Theorem lplncmp 35638

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 1173	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝑋 ∈ 𝑃)
2		simp1 1172	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝐾 ∈ HL)
3		eqid 2826	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		lplncmp.p	. . . . . . 7 ⊢ 𝑃 = (LPlanes‘𝐾)
5	3, 4	lplnbase 35610	. . . . . 6 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾))
6	5	3ad2ant2 1170	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝑋 ∈ (Base‘𝐾))
7		eqid 2826	. . . . . 6 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
8		eqid 2826	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
9	3, 7, 8, 4	islpln4 35607	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝑃 ↔ ∃𝑧 ∈ (LLines‘𝐾)𝑧( ⋖ ‘𝐾)𝑋))
10	2, 6, 9	syl2anc 581	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ∈ 𝑃 ↔ ∃𝑧 ∈ (LLines‘𝐾)𝑧( ⋖ ‘𝐾)𝑋))
11	1, 10	mpbid 224	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ∃𝑧 ∈ (LLines‘𝐾)𝑧( ⋖ ‘𝐾)𝑋)
12		simpr3 1258	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ≤ 𝑌)
13		hlpos 35442	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
14	13	3ad2ant1 1169	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝐾 ∈ Poset)
15	14	adantr 474	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ Poset)
16	6	adantr 474	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ∈ (Base‘𝐾))
17		simpl3 1252	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ 𝑃)
18	3, 4	lplnbase 35610	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 → 𝑌 ∈ (Base‘𝐾))
19	17, 18	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ (Base‘𝐾))
20		simpr1 1254	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ∈ (LLines‘𝐾))
21	3, 8	llnbase 35585	. . . . . . . 8 ⊢ (𝑧 ∈ (LLines‘𝐾) → 𝑧 ∈ (Base‘𝐾))
22	20, 21	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ∈ (Base‘𝐾))
23		simpr2 1256	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧( ⋖ ‘𝐾)𝑋)
24		simpl1 1248	. . . . . . . 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 35354	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑧( ⋖ ‘𝐾)𝑋) → 𝑧 ≤ 𝑋)
27	24, 22, 16, 23, 26	syl31anc 1498	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ≤ 𝑋)
28	3, 25	postr 17307	. . . . . . . . . 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 1497	. . . . . . . . 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 692	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ≤ 𝑌)
31	25, 7, 8, 4	llncvrlpln2 35633	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ (LLines‘𝐾) ∧ 𝑌 ∈ 𝑃) ∧ 𝑧 ≤ 𝑌) → 𝑧( ⋖ ‘𝐾)𝑌)
32	24, 20, 17, 30, 31	syl31anc 1498	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧( ⋖ ‘𝐾)𝑌)
33	3, 25, 7	cvrcmp 35359	. . . . . . 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 1513	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
35	12, 34	mpbid 224	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ (𝑧 ∈ (LLines‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 = 𝑌)
36	35	3exp2 1469	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑧 ∈ (LLines‘𝐾) → (𝑧( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))))
37	36	rexlimdv 3240	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (∃𝑧 ∈ (LLines‘𝐾)𝑧( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌)))
38	11, 37	mpd 15	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
39	3, 25	posref 17305	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋 ≤ 𝑋)
40	14, 6, 39	syl2anc 581	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝑋 ≤ 𝑋)
41		breq2 4878	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
42	40, 41	syl5ibcom 237	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
43	38, 42	impbid 204	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∃wrex 3119   class class class wbr 4874  ‘cfv 6124  Basecbs 16223  lecple 16313  Posetcpo 17294   ⋖ ccvr 35338  HLchlt 35426  LLinesclln 35567  LPlanesclpl 35568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-proset 17282  df-poset 17300  df-plt 17312  df-lub 17328  df-glb 17329  df-join 17330  df-meet 17331  df-p0 17393  df-lat 17400  df-clat 17462  df-oposet 35252  df-ol 35254  df-oml 35255  df-covers 35342  df-ats 35343  df-atl 35374  df-cvlat 35398  df-hlat 35427  df-llines 35574  df-lplanes 35575

This theorem is referenced by:  lplnexllnN  35640  lplnnlt  35641  2llnjaN  35642  dalem-cly  35747  dalem44  35792