dihglblem5 - Mathbox for Norm Megill

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Theorem dihglblem5 39308

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 6784	. . 3 ⊢ (𝐼‘𝑥) ∈ V
2	1	dfiin2 4969	. 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 483	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	3, 4, 5	dvhlmod 39120	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → 𝑈 ∈ LMod)
7		simpll 764	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		simplrl 774	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → 𝑇 ⊆ 𝐵)
9		simpr 485	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑇)
10	8, 9	sseldd 3927	. . . . . 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 39260	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝑆)
15	7, 10, 14	syl2anc 584	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → (𝐼‘𝑥) ∈ 𝑆)
16	15	ralrimiva 3110	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∀𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆)
17		uniiunlem 4024	. . . . 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 231	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ⊆ 𝑆)
20		simprr 770	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → 𝑇 ≠ ∅)
21		n0 4286	. . . . 5 ⊢ (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑇)
22	20, 21	sylib 217	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∃𝑥 𝑥 ∈ 𝑇)
23		nfre1 3237	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)
24	23	nfab 2915	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)}
25		nfcv 2909	. . . . . 6 ⊢ Ⅎ𝑥∅
26	24, 25	nfne 3047	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅
27	1	elabrex 7113	. . . . . 6 ⊢ (𝑥 ∈ 𝑇 → (𝐼‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)})
28	27	ne0d 4275	. . . . 5 ⊢ (𝑥 ∈ 𝑇 → {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅)
29	26, 28	exlimi 2214	. . . 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 20224	. . 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 1370	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ∈ 𝑆)
33	2, 32	eqeltrid 2845	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∃wex 1786 ∈ wcel 2110 {cab 2717 ≠ wne 2945 ∀wral 3066 ∃wrex 3067 ⊆ wss 3892 ∅c0 4262 ∩ cint 4885 ∩ ciin 4931 ‘cfv 6432 Basecbs 16910 glbcglb 18026 LModclmod 20121 LSubSpclss 20191 HLchlt 37360 LHypclh 37994 DVecHcdvh 39088 DIsoHcdih 39238

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-riotaBAD 36963

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-tpos 8033 df-undef 8080 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-0g 17150 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 df-oposet 37186 df-ol 37188 df-oml 37189 df-covers 37276 df-ats 37277 df-atl 37308 df-cvlat 37332 df-hlat 37361 df-llines 37508 df-lplanes 37509 df-lvols 37510 df-lines 37511 df-psubsp 37513 df-pmap 37514 df-padd 37806 df-lhyp 37998 df-laut 37999 df-ldil 38114 df-ltrn 38115 df-trl 38169 df-tendo 38765 df-edring 38767 df-disoa 39039 df-dvech 39089 df-dib 39149 df-dic 39183 df-dih 39239

This theorem is referenced by: dihglblem6 39350

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