dochexmidlem6 - Mathbox for Norm Megill

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Theorem dochexmidlem6 41925

Description: Lemma for dochexmid 41928. (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 41924	. . . . . 6 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 })
17	16	fveq2d 6838	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = ( ⊥ ‘{ 0 }))
18	1, 3, 2, 4, 12	doch0 41818	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = 𝑉)
19	9, 18	syl 17	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{ 0 }) = 𝑉)
20	17, 19	eqtrd 2772	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) = 𝑉)
21	20	ineq1d 4160	. . 3 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = (𝑉 ∩ 𝑀))
22		eqid 2737	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
23		dochexmidlem6.c	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
24	4, 5	lssss 20922	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉)
25	10, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
26	1, 3, 4, 2	dochssv 41815	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉)
27	9, 25, 26	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ 𝑉)
28	1, 22, 3, 4, 2	dochcl 41813	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ 𝑉) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
29	9, 27, 28	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
30	23, 29	eqeltrrd 2838	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ran ((DIsoH‘𝐾)‘𝑊))
31	1, 22, 3, 7, 8, 9, 30, 11	dihsmatrn 41896	. . . . . 6 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	13, 31	eqeltrid 2841	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	1, 3, 22, 5	dihrnlss 41737	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑀 ∈ 𝑆)
34	9, 32, 33	syl2anc 585	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑆)
35	1, 3, 9	dvhlmod 41570	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
36	5, 8, 35, 11	lsatlssel 39457	. . . . . . . . 9 ⊢ (𝜑 → 𝑝 ∈ 𝑆)
37	5, 7	lsmcl 21070	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑝 ∈ 𝑆) → (𝑋 ⊕ 𝑝) ∈ 𝑆)
38	35, 10, 36, 37	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ 𝑆)
39	4, 5	lssss 20922	. . . . . . . 8 ⊢ ((𝑋 ⊕ 𝑝) ∈ 𝑆 → (𝑋 ⊕ 𝑝) ⊆ 𝑉)
40	38, 39	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ⊆ 𝑉)
41	13, 40	eqsstrid 3961	. . . . . 6 ⊢ (𝜑 → 𝑀 ⊆ 𝑉)
42	1, 22, 3, 4, 2, 9, 41	dochoccl 41829	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀))
43	32, 42	mpbid 232	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑀)) = 𝑀)
44	5	lsssssubg 20944	. . . . . . . 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 19623	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑈) ∧ 𝑝 ∈ (SubGrp‘𝑈)) → 𝑋 ⊆ (𝑋 ⊕ 𝑝))
49	46, 47, 48	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ (𝑋 ⊕ 𝑝))
50	49, 13	sseqtrrdi 3964	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑀)
51	1, 3, 5, 2, 9, 10, 34, 43, 50	dihoml4 41837	. . 3 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑀)) ∩ 𝑀) = ( ⊥ ‘( ⊥ ‘𝑋)))
52		sseqin2 4164	. . . 4 ⊢ (𝑀 ⊆ 𝑉 ↔ (𝑉 ∩ 𝑀) = 𝑀)
53	41, 52	sylib 218	. . 3 ⊢ (𝜑 → (𝑉 ∩ 𝑀) = 𝑀)
54	21, 51, 53	3eqtr3rd 2781	. 2 ⊢ (𝜑 → 𝑀 = ( ⊥ ‘( ⊥ ‘𝑋)))
55	54, 23	eqtrd 2772	1 ⊢ (𝜑 → 𝑀 = 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∩ cin 3889 ⊆ wss 3890 {csn 4568 ran crn 5625 ‘cfv 6492 (class class class)co 7360 Basecbs 17170 0_gc0g 17393 SubGrpcsubg 19087 LSSumclsm 19600 LModclmod 20846 LSubSpclss 20917 LSpanclspn 20957 LSAtomsclsa 39434 HLchlt 39810 LHypclh 40444 DVecHcdvh 41538 DIsoHcdih 41688 ocHcoch 41807

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-riotaBAD 39413

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 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 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8169 df-undef 8216 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-0g 17395 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18251 df-poset 18270 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18389 df-clat 18456 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-cntz 19283 df-oppg 19312 df-lsm 19602 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-drng 20699 df-lmod 20848 df-lss 20918 df-lsp 20958 df-lvec 21090 df-lsatoms 39436 df-lcv 39479 df-oposet 39636 df-ol 39638 df-oml 39639 df-covers 39726 df-ats 39727 df-atl 39758 df-cvlat 39782 df-hlat 39811 df-llines 39958 df-lplanes 39959 df-lvols 39960 df-lines 39961 df-psubsp 39963 df-pmap 39964 df-padd 40256 df-lhyp 40448 df-laut 40449 df-ldil 40564 df-ltrn 40565 df-trl 40619 df-tgrp 41203 df-tendo 41215 df-edring 41217 df-dveca 41463 df-disoa 41489 df-dvech 41539 df-dib 41599 df-dic 41633 df-dih 41689 df-doch 41808 df-djh 41855

This theorem is referenced by: dochexmidlem8 41927

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