dochexmidlem8 - Mathbox for Norm Megill

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Theorem dochexmidlem8 42716

Description: Lemma for dochexmid 42717. The contradiction of dochexmidlem6 42714 and dochexmidlem7 42715 shows that there can be no atom 𝑝 that is not in 𝑋 + ( ⊥ ‘𝑋), which is therefore the whole atom space. (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)
dochexmidlem8.z	⊢ 0 = (0_g‘𝑈)
dochexmidlem8.xn	⊢ (𝜑 → 𝑋 ≠ { 0 })
dochexmidlem8.c	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)

**Assertion**
Ref	Expression
dochexmidlem8	⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉)

Proof of Theorem dochexmidlem8 Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nonconne 2971	. 2 ⊢ ¬ (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)
2		dochexmidlem1.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
3		dochexmidlem1.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		dochexmidlem1.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
5	2, 3, 4	dvhlmod 42359	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
6		dochexmidlem1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
7		dochexmidlem1.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
8		dochexmidlem1.s	. . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑈)
9	7, 8	lssss 21241	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉)
10	6, 9	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
11		dochexmidlem1.o	. . . . . . . 8 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
12	2, 3, 7, 8, 11	dochlss 42603	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆)
13	4, 10, 12	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆)
14		dochexmidlem1.p	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝑈)
15	8, 14	lsmcl 21388	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘𝑋) ∈ 𝑆) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆)
16	5, 6, 13, 15	syl3anc 1399	. . . . 5 ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆)
17	7, 8	lssss 21241	. . . . 5 ⊢ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆 → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉)
18	16, 17	syl 18	. . . 4 ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉)
19		dochexmidlem1.a	. . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑈)
20	5	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → 𝑈 ∈ LMod)
21	16	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆)
22	7, 8	lss1 21243	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑉 ∈ 𝑆)
23	5, 22	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑆)
24	23	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → 𝑉 ∈ 𝑆)
25		df-pss 4017	. . . . . . . 8 ⊢ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊊ 𝑉 ↔ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉))
26	25	bilanri 512	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊊ 𝑉)
27	8, 19, 20, 21, 24, 26	lpssat 40262	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))))
28	27	ex 418	. . . . 5 ⊢ (𝜑 → (((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))))
29		dochexmidlem1.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈)
30	4	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	6	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑋 ∈ 𝑆)
32		simp2 1155	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑝 ∈ 𝐴)
33		dochexmidlem8.z	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑈)
34		eqid 2763	. . . . . . . . 9 ⊢ (𝑋 ⊕ 𝑝) = (𝑋 ⊕ 𝑝)
35		dochexmidlem8.xn	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ { 0 })
36	35	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑋 ≠ { 0 })
37		dochexmidlem8.c	. . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
38	37	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
39		simp3 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
40	2, 11, 3, 7, 8, 29, 14, 19, 30, 31, 32, 33, 34, 36, 38, 39	dochexmidlem6 42714	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 ⊕ 𝑝) = 𝑋)
41	2, 11, 3, 7, 8, 29, 14, 19, 30, 31, 32, 33, 34, 36, 38, 39	dochexmidlem7 42715	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 ⊕ 𝑝) ≠ 𝑋)
42	40, 41	pm2.21ddne 3043	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))
43	42	3adant3l 1199	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))
44	43	rexlimdv3a 3171	. . . . 5 ⊢ (𝜑 → (∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))
45	28, 44	syld 48	. . . 4 ⊢ (𝜑 → (((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))
46	18, 45	mpand 708	. . 3 ⊢ (𝜑 → ((𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉 → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))
47	46	necon1bd 2977	. 2 ⊢ (𝜑 → (¬ (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋) → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉))
48	1, 47	mpi 21	1 ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∃wrex 3090 ⊆ wss 3900 ⊊ wpss 4016 {csn 4588 ‘cfv 6558 (class class class)co 7443 Basecbs 17416 0_gc0g 17639 LSSumclsm 19878 LModclmod 21165 LSubSpclss 21236 LSpanclspn 21276 LSAtomsclsa 40223 HLchlt 40599 LHypclh 41233 DVecHcdvh 42327 ocHcoch 42596

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-riotaBAD 40202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-tpos 8271 df-undef 8318 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-n0 12634 df-z 12721 df-uz 12993 df-fz 13667 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-sca 17473 df-vsca 17474 df-0g 17641 df-mre 17785 df-mrc 17786 df-acs 17788 df-proset 18497 df-poset 18516 df-plt 18531 df-lub 18547 df-glb 18548 df-join 18549 df-meet 18550 df-p0 18626 df-p1 18627 df-lat 18635 df-clat 18702 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-subg 19362 df-cntz 19561 df-oppg 19590 df-lsm 19880 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-ring 20491 df-oppr 20597 df-dvdsr 20617 df-unit 20618 df-invr 20648 df-dvr 20661 df-drng 21012 df-lmod 21167 df-lss 21237 df-lsp 21277 df-lvec 21408 df-lsatoms 40225 df-lcv 40268 df-oposet 40425 df-ol 40427 df-oml 40428 df-covers 40515 df-ats 40516 df-atl 40547 df-cvlat 40571 df-hlat 40600 df-llines 40747 df-lplanes 40748 df-lvols 40749 df-lines 40750 df-psubsp 40752 df-pmap 40753 df-padd 41045 df-lhyp 41237 df-laut 41238 df-ldil 41353 df-ltrn 41354 df-trl 41408 df-tgrp 41992 df-tendo 42004 df-edring 42006 df-dveca 42252 df-disoa 42278 df-dvech 42328 df-dib 42388 df-dic 42422 df-dih 42478 df-doch 42597 df-djh 42644

This theorem is used by: dochexmid 42717

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