dochexmidlem4 - Mathbox for Norm Megill

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Theorem dochexmidlem4 41833

Description: Lemma for dochexmid 41838. (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)
dochexmidlem4.pp	⊢ (𝜑 → 𝑝 ∈ 𝐴)
dochexmidlem4.qq	⊢ (𝜑 → 𝑞 ∈ 𝐴)
dochexmidlem4.z	⊢ 0 = (0_g‘𝑈)
dochexmidlem4.m	⊢ 𝑀 = (𝑋 ⊕ 𝑝)
dochexmidlem4.xn	⊢ (𝜑 → 𝑋 ≠ { 0 })
dochexmidlem4.pl	⊢ (𝜑 → 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀))

**Assertion**
Ref	Expression
dochexmidlem4	⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))

Proof of Theorem dochexmidlem4 Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dochexmidlem4.z	. . 3 ⊢ 0 = (0_g‘𝑈)
2		dochexmidlem1.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑈)
3		dochexmidlem1.p	. . 3 ⊢ ⊕ = (LSSum‘𝑈)
4		dochexmidlem1.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑈)
5		dochexmidlem1.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
6		dochexmidlem1.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		dochexmidlem1.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8	5, 6, 7	dvhlmod 41480	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
9		dochexmidlem1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
10		dochexmidlem4.pp	. . . 4 ⊢ (𝜑 → 𝑝 ∈ 𝐴)
11	2, 4, 8, 10	lsatlssel 39367	. . 3 ⊢ (𝜑 → 𝑝 ∈ 𝑆)
12		dochexmidlem4.qq	. . 3 ⊢ (𝜑 → 𝑞 ∈ 𝐴)
13		dochexmidlem4.xn	. . 3 ⊢ (𝜑 → 𝑋 ≠ { 0 })
14		dochexmidlem4.pl	. . . . 5 ⊢ (𝜑 → 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀))
15		inss2 4192	. . . . 5 ⊢ (( ⊥ ‘𝑋) ∩ 𝑀) ⊆ 𝑀
16	14, 15	sstrdi 3948	. . . 4 ⊢ (𝜑 → 𝑞 ⊆ 𝑀)
17		dochexmidlem4.m	. . . 4 ⊢ 𝑀 = (𝑋 ⊕ 𝑝)
18	16, 17	sseqtrdi 3976	. . 3 ⊢ (𝜑 → 𝑞 ⊆ (𝑋 ⊕ 𝑝))
19	1, 2, 3, 4, 8, 9, 11, 12, 13, 18	lsmsat 39378	. 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝐴 (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ (𝑟 ⊕ 𝑝)))
20		dochexmidlem1.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
21		dochexmidlem1.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
22		dochexmidlem1.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
23	7	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ (𝑟 ⊕ 𝑝))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
24	9	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ (𝑟 ⊕ 𝑝))) → 𝑋 ∈ 𝑆)
25	10	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ (𝑟 ⊕ 𝑝))) → 𝑝 ∈ 𝐴)
26	12	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ (𝑟 ⊕ 𝑝))) → 𝑞 ∈ 𝐴)
27		simp2 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ (𝑟 ⊕ 𝑝))) → 𝑟 ∈ 𝐴)
28		inss1 4191	. . . . . 6 ⊢ (( ⊥ ‘𝑋) ∩ 𝑀) ⊆ ( ⊥ ‘𝑋)
29	14, 28	sstrdi 3948	. . . . 5 ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋))
30	29	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ (𝑟 ⊕ 𝑝))) → 𝑞 ⊆ ( ⊥ ‘𝑋))
31		simp3l 1203	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ (𝑟 ⊕ 𝑝))) → 𝑟 ⊆ 𝑋)
32		simp3r 1204	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ (𝑟 ⊕ 𝑝))) → 𝑞 ⊆ (𝑟 ⊕ 𝑝))
33	5, 20, 6, 21, 2, 22, 3, 4, 23, 24, 25, 26, 27, 30, 31, 32	dochexmidlem3 41832	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐴 ∧ (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ (𝑟 ⊕ 𝑝))) → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
34	33	rexlimdv3a 3143	. 2 ⊢ (𝜑 → (∃𝑟 ∈ 𝐴 (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ (𝑟 ⊕ 𝑝)) → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))))
35	19, 34	mpd 15	1 ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 ∩ cin 3902 ⊆ wss 3903 {csn 4582 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 0_gc0g 17371 LSSumclsm 19575 LSubSpclss 20894 LSpanclspn 20934 LSAtomsclsa 39344 HLchlt 39720 LHypclh 40354 DVecHcdvh 41448 ocHcoch 41717

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39323

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-undef 8225 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-0g 17373 df-mre 17517 df-mrc 17518 df-acs 17520 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cntz 19258 df-oppg 19287 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-drng 20676 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lvec 21067 df-lsatoms 39346 df-lcv 39389 df-oposet 39546 df-ol 39548 df-oml 39549 df-covers 39636 df-ats 39637 df-atl 39668 df-cvlat 39692 df-hlat 39721 df-llines 39868 df-lplanes 39869 df-lvols 39870 df-lines 39871 df-psubsp 39873 df-pmap 39874 df-padd 40166 df-lhyp 40358 df-laut 40359 df-ldil 40474 df-ltrn 40475 df-trl 40529 df-tendo 41125 df-edring 41127 df-disoa 41399 df-dvech 41449 df-dib 41509 df-dic 41543 df-dih 41599 df-doch 41718

This theorem is referenced by: dochexmidlem5 41834

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