dihglblem5 - Mathbox for Norm Megill

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Theorem dihglblem5 41761

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 6848	. . 3 ⊢ (𝐼‘𝑥) ∈ V
2	1	dfiin2 4976	. 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 482	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	3, 4, 5	dvhlmod 41573	. . 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 484	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑇)
10	8, 9	sseldd 3923	. . . . . 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 41713	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝑆)
15	7, 10, 14	syl2anc 585	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → (𝐼‘𝑥) ∈ 𝑆)
16	15	ralrimiva 3130	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∀𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆)
17		uniiunlem 4028	. . . . 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 232	. . 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 4294	. . . . 5 ⊢ (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑇)
22	20, 21	sylib 218	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∃𝑥 𝑥 ∈ 𝑇)
23		nfre1 3263	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)
24	23	nfab 2905	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)}
25		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥∅
26	24, 25	nfne 3034	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅
27	1	elabrex 7191	. . . . . 6 ⊢ (𝑥 ∈ 𝑇 → (𝐼‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)})
28	27	ne0d 4283	. . . . 5 ⊢ (𝑥 ∈ 𝑇 → {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅)
29	26, 28	exlimi 2225	. . . 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 20953	. . 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 1374	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ∈ 𝑆)
33	2, 32	eqeltrid 2841	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 {cab 2715 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ⊆ wss 3890 ∅c0 4274 ∩ cint 4890 ∩ ciin 4935 ‘cfv 6493 Basecbs 17173 glbcglb 18270 LModclmod 20849 LSubSpclss 20920 HLchlt 39813 LHypclh 40447 DVecHcdvh 41541 DIsoHcdih 41691

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-riotaBAD 39416

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8170 df-undef 8217 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-n0 12432 df-z 12519 df-uz 12783 df-fz 13456 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-sca 17230 df-vsca 17231 df-0g 17398 df-proset 18254 df-poset 18273 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18392 df-clat 18459 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19093 df-cntz 19286 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-drng 20702 df-lmod 20851 df-lss 20921 df-lsp 20961 df-lvec 21093 df-oposet 39639 df-ol 39641 df-oml 39642 df-covers 39729 df-ats 39730 df-atl 39761 df-cvlat 39785 df-hlat 39814 df-llines 39961 df-lplanes 39962 df-lvols 39963 df-lines 39964 df-psubsp 39966 df-pmap 39967 df-padd 40259 df-lhyp 40451 df-laut 40452 df-ldil 40567 df-ltrn 40568 df-trl 40622 df-tendo 41218 df-edring 41220 df-disoa 41492 df-dvech 41542 df-dib 41602 df-dic 41636 df-dih 41692

This theorem is referenced by: dihglblem6 41803

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