: Lemma for dochexmid 40941. (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 40937	. . . . . 6 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 })
17	16	fveq2d 6901	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = ( ⊥ ‘{ 0 }))
18	1, 3, 2, 4, 12	doch0 40831	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = 𝑉)
19	9, 18	syl 17	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{ 0 }) = 𝑉)
20	17, 19	eqtrd 2768	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = 𝑉)
21	20	ineq1d 4211	. . 3 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = (𝑉 ∩ 𝑀))
22		eqid 2728	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
23		dochexmidlem6.c	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
24	4, 5	lssss 20819	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉)
25	10, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
26	1, 3, 4, 2	dochssv 40828	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉)
27	9, 25, 26	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ 𝑉)
28	1, 22, 3, 4, 2	dochcl 40826	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
29	9, 27, 28	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
30	23, 29	eqeltrrd 2830	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ran ((DIsoH‘𝐾)‘𝑊))
31	1, 22, 3, 7, 8, 9, 30, 11	dihsmatrn 40909	. . . . . 6 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	13, 31	eqeltrid 2833	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	1, 3, 22, 5	dihrnlss 40750	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑀 ∈ 𝑆)
34	9, 32, 33	syl2anc 583	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑆)
35	1, 3, 9	dvhlmod 40583	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
36	5, 8, 35, 11	lsatlssel 38469	. . . . . . . . 9 ⊢ (𝜑 → 𝑝 ∈ 𝑆)
37	5, 7	lsmcl 20967	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑝 ∈ 𝑆) → (𝑋 ⊕ 𝑝) ∈ 𝑆)
38	35, 10, 36, 37	syl3anc 1369	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ 𝑆)
39	4, 5	lssss 20819	. . . . . . . 8 ⊢ ((𝑋 ⊕ 𝑝) ∈ 𝑆 → (𝑋 ⊕ 𝑝) ⊆ 𝑉)
40	38, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ⊆ 𝑉)
41	13, 40	eqsstrid 4028	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ 𝑉)
42	1, 22, 3, 4, 2, 9, 41	dochoccl 40842	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀))
43	32, 42	mpbid 231	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀)
44	5	lsssssubg 20841	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑈))
45	35, 44	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑈))
46	45, 10	sseldd 3981	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘𝑈))
47	45, 36	sseldd 3981	. . . . . 6 ⊢ (𝜑 → 𝑝 ∈ (SubGrp‘𝑈))
48	7	lsmub1 19611	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑈) ∧ 𝑝 ∈ (SubGrp‘𝑈)) → 𝑋 ⊆ (𝑋 ⊕ 𝑝))
49	46, 47, 48	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ (𝑋 ⊕ 𝑝))
50	49, 13	sseqtrrdi 4031	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑀)
51	1, 3, 5, 2, 9, 10, 34, 43, 50	dihoml4 40850	. . 3 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = ( ⊥ ‘( ⊥ ‘𝑋)))
52		sseqin2 4215	. . . 4 ⊢ (𝑀 ⊆ 𝑉 ↔ (𝑉 ∩ 𝑀) = 𝑀)
53	41, 52	sylib 217	. . 3 ⊢ (𝜑 → (𝑉 ∩ 𝑀) = 𝑀)
54	21, 51, 53	3eqtr3rd 2777	. 2 ⊢ (𝜑 → 𝑀 = ( ⊥ ‘( ⊥ ‘𝑋)))
55	54, 23	eqtrd 2768	1 ⊢ (𝜑 → 𝑀 = 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∩ cin 3946 ⊆ wss 3947 {csn 4629 ran crn 5679 ‘cfv 6548 (class class class)co 7420 Basecbs 17179 0_gc0g 17420 SubGrpcsubg 19074 LSSumclsm 19588 LModclmod 20742 LSubSpclss 20814 LSpanclspn 20854 LSAtomsclsa 38446 HLchlt 38822 LHypclh 39457 DVecHcdvh 40551 DIsoHcdih 40701 ocHcoch 40820

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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-riotaBAD 38425

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8231 df-undef 8278 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-0g 17422 df-mre 17565 df-mrc 17566 df-acs 17568 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-cntz 19267 df-oppg 19296 df-lsm 19590 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-drng 20625 df-lmod 20744 df-lss 20815 df-lsp 20855 df-lvec 20987 df-lsatoms 38448 df-lcv 38491 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-tgrp 40216 df-tendo 40228 df-edring 40230 df-dveca 40476 df-disoa 40502 df-dvech 40552 df-dib 40612 df-dic 40646 df-dih 40702 df-doch 40821 df-djh 40868

This theorem is referenced by: dochexmidlem8 40940

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