dihglblem5 - Mathbox for Norm Megill

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Theorem dihglblem5 39761

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 6855	. . 3 ⊢ (𝐼‘𝑥) ∈ V
2	1	dfiin2 4994	. 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 39573	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → 𝑈 ∈ LMod)
7		simpll 765	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		simplrl 775	. . . . . . 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 3945	. . . . . 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 39713	. . . . . 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 3143	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∀𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆)
17		uniiunlem 4044	. . . . 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 771	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → 𝑇 ≠ ∅)
21		n0 4306	. . . . 5 ⊢ (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑇)
22	20, 21	sylib 217	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∃𝑥 𝑥 ∈ 𝑇)
23		nfre1 3268	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)
24	23	nfab 2913	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)}
25		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥∅
26	24, 25	nfne 3045	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅
27	1	elabrex 7190	. . . . . 6 ⊢ (𝑥 ∈ 𝑇 → (𝐼‘𝑥) ∈ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)})
28	27	ne0d 4295	. . . . 5 ⊢ (𝑥 ∈ 𝑇 → {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅)
29	26, 28	exlimi 2210	. . . 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 20425	. . 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 1371	. 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 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2713 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 ⊆ wss 3910 ∅c0 4282 ∩ cint 4907 ∩ ciin 4955 ‘cfv 6496 Basecbs 17083 glbcglb 18199 LModclmod 20322 LSubSpclss 20392 HLchlt 37812 LHypclh 38447 DVecHcdvh 39541 DIsoHcdih 39691

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-riotaBAD 37415

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-undef 8204 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-0g 17323 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tendo 39218 df-edring 39220 df-disoa 39492 df-dvech 39542 df-dib 39602 df-dic 39636 df-dih 39692

This theorem is referenced by: dihglblem6 39803

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