dihglblem5 - Mathbox for Norm Megill

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Theorem dihglblem5 42132

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 6898	. . 3 ⊢ (𝐼‘𝑥) ∈ V
2	1	dfiin2 4999	. 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 488	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	3, 4, 5	dvhlmod 41944	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → 𝑈 ∈ LMod)
7		simpll 779	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		simplrl 789	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → 𝑇 ⊆ 𝐵)
9		simpr 490	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑇)
10	8, 9	sseldd 3939	. . . . . 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 42084	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝑆)
15	7, 10, 14	syl2anc 596	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) ∧ 𝑥 ∈ 𝑇) → (𝐼‘𝑥) ∈ 𝑆)
16	15	ralrimiva 3159	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∀𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆)
17		uniiunlem 4042	. . . . 5 ⊢ (∀𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆 → (∀𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ⊆ 𝑆))
18	16, 17	syl 18	. . . 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 785	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → 𝑇 ≠ ∅)
21		n0 4307	. . . . 5 ⊢ (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑇)
22	20, 21	sylib 221	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∃𝑥 𝑥 ∈ 𝑇)
23		nfre1 3292	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)
24	23	nfab 2933	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)}
25		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑥∅
26	24, 25	nfne 3063	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅
27	1	elabrex 7245	. . . . . 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 2256	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝑇 → {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅)
30	22, 29	syl 18	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ≠ ∅)
31	13	lssintcl 21137	. . 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 1398	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝑇 𝑦 = (𝐼‘𝑥)} ∈ 𝑆)
33	2, 32	eqeltrid 2869	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2146 {cab 2743 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 ⊆ wss 3906 ∅c0 4286 ∩ cint 4914 ∩ ciin 4959 ‘cfv 6540 Basecbs 17293 glbcglb 18390 LModclmod 21033 LSubSpclss 21104 HLchlt 40184 LHypclh 40818 DVecHcdvh 41912 DIsoHcdih 42062

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-riotaBAD 39787

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8228 df-undef 8275 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-n0 12522 df-z 12609 df-uz 12881 df-fz 13554 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-sca 17350 df-vsca 17351 df-0g 17518 df-proset 18374 df-poset 18393 df-plt 18408 df-lub 18424 df-glb 18425 df-join 18426 df-meet 18427 df-p0 18503 df-p1 18504 df-lat 18512 df-clat 18579 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-submnd 18881 df-grp 19049 df-minusg 19050 df-sbg 19051 df-subg 19235 df-cntz 19433 df-lsm 19752 df-cmn 19898 df-abl 19899 df-mgp 20263 df-rng 20277 df-ur 20310 df-ring 20363 df-oppr 20467 df-dvdsr 20487 df-unit 20488 df-invr 20518 df-dvr 20531 df-drng 20881 df-lmod 21035 df-lss 21105 df-lsp 21145 df-lvec 21276 df-oposet 40010 df-ol 40012 df-oml 40013 df-covers 40100 df-ats 40101 df-atl 40132 df-cvlat 40156 df-hlat 40185 df-llines 40332 df-lplanes 40333 df-lvols 40334 df-lines 40335 df-psubsp 40337 df-pmap 40338 df-padd 40630 df-lhyp 40822 df-laut 40823 df-ldil 40938 df-ltrn 40939 df-trl 40993 df-tendo 41589 df-edring 41591 df-disoa 41863 df-dvech 41913 df-dib 41973 df-dic 42007 df-dih 42063

This theorem is used by: dihglblem6 42174

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