Theorem lvolcmp 39574

Description: If two lattice planes are comparable, they are equal. (Contributed by NM, 12-Jul-2012.)

Hypotheses

Ref	Expression
lvolcmp.l	⊢ ≤ = (le‘𝐾)
lvolcmp.v	⊢ 𝑉 = (LVols‘𝐾)

Assertion

Ref	Expression
lvolcmp	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Proof of Theorem lvolcmp

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1137	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ 𝑉)
2		simp1 1136	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝐾 ∈ HL)
3		eqid 2740	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		lvolcmp.v	. . . . . . 7 ⊢ 𝑉 = (LVols‘𝐾)
5	3, 4	lvolbase 39535	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Base‘𝐾))
6	5	3ad2ant2 1134	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ (Base‘𝐾))
7		eqid 2740	. . . . . 6 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
8		eqid 2740	. . . . . 6 ⊢ (LPlanes‘𝐾) = (LPlanes‘𝐾)
9	3, 7, 8, 4	islvol4 39531	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝑉 ↔ ∃𝑧 ∈ (LPlanes‘𝐾)𝑧( ⋖ ‘𝐾)𝑋))
10	2, 6, 9	syl2anc 583	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∈ 𝑉 ↔ ∃𝑧 ∈ (LPlanes‘𝐾)𝑧( ⋖ ‘𝐾)𝑋))
11	1, 10	mpbid 232	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑧 ∈ (LPlanes‘𝐾)𝑧( ⋖ ‘𝐾)𝑋)
12		simpr3 1196	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ≤ 𝑌)
13		hlpos 39322	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
14	13	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝐾 ∈ Poset)
15	14	adantr 480	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ Poset)
16	6	adantr 480	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ∈ (Base‘𝐾))
17		simpl3 1193	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ 𝑉)
18	3, 4	lvolbase 39535	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ (Base‘𝐾))
19	17, 18	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ (Base‘𝐾))
20		simpr1 1194	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ∈ (LPlanes‘𝐾))
21	3, 8	lplnbase 39491	. . . . . . . 8 ⊢ (𝑧 ∈ (LPlanes‘𝐾) → 𝑧 ∈ (Base‘𝐾))
22	20, 21	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ∈ (Base‘𝐾))
23		simpr2 1195	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧( ⋖ ‘𝐾)𝑋)
24		simpl1 1191	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ HL)
25		lvolcmp.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
26	3, 25, 7	cvrle 39234	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑧( ⋖ ‘𝐾)𝑋) → 𝑧 ≤ 𝑋)
27	24, 22, 16, 23, 26	syl31anc 1373	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ≤ 𝑋)
28	3, 25	postr 18390	. . . . . . . . . 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 1372	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → ((𝑧 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑧 ≤ 𝑌))
30	27, 12, 29	mp2and 698	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ≤ 𝑌)
31	25, 7, 8, 4	lplncvrlvol2 39572	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑌 ∈ 𝑉) ∧ 𝑧 ≤ 𝑌) → 𝑧( ⋖ ‘𝐾)𝑌)
32	24, 20, 17, 30, 31	syl31anc 1373	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧( ⋖ ‘𝐾)𝑌)
33	3, 25, 7	cvrcmp 39239	. . . . . . 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 1388	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
35	12, 34	mpbid 232	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 = 𝑌)
36	35	3exp2 1354	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑧 ∈ (LPlanes‘𝐾) → (𝑧( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))))
37	36	rexlimdv 3159	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (∃𝑧 ∈ (LPlanes‘𝐾)𝑧( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌)))
38	11, 37	mpd 15	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
39	3, 25	posref 18388	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋 ≤ 𝑋)
40	14, 6, 39	syl2anc 583	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ≤ 𝑋)
41		breq2 5170	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
42	40, 41	syl5ibcom 245	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
43	38, 42	impbid 212	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   class class class wbr 5166  ‘cfv 6573  Basecbs 17258  lecple 17318  Posetcpo 18377   ⋖ ccvr 39218  HLchlt 39306  LPlanesclpl 39449  LVolsclvol 39450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-proset 18365  df-poset 18383  df-plt 18400  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-p0 18495  df-lat 18502  df-clat 18569  df-oposet 39132  df-ol 39134  df-oml 39135  df-covers 39222  df-ats 39223  df-atl 39254  df-cvlat 39278  df-hlat 39307  df-llines 39455  df-lplanes 39456  df-lvols 39457

This theorem is referenced by:  lvolnltN  39575  2lplnja  39576