Theorem lvolcmp 40189

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 1146	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ 𝑉)
2		simp1 1145	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝐾 ∈ HL)
3		eqid 2756	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		lvolcmp.v	. . . . . . 7 ⊢ 𝑉 = (LVols‘𝐾)
5	3, 4	lvolbase 40150	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Base‘𝐾))
6	5	3ad2ant2 1143	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ (Base‘𝐾))
7		eqid 2756	. . . . . 6 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
8		eqid 2756	. . . . . 6 ⊢ (LPlanes‘𝐾) = (LPlanes‘𝐾)
9	3, 7, 8, 4	islvol4 40146	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝑉 ↔ ∃𝑧 ∈ (LPlanes‘𝐾)𝑧( ⋖ ‘𝐾)𝑋))
10	2, 6, 9	syl2anc 592	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ∈ 𝑉 ↔ ∃𝑧 ∈ (LPlanes‘𝐾)𝑧( ⋖ ‘𝐾)𝑋))
11	1, 10	mpbid 234	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑧 ∈ (LPlanes‘𝐾)𝑧( ⋖ ‘𝐾)𝑋)
12		simpr3 1206	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ≤ 𝑌)
13		hlpos 39938	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
14	13	3ad2ant1 1142	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝐾 ∈ Poset)
15	14	adantr 483	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ Poset)
16	6	adantr 483	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ∈ (Base‘𝐾))
17		simpl3 1203	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ 𝑉)
18	3, 4	lvolbase 40150	. . . . . . . 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 1204	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ∈ (LPlanes‘𝐾))
21	3, 8	lplnbase 40106	. . . . . . . 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 1205	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧( ⋖ ‘𝐾)𝑋)
24		simpl1 1201	. . . . . . . 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 39850	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑧( ⋖ ‘𝐾)𝑋) → 𝑧 ≤ 𝑋)
27	24, 22, 16, 23, 26	syl31anc 1388	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ≤ 𝑋)
28	3, 25	postr 18328	. . . . . . . . . 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 1387	. . . . . . . . 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 707	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧 ≤ 𝑌)
31	25, 7, 8, 4	lplncvrlvol2 40187	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑌 ∈ 𝑉) ∧ 𝑧 ≤ 𝑌) → 𝑧( ⋖ ‘𝐾)𝑌)
32	24, 20, 17, 30, 31	syl31anc 1388	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑧( ⋖ ‘𝐾)𝑌)
33	3, 25, 7	cvrcmp 39855	. . . . . . 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 1403	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
35	12, 34	mpbid 234	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑧 ∈ (LPlanes‘𝐾) ∧ 𝑧( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 = 𝑌)
36	35	3exp2 1364	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑧 ∈ (LPlanes‘𝐾) → (𝑧( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))))
37	36	rexlimdv 3155	. . 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 18326	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋 ≤ 𝑋)
40	14, 6, 39	syl2anc 592	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ≤ 𝑋)
41		breq2 5098	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
42	40, 41	syl5ibcom 247	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
43	38, 42	impbid 214	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ∃wrex 3080   class class class wbr 5094  ‘cfv 6510  Basecbs 17221  lecple 17269  Posetcpo 18315   ⋖ ccvr 39834  HLchlt 39922  LPlanesclpl 40064  LVolsclvol 40065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-riota 7342  df-ov 7388  df-oprab 7389  df-proset 18302  df-poset 18321  df-plt 18336  df-lub 18352  df-glb 18353  df-join 18354  df-meet 18355  df-p0 18431  df-lat 18440  df-clat 18507  df-oposet 39748  df-ol 39750  df-oml 39751  df-covers 39838  df-ats 39839  df-atl 39870  df-cvlat 39894  df-hlat 39923  df-llines 40070  df-lplanes 40071  df-lvols 40072

This theorem is referenced by:  lvolnltN  40190  2lplnja  40191