dochexmidlem6 - Mathbox for Norm Megill

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Theorem dochexmidlem6 42037

Description: Lemma for dochexmid 42040. (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 42036	. . . . . 6 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 })
17	16	fveq2d 6860	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = ( ⊥ ‘{ 0 }))
18	1, 3, 2, 4, 12	doch0 41930	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = 𝑉)
19	9, 18	syl 17	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{ 0 }) = 𝑉)
20	17, 19	eqtrd 2791	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = 𝑉)
21	20	ineq1d 4166	. . 3 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = (𝑉 ∩ 𝑀))
22		eqid 2756	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
23		dochexmidlem6.c	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
24	4, 5	lssss 20976	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉)
25	10, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
26	1, 3, 4, 2	dochssv 41927	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉)
27	9, 25, 26	syl2anc 592	. . . . . . . . 9 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ 𝑉)
28	1, 22, 3, 4, 2	dochcl 41925	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
29	9, 27, 28	syl2anc 592	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
30	23, 29	eqeltrrd 2857	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ran ((DIsoH‘𝐾)‘𝑊))
31	1, 22, 3, 7, 8, 9, 30, 11	dihsmatrn 42008	. . . . . 6 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	13, 31	eqeltrid 2860	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	1, 3, 22, 5	dihrnlss 41849	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑀 ∈ 𝑆)
34	9, 32, 33	syl2anc 592	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑆)
35	1, 3, 9	dvhlmod 41682	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
36	5, 8, 35, 11	lsatlssel 39569	. . . . . . . . 9 ⊢ (𝜑 → 𝑝 ∈ 𝑆)
37	5, 7	lsmcl 21123	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑝 ∈ 𝑆) → (𝑋 ⊕ 𝑝) ∈ 𝑆)
38	35, 10, 36, 37	syl3anc 1386	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ 𝑆)
39	4, 5	lssss 20976	. . . . . . . 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 41941	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀))
43	32, 42	mpbid 234	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀)
44	5	lsssssubg 20998	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑈))
45	35, 44	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑈))
46	45, 10	sseldd 3932	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘𝑈))
47	45, 36	sseldd 3932	. . . . . 6 ⊢ (𝜑 → 𝑝 ∈ (SubGrp‘𝑈))
48	7	lsmub1 19673	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑈) ∧ 𝑝 ∈ (SubGrp‘𝑈)) → 𝑋 ⊆ (𝑋 ⊕ 𝑝))
49	46, 47, 48	syl2anc 592	. . . . 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 41949	. . 3 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = ( ⊥ ‘( ⊥ ‘𝑋)))
52		sseqin2 4170	. . . 4 ⊢ (𝑀 ⊆ 𝑉 ↔ (𝑉 ∩ 𝑀) = 𝑀)
53	41, 52	sylib 220	. . 3 ⊢ (𝜑 → (𝑉 ∩ 𝑀) = 𝑀)
54	21, 51, 53	3eqtr3rd 2800	. 2 ⊢ (𝜑 → 𝑀 = ( ⊥ ‘( ⊥ ‘𝑋)))
55	54, 23	eqtrd 2791	1 ⊢ (𝜑 → 𝑀 = 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1554 ∈ wcel 2136 ≠ wne 2951 ∩ cin 3898 ⊆ wss 3899 {csn 4576 ran crn 5641 ‘cfv 6510 (class class class)co 7385 Basecbs 17221 0_gc0g 17444 SubGrpcsubg 19138 LSSumclsm 19650 LModclmod 20900 LSubSpclss 20971 LSpanclspn 21011 LSAtomsclsa 39546 HLchlt 39922 LHypclh 40556 DVecHcdvh 41650 DIsoHcdih 41800 ocHcoch 41919

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140 ax-riotaBAD 39525

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-iin 4946 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-om 7836 df-1st 7959 df-2nd 7960 df-tpos 8194 df-undef 8241 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-2o 8426 df-er 8666 df-map 8798 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-nn 12201 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-n0 12472 df-z 12559 df-uz 12830 df-fz 13503 df-struct 17159 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-mulr 17276 df-sca 17278 df-vsca 17279 df-0g 17446 df-mre 17590 df-mrc 17591 df-acs 17593 df-proset 18302 df-poset 18321 df-plt 18336 df-lub 18352 df-glb 18353 df-join 18354 df-meet 18355 df-p0 18431 df-p1 18432 df-lat 18440 df-clat 18507 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-submnd 18794 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19333 df-oppg 19362 df-lsm 19652 df-cmn 19798 df-abl 19799 df-mgp 20163 df-rng 20175 df-ur 20204 df-ring 20257 df-oppr 20358 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-dvr 20422 df-drng 20753 df-lmod 20902 df-lss 20972 df-lsp 21012 df-lvec 21143 df-lsatoms 39548 df-lcv 39591 df-oposet 39748 df-ol 39750 df-oml 39751 df-covers 39838 df-ats 39839 df-atl 39870 df-cvlat 39894 df-hlat 39923 df-llines 40070 df-lplanes 40071 df-lvols 40072 df-lines 40073 df-psubsp 40075 df-pmap 40076 df-padd 40368 df-lhyp 40560 df-laut 40561 df-ldil 40676 df-ltrn 40677 df-trl 40731 df-tgrp 41315 df-tendo 41327 df-edring 41329 df-dveca 41575 df-disoa 41601 df-dvech 41651 df-dib 41711 df-dic 41745 df-dih 41801 df-doch 41920 df-djh 41967

This theorem is referenced by: dochexmidlem8 42039

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