Theorem lplncmp 40314

Description: If two lattice planes are comparable, they are equal. (Contributed by NM, 24-Jun-2012.)

Hypotheses

Ref	Expression
lplncmp.l	⊢ ≤ = (le‘𝐾)
lplncmp.p	⊢ 𝑃 = (LPlanes‘𝐾)

Assertion

Ref	Expression
lplncmp	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Proof of Theorem lplncmp

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1155	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝑋 ∈ 𝑃)
2		simp1 1154	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝐾 ∈ HL)
3		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		lplncmp.p	. . . . . . 7 ⊢ 𝑃 = (LPlanes‘𝐾)
5	3, 4	lplnbase 40286	. . . . . 6 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾))
6	5	3ad2ant2 1152	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝑋 ∈ (Base‘𝐾))
7		eqid 2763	. . . . . 6 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
8		eqid 2763	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
9	3, 7, 8, 4	islpln4 40283	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝑃 ↔ ∃𝑧 ∈ (LLines‘𝐾)𝑧( ⋖ ‘𝐾)𝑋))
10	2, 6, 9	syl2anc 595	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ∈ 𝑃 ↔ ∃𝑧 ∈ (LLines‘𝐾)𝑧( ⋖ ‘𝐾)𝑋))
11	1, 10	mpbid 235	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ∃𝑧 ∈ (LLines‘𝐾)𝑧( ⋖ ‘𝐾)𝑋)
12		simpr3 1215	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ≤ 𝑌)
13		hlpos 40118	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
14	13	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝐾 ∈ Poset)
15	14	adantr 485	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ Poset)
16	6	adantr 485	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ∈ (Base‘𝐾))
17		simpl3 1212	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ 𝑃)
18	3, 4	lplnbase 40286	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 → 𝑌 ∈ (Base‘𝐾))
19	17, 18	syl 18	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ (Base‘𝐾))
20		simpr1 1213	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ∈ (LLines‘𝐾))
21	3, 8	llnbase 40261	. . . . . . . 8 ⊢ (𝑧 ∈ (LLines‘𝐾) → 𝑧 ∈ (Base‘𝐾))
22	20, 21	syl 18	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ∈ (Base‘𝐾))
23		simpr2 1214	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧( ⋖ ‘𝐾)𝑋)
24		simpl1 1210	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ HL)
25		lplncmp.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
26	3, 25, 7	cvrle 40030	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑧( ⋖ ‘𝐾)𝑋) → 𝑧 ≤ 𝑋)
27	24, 22, 16, 23, 26	syl31anc 1400	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ≤ 𝑋)
28	3, 25	postr 18377	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑧 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑧 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑧 ≤ 𝑌))
29	15, 22, 16, 19, 28	syl13anc 1399	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → ((𝑧 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑧 ≤ 𝑌))
30	27, 12, 29	mp2and 711	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ≤ 𝑌)
31	25, 7, 8, 4	llncvrlpln2 40309	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ (LLines‘𝐾) ∧ 𝑌 ∈ 𝑃) ∧ 𝑧 ≤ 𝑌) → 𝑧( ⋖ ‘𝐾)𝑌)
32	24, 20, 17, 30, 31	syl31anc 1400	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧( ⋖ ‘𝐾)𝑌)
33	3, 25, 7	cvrcmp 40035	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) ∧ (𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑧( ⋖ ‘𝐾)𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
34	15, 16, 19, 22, 23, 32, 33	syl132anc 1415	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
35	12, 34	mpbid 235	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 = 𝑌)
36	35	3exp2 1373	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑧 ∈ (LLines‘𝐾) → (𝑧( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))))
37	36	rexlimdv 3164	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (∃𝑧 ∈ (LLines‘𝐾)𝑧( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌)))
38	11, 37	mpd 16	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
39	3, 25	posref 18375	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋 ≤ 𝑋)
40	14, 6, 39	syl2anc 595	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝑋 ≤ 𝑋)
41		breq2 5114	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
42	40, 41	syl5ibcom 248	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
43	38, 42	impbid 215	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   class class class wbr 5110  ‘cfv 6538  Basecbs 17270  lecple 17318  Posetcpo 18364   ⋖ ccvr 40014  HLchlt 40102  LLinesclln 40243  LPlanesclpl 40244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  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-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-proset 18351  df-poset 18370  df-plt 18385  df-lub 18401  df-glb 18402  df-join 18403  df-meet 18404  df-p0 18480  df-lat 18489  df-clat 18556  df-oposet 39928  df-ol 39930  df-oml 39931  df-covers 40018  df-ats 40019  df-atl 40050  df-cvlat 40074  df-hlat 40103  df-llines 40250  df-lplanes 40251

This theorem is referenced by:  lplnexllnN  40316  lplnnlt  40317  2llnjaN  40318  dalem-cly  40423  dalem44  40468