dia2dimlem10 - Mathbox for Norm Megill

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Theorem dia2dimlem10 39467

Description: Lemma for dia2dim 39471. Convert membership in closed subspace (𝐼‘(𝑈 ∨ 𝑉)) to a lattice ordering. (Contributed by NM, 8-Sep-2014.)

**Hypotheses**
Ref	Expression
dia2dimlem10.l	⊢ ≤ = (le‘𝐾)
dia2dimlem10.j	⊢ ∨ = (join‘𝐾)
dia2dimlem10.a	⊢ 𝐴 = (Atoms‘𝐾)
dia2dimlem10.h	⊢ 𝐻 = (LHyp‘𝐾)
dia2dimlem10.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dia2dimlem10.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dia2dimlem10.y	⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
dia2dimlem10.s	⊢ 𝑆 = (LSubSp‘𝑌)
dia2dimlem10.n	⊢ 𝑁 = (LSpan‘𝑌)
dia2dimlem10.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
dia2dimlem10.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dia2dimlem10.u	⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
dia2dimlem10.v	⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
dia2dimlem10.f	⊢ (𝜑 → 𝐹 ∈ 𝑇)
dia2dimlem10.fe	⊢ (𝜑 → 𝐹 ∈ (𝐼‘(𝑈 ∨ 𝑉)))

**Assertion**
Ref	Expression
dia2dimlem10	⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))

**Proof of Theorem dia2dimlem10**
Step	Hyp	Ref	Expression
1		dia2dimlem10.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dia2dimlem10.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
3		dia2dimlem10.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		dia2dimlem10.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5		dia2dimlem10.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
6		dia2dimlem10.y	. . . . 5 ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
7		dia2dimlem10.i	. . . . 5 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
8		dia2dimlem10.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑌)
9	3, 4, 5, 6, 7, 8	dia1dim2 39456	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = (𝑁‘{𝐹}))
10	1, 2, 9	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐼‘(𝑅‘𝐹)) = (𝑁‘{𝐹}))
11		dia2dimlem10.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑌)
12	3, 6	dvalvec 39420	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑌 ∈ LVec)
13		lveclmod 20496	. . . . 5 ⊢ (𝑌 ∈ LVec → 𝑌 ∈ LMod)
14	1, 12, 13	3syl 18	. . . 4 ⊢ (𝜑 → 𝑌 ∈ LMod)
15	1	simpld 496	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
16		dia2dimlem10.u	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
17	16	simpld 496	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
18		dia2dimlem10.v	. . . . . . 7 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
19	18	simpld 496	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
20		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
21		dia2dimlem10.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
22		dia2dimlem10.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
23	20, 21, 22	hlatjcl 37760	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
24	15, 17, 19, 23	syl3anc 1372	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
25	16	simprd 497	. . . . . 6 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
26	18	simprd 497	. . . . . 6 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
27	15	hllatd 37757	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
28	20, 22	atbase 37682	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
29	17, 28	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
30	20, 22	atbase 37682	. . . . . . . 8 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
31	19, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
32	1	simprd 497	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
33	20, 3	lhpbase 38392	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
34	32, 33	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
35		dia2dimlem10.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
36	20, 35, 21	latjle12 18275	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑈 ≤ 𝑊 ∧ 𝑉 ≤ 𝑊) ↔ (𝑈 ∨ 𝑉) ≤ 𝑊))
37	27, 29, 31, 34, 36	syl13anc 1373	. . . . . 6 ⊢ (𝜑 → ((𝑈 ≤ 𝑊 ∧ 𝑉 ≤ 𝑊) ↔ (𝑈 ∨ 𝑉) ≤ 𝑊))
38	25, 26, 37	mpbi2and 711	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ 𝑊)
39	20, 35, 3, 6, 7, 11	dialss 39440	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑈 ∨ 𝑉) ∈ (Base‘𝐾) ∧ (𝑈 ∨ 𝑉) ≤ 𝑊)) → (𝐼‘(𝑈 ∨ 𝑉)) ∈ 𝑆)
40	1, 24, 38, 39	syl12anc 836	. . . 4 ⊢ (𝜑 → (𝐼‘(𝑈 ∨ 𝑉)) ∈ 𝑆)
41		dia2dimlem10.fe	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐼‘(𝑈 ∨ 𝑉)))
42	11, 8, 14, 40, 41	lspsnel5a 20386	. . 3 ⊢ (𝜑 → (𝑁‘{𝐹}) ⊆ (𝐼‘(𝑈 ∨ 𝑉)))
43	10, 42	eqsstrd 3981	. 2 ⊢ (𝜑 → (𝐼‘(𝑅‘𝐹)) ⊆ (𝐼‘(𝑈 ∨ 𝑉)))
44	20, 3, 4, 5	trlcl 38558	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
45	1, 2, 44	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑅‘𝐹) ∈ (Base‘𝐾))
46	35, 3, 4, 5	trlle 38578	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊)
47	1, 2, 46	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑅‘𝐹) ≤ 𝑊)
48	20, 35, 3, 7	diaord 39441	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑅‘𝐹) ≤ 𝑊) ∧ ((𝑈 ∨ 𝑉) ∈ (Base‘𝐾) ∧ (𝑈 ∨ 𝑉) ≤ 𝑊)) → ((𝐼‘(𝑅‘𝐹)) ⊆ (𝐼‘(𝑈 ∨ 𝑉)) ↔ (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)))
49	1, 45, 47, 24, 38, 48	syl122anc 1380	. 2 ⊢ (𝜑 → ((𝐼‘(𝑅‘𝐹)) ⊆ (𝐼‘(𝑈 ∨ 𝑉)) ↔ (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)))
50	43, 49	mpbid 231	1 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3909 {csn 4585 class class class wbr 5104 ‘cfv 6492 (class class class)co 7350 Basecbs 17019 lecple 17076 joincjn 18136 Latclat 18256 LModclmod 20251 LSubSpclss 20321 LSpanclspn 20361 LVecclvec 20492 Atomscatm 37656 HLchlt 37743 LHypclh 38378 LTrncltrn 38495 trLctrl 38552 DVecAcdveca 39396 DIsoAcdia 39422

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-riotaBAD 37346

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-tpos 8125 df-undef 8172 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-n0 12348 df-z 12434 df-uz 12698 df-fz 13355 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-mulr 17083 df-sca 17085 df-vsca 17086 df-0g 17259 df-proset 18120 df-poset 18138 df-plt 18155 df-lub 18171 df-glb 18172 df-join 18173 df-meet 18174 df-p0 18250 df-p1 18251 df-lat 18257 df-clat 18324 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-grp 18687 df-minusg 18688 df-sbg 18689 df-cmn 19499 df-abl 19500 df-mgp 19832 df-ur 19849 df-ring 19896 df-oppr 19978 df-dvdsr 19999 df-unit 20000 df-invr 20030 df-dvr 20041 df-drng 20116 df-lmod 20253 df-lss 20322 df-lsp 20362 df-lvec 20493 df-oposet 37569 df-ol 37571 df-oml 37572 df-covers 37659 df-ats 37660 df-atl 37691 df-cvlat 37715 df-hlat 37744 df-llines 37892 df-lplanes 37893 df-lvols 37894 df-lines 37895 df-psubsp 37897 df-pmap 37898 df-padd 38190 df-lhyp 38382 df-laut 38383 df-ldil 38498 df-ltrn 38499 df-trl 38553 df-tgrp 39137 df-tendo 39149 df-edring 39151 df-dveca 39397 df-disoa 39423

This theorem is referenced by: dia2dimlem11 39468

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