Theorem atlelt 40232

Description: Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)

Hypotheses

Ref	Expression
atlelt.b	⊢ 𝐵 = (Base‘𝐾)
atlelt.l	⊢ ≤ = (le‘𝐾)
atlelt.s	⊢ < = (lt‘𝐾)
atlelt.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
atlelt	⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑃 < 𝑋)

Proof of Theorem atlelt

Step	Hyp	Ref	Expression
1		simp3r 1221	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑄 < 𝑋)
2		breq1 5112	. . 3 ⊢ (𝑃 = 𝑄 → (𝑃 < 𝑋 ↔ 𝑄 < 𝑋))
3	1, 2	syl5ibrcom 250	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃 = 𝑄 → 𝑃 < 𝑋))
4		simp1 1154	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝐾 ∈ HL)
5		simp21 1225	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑃 ∈ 𝐴)
6		simp22 1226	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑄 ∈ 𝐴)
7		atlelt.s	. . . . 5 ⊢ < = (lt‘𝐾)
8		eqid 2763	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
9		atlelt.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
10	7, 8, 9	atlt 40231	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃 ≠ 𝑄))
11	4, 5, 6, 10	syl3anc 1398	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) ↔ 𝑃 ≠ 𝑄))
12		simp3l 1220	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑃 ≤ 𝑋)
13		simp23 1227	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑋 ∈ 𝐵)
14	4, 6, 13	3jca 1146	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵))
15		atlelt.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
16	15, 7	pltle 18382	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑄 < 𝑋 → 𝑄 ≤ 𝑋))
17	14, 1, 16	sylc 66	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑄 ≤ 𝑋)
18		hllat 40157	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
19	18	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝐾 ∈ Lat)
20		atlelt.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
21	20, 9	atbase 40083	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
22	5, 21	syl 18	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑃 ∈ 𝐵)
23	20, 9	atbase 40083	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
24	6, 23	syl 18	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑄 ∈ 𝐵)
25	20, 15, 8	latjle12 18501	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋) ↔ (𝑃(join‘𝐾)𝑄) ≤ 𝑋))
26	19, 22, 24, 13, 25	syl13anc 1399	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → ((𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋) ↔ (𝑃(join‘𝐾)𝑄) ≤ 𝑋))
27	12, 17, 26	mpbi2and 724	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) ≤ 𝑋)
28		hlpos 40160	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
29	28	3ad2ant1 1151	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝐾 ∈ Poset)
30	20, 8	latjcl 18490	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
31	19, 22, 24, 30	syl3anc 1398	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃(join‘𝐾)𝑄) ∈ 𝐵)
32	20, 15, 7	pltletr 18392	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑃 ∈ 𝐵 ∧ (𝑃(join‘𝐾)𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 < (𝑃(join‘𝐾)𝑄) ∧ (𝑃(join‘𝐾)𝑄) ≤ 𝑋) → 𝑃 < 𝑋))
33	29, 22, 31, 13, 32	syl13anc 1399	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → ((𝑃 < (𝑃(join‘𝐾)𝑄) ∧ (𝑃(join‘𝐾)𝑄) ≤ 𝑋) → 𝑃 < 𝑋))
34	27, 33	mpan2d 706	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑄) → 𝑃 < 𝑋))
35	11, 34	sylbird 263	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → (𝑃 ≠ 𝑄 → 𝑃 < 𝑋))
36	3, 35	pm2.61dne 3044	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋)) → 𝑃 < 𝑋)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  lecple 17312  Posetcpo 18358  ltcplt 18359  joincjn 18362  Latclat 18482  Atomscatm 40057  HLchlt 40144

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 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

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 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-proset 18345  df-poset 18364  df-plt 18379  df-lub 18395  df-glb 18396  df-join 18397  df-meet 18398  df-p0 18474  df-lat 18483  df-clat 18550  df-oposet 39970  df-ol 39972  df-oml 39973  df-covers 40060  df-ats 40061  df-atl 40092  df-cvlat 40116  df-hlat 40145

This theorem is referenced by:  1cvratlt  40268