dihglblem4 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dihglblem4		Structured version Visualization version GIF version

Theorem dihglblem4 37106

Description: Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)

**Hypotheses**
Ref	Expression
dihglblem.b	⊢ 𝐵 = (Base‘𝐾)
dihglblem.l	⊢ ≤ = (le‘𝐾)
dihglblem.m	⊢ ∧ = (meet‘𝐾)
dihglblem.g	⊢ 𝐺 = (glb‘𝐾)
dihglblem.h	⊢ 𝐻 = (LHyp‘𝐾)
dihglblem.t	⊢ 𝑇 = {𝑢 ∈ 𝐵 ∣ ∃𝑣 ∈ 𝑆 𝑢 = (𝑣 ∧ 𝑊)}
dihglblem.i	⊢ 𝐽 = ((DIsoB‘𝐾)‘𝑊)
dihglblem.ih	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
dihglblem4	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) ⊆ ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))

Distinct variable groups: 𝑥,𝑢,𝑣, ∧ 𝑥, ≤ 𝑥,𝐵,𝑢 𝑥,𝐺 𝑥,𝐻 𝑥,𝐾 𝑥,𝑆,𝑢,𝑣 𝑥,𝑇 𝑥,𝑊,𝑢,𝑣 𝑢, ≤ ,𝑣 𝑣,𝐵 𝑢,𝐺,𝑣 𝑢,𝐻,𝑣 𝑢,𝐾,𝑣 𝑥,𝐼 Allowed substitution hints: 𝑇(𝑣,𝑢) 𝐼(𝑣,𝑢) 𝐽(𝑥,𝑣,𝑢)

**Proof of Theorem dihglblem4**
Step	Hyp	Ref	Expression
1		hlclat 35166	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
2	1	ad3antrrr 768	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝐾 ∈ CLat)
3		simplrl 819	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ 𝐵)
4		simpr 479	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
5		dihglblem.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6		dihglblem.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
7		dihglblem.g	. . . . . 6 ⊢ 𝐺 = (glb‘𝐾)
8	5, 6, 7	clatglble 17346	. . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆) ≤ 𝑥)
9	2, 3, 4, 8	syl3anc 1477	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆) ≤ 𝑥)
10		simpll 807	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11	5, 7	clatglbcl 17335	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) ∈ 𝐵)
12	2, 3, 11	syl2anc 696	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆) ∈ 𝐵)
13	3, 4	sseldd 3745	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
14		dihglblem.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
15		dihglblem.ih	. . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
16	5, 6, 14, 15	dihord 37073	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺‘𝑆) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝐼‘(𝐺‘𝑆)) ⊆ (𝐼‘𝑥) ↔ (𝐺‘𝑆) ≤ 𝑥))
17	10, 12, 13, 16	syl3anc 1477	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → ((𝐼‘(𝐺‘𝑆)) ⊆ (𝐼‘𝑥) ↔ (𝐺‘𝑆) ≤ 𝑥))
18	9, 17	mpbird 247	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐼‘(𝐺‘𝑆)) ⊆ (𝐼‘𝑥))
19	18	ralrimiva 3104	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) → ∀𝑥 ∈ 𝑆 (𝐼‘(𝐺‘𝑆)) ⊆ (𝐼‘𝑥))
20		ssiin 4722	. 2 ⊢ ((𝐼‘(𝐺‘𝑆)) ⊆ ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝐼‘(𝐺‘𝑆)) ⊆ (𝐼‘𝑥))
21	19, 20	sylibr 224	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) ⊆ ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∀wral 3050 ∃wrex 3051 {crab 3054 ⊆ wss 3715 ∅c0 4058 ∩ ciin 4673 class class class wbr 4804 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 lecple 16170 glbcglb 17164 meetcmee 17166 CLatccla 17328 HLchlt 35158 LHypclh 35791 DIsoBcdib 36947 DIsoHcdih 37037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-riotaBAD 34760

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-tpos 7522 df-undef 7569 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-n0 11505 df-z 11590 df-uz 11900 df-fz 12540 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-sca 16179 df-vsca 16180 df-0g 16324 df-preset 17149 df-poset 17167 df-plt 17179 df-lub 17195 df-glb 17196 df-join 17197 df-meet 17198 df-p0 17260 df-p1 17261 df-lat 17267 df-clat 17329 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-submnd 17557 df-grp 17646 df-minusg 17647 df-sbg 17648 df-subg 17812 df-cntz 17970 df-lsm 18271 df-cmn 18415 df-abl 18416 df-mgp 18710 df-ur 18722 df-ring 18769 df-oppr 18843 df-dvdsr 18861 df-unit 18862 df-invr 18892 df-dvr 18903 df-drng 18971 df-lmod 19087 df-lss 19155 df-lsp 19194 df-lvec 19325 df-oposet 34984 df-ol 34986 df-oml 34987 df-covers 35074 df-ats 35075 df-atl 35106 df-cvlat 35130 df-hlat 35159 df-llines 35305 df-lplanes 35306 df-lvols 35307 df-lines 35308 df-psubsp 35310 df-pmap 35311 df-padd 35603 df-lhyp 35795 df-laut 35796 df-ldil 35911 df-ltrn 35912 df-trl 35967 df-tendo 36563 df-edring 36565 df-disoa 36838 df-dvech 36888 df-dib 36948 df-dic 36982 df-dih 37038

This theorem is referenced by: dihglblem6 37149

W3C validator