dihmeetlem15N - Mathbox for Norm Megill

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Theorem dihmeetlem15N 39715

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
dihmeetlem14.b	⊢ 𝐵 = (Base‘𝐾)
dihmeetlem14.l	⊢ ≤ = (le‘𝐾)
dihmeetlem14.h	⊢ 𝐻 = (LHyp‘𝐾)
dihmeetlem14.j	⊢ ∨ = (join‘𝐾)
dihmeetlem14.m	⊢ ∧ = (meet‘𝐾)
dihmeetlem14.a	⊢ 𝐴 = (Atoms‘𝐾)
dihmeetlem14.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihmeetlem14.s	⊢ ⊕ = (LSSum‘𝑈)
dihmeetlem14.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihmeetlem15.z	⊢ 0 = (0_g‘𝑈)

**Assertion**
Ref	Expression
dihmeetlem15N	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → ((𝐼‘𝑟) ∩ (𝐼‘𝑝)) = { 0 })

Proof of Theorem dihmeetlem15N Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1192	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpr1 1195	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊))
3		simpl3 1194	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊))
4		simpl3r 1230	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → ¬ 𝑝 ≤ 𝑊)
5		simp3 1139	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → 𝑟 = 𝑝)
6		simp22 1208	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → 𝑟 ≤ 𝑌)
7	5, 6	eqbrtrrd 5128	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → 𝑝 ≤ 𝑌)
8		simp11l 1285	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → 𝐾 ∈ HL)
9	8	hllatd 37757	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → 𝐾 ∈ Lat)
10		simp13l 1289	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → 𝑝 ∈ 𝐴)
11		dihmeetlem14.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
12		dihmeetlem14.a	. . . . . . . . . 10 ⊢ 𝐴 = (Atoms‘𝐾)
13	11, 12	atbase 37682	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
14	10, 13	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → 𝑝 ∈ 𝐵)
15		simp12 1205	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → 𝑌 ∈ 𝐵)
16		dihmeetlem14.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
17		dihmeetlem14.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
18	11, 16, 17	latleeqm2 18293	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑝 ≤ 𝑌 ↔ (𝑌 ∧ 𝑝) = 𝑝))
19	9, 14, 15, 18	syl3anc 1372	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → (𝑝 ≤ 𝑌 ↔ (𝑌 ∧ 𝑝) = 𝑝))
20	7, 19	mpbid 231	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → (𝑌 ∧ 𝑝) = 𝑝)
21		simp23 1209	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → (𝑌 ∧ 𝑝) ≤ 𝑊)
22	20, 21	eqbrtrrd 5128	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊) ∧ 𝑟 = 𝑝) → 𝑝 ≤ 𝑊)
23	22	3expia 1122	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → (𝑟 = 𝑝 → 𝑝 ≤ 𝑊))
24	23	necon3bd 2956	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → (¬ 𝑝 ≤ 𝑊 → 𝑟 ≠ 𝑝))
25	4, 24	mpd 15	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → 𝑟 ≠ 𝑝)
26		dihmeetlem14.j	. . 3 ⊢ ∨ = (join‘𝐾)
27		dihmeetlem14.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
28		eqid 2738	. . 3 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
29		eqid 2738	. . 3 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
30		eqid 2738	. . 3 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
31		eqid 2738	. . 3 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
32		dihmeetlem14.i	. . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
33		dihmeetlem14.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
34		dihmeetlem15.z	. . 3 ⊢ 0 = (0_g‘𝑈)
35		eqid 2738	. . 3 ⊢ (℩ℎ ∈ ((LTrn‘𝐾)‘𝑊)(ℎ‘((oc‘𝐾)‘𝑊)) = 𝑟) = (℩ℎ ∈ ((LTrn‘𝐾)‘𝑊)(ℎ‘((oc‘𝐾)‘𝑊)) = 𝑟)
36		eqid 2738	. . 3 ⊢ (℩ℎ ∈ ((LTrn‘𝐾)‘𝑊)(ℎ‘((oc‘𝐾)‘𝑊)) = 𝑝) = (℩ℎ ∈ ((LTrn‘𝐾)‘𝑊)(ℎ‘((oc‘𝐾)‘𝑊)) = 𝑝)
37	11, 16, 26, 12, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36	dihmeetlem13N 39713	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ 𝑟 ≠ 𝑝) → ((𝐼‘𝑟) ∩ (𝐼‘𝑝)) = { 0 })
38	1, 2, 3, 25, 37	syl121anc 1376	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → ((𝐼‘𝑟) ∩ (𝐼‘𝑝)) = { 0 })

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∩ cin 3908 {csn 4585 class class class wbr 5104 ↦ cmpt 5187 I cid 5528 ↾ cres 5633 ‘cfv 6492 ℩crio 7305 (class class class)co 7350 Basecbs 17019 lecple 17076 occoc 17077 0_gc0g 17257 joincjn 18136 meetcmee 18137 Latclat 18256 LSSumclsm 19351 Atomscatm 37656 HLchlt 37743 LHypclh 38378 LTrncltrn 38495 TEndoctendo 39146 DVecHcdvh 39472 DIsoHcdih 39622

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-riotaBAD 37346

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-tpos 8125 df-undef 8172 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-n0 12348 df-z 12434 df-uz 12698 df-fz 13355 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-mulr 17083 df-sca 17085 df-vsca 17086 df-0g 17259 df-proset 18120 df-poset 18138 df-plt 18155 df-lub 18171 df-glb 18172 df-join 18173 df-meet 18174 df-p0 18250 df-p1 18251 df-lat 18257 df-clat 18324 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-submnd 18538 df-grp 18687 df-minusg 18688 df-sbg 18689 df-subg 18860 df-cntz 19032 df-lsm 19353 df-cmn 19499 df-abl 19500 df-mgp 19832 df-ur 19849 df-ring 19896 df-oppr 19978 df-dvdsr 19999 df-unit 20000 df-invr 20030 df-dvr 20041 df-drng 20116 df-lmod 20253 df-lss 20322 df-lsp 20362 df-lvec 20493 df-oposet 37569 df-ol 37571 df-oml 37572 df-covers 37659 df-ats 37660 df-atl 37691 df-cvlat 37715 df-hlat 37744 df-llines 37892 df-lplanes 37893 df-lvols 37894 df-lines 37895 df-psubsp 37897 df-pmap 37898 df-padd 38190 df-lhyp 38382 df-laut 38383 df-ldil 38498 df-ltrn 38499 df-trl 38553 df-tendo 39149 df-edring 39151 df-disoa 39423 df-dvech 39473 df-dib 39533 df-dic 39567 df-dih 39623

This theorem is referenced by: dihmeetlem16N 39716

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