dihglblem5 - Mathbox for Norm Megill

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Theorem dihglblem5 39049

Description: Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)

**Hypotheses**
Ref	Expression
dihglblem5.b	⊢ 𝐵 = (Base‘𝐾)
dihglblem5.g	⊢ 𝐺 = (glb‘𝐾)
dihglblem5.h	⊢ 𝐻 = (LHyp‘𝐾)
dihglblem5.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihglblem5.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihglblem5.s	⊢ 𝑆 = (LSubSp‘𝑈)

**Assertion**
Ref	Expression
dihglblem5	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆)

Distinct variable groups: 𝑥,𝐵 𝑥,𝐻 𝑥,𝐾 𝑥,𝑆 𝑥,𝑇 𝑥,𝑊 Allowed substitution hints: 𝑈(𝑥) 𝐺(𝑥) 𝐼(𝑥)

Proof of Theorem dihglblem5 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6730	. . 3 ⊢ (𝐼‘𝑥) ∈ V
2	1	dfiin2 4943	. 2 ⊢ ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)}
3		dihglblem5.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4		dihglblem5.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		simpl 486	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	3, 4, 5	dvhlmod 38861	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → 𝑈 ∈ LMod)
7		simpll 767	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		simplrl 777	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → 𝑇 ⊆ 𝐵)
9		simpr 488	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑇)
10	8, 9	sseldd 3902	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝐵)
11		dihglblem5.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
12		dihglblem5.i	. . . . . . 7 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
13		dihglblem5.s	. . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈)
14	11, 3, 12, 4, 13	dihlss 39001	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝑆)
15	7, 10, 14	syl2anc 587	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → (𝐼‘𝑥) ∈ 𝑆)
16	15	ralrimiva 3105	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∀𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆)
17		uniiunlem 3999	. . . . 5 ⊢ (∀𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆 → (∀𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ⊆ 𝑆))
18	16, 17	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → (∀𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ⊆ 𝑆))
19	16, 18	mpbid 235	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ⊆ 𝑆)
20		simprr 773	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → 𝑇 ≠ ∅)
21		n0 4261	. . . . 5 ⊢ (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑇)
22	20, 21	sylib 221	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∃𝑥 𝑥 ∈ 𝑇)
23		nfre1 3225	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)
24	23	nfab 2910	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)}
25		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑥∅
26	24, 25	nfne 3042	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅
27	1	elabrex 7056	. . . . . 6 ⊢ (𝑥 ∈ 𝑇 → (𝐼‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)})
28	27	ne0d 4250	. . . . 5 ⊢ (𝑥 ∈ 𝑇 → {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅)
29	26, 28	exlimi 2215	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝑇 → {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅)
30	22, 29	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅)
31	13	lssintcl 20001	. . 3 ⊢ ((𝑈 ∈ LMod ∧ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ⊆ 𝑆 ∧ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅) → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ∈ 𝑆)
32	6, 19, 30, 31	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ∈ 𝑆)
33	2, 32	eqeltrid 2842	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2110 {cab 2714 ≠ wne 2940 ∀wral 3061 ∃wrex 3062 ⊆ wss 3866 ∅c0 4237 ∩ cint 4859 ∩ ciin 4905 ‘cfv 6380 Basecbs 16760 glbcglb 17817 LModclmod 19899 LSubSpclss 19968 HLchlt 37101 LHypclh 37735 DVecHcdvh 38829 DIsoHcdih 38979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-riotaBAD 36704

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-undef 8015 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-0g 16946 df-proset 17802 df-poset 17820 df-plt 17836 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p0 17931 df-p1 17932 df-lat 17938 df-clat 18005 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-cntz 18711 df-lsm 19025 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-drng 19769 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lvec 20140 df-oposet 36927 df-ol 36929 df-oml 36930 df-covers 37017 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-llines 37249 df-lplanes 37250 df-lvols 37251 df-lines 37252 df-psubsp 37254 df-pmap 37255 df-padd 37547 df-lhyp 37739 df-laut 37740 df-ldil 37855 df-ltrn 37856 df-trl 37910 df-tendo 38506 df-edring 38508 df-disoa 38780 df-dvech 38830 df-dib 38890 df-dic 38924 df-dih 38980

This theorem is referenced by: dihglblem6 39091

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