Theorem paddasslem5 36975

Description: Lemma for paddass 36989. Show 𝑠 ≠ 𝑧 by contradiction. (Contributed by NM, 8-Jan-2012.)

Hypotheses

Ref	Expression
paddasslem.l	⊢ ≤ = (le‘𝐾)
paddasslem.j	⊢ ∨ = (join‘𝐾)
paddasslem.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
paddasslem5	⊢ (((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧) ∧ 𝑠 ≤ (𝑥 ∨ 𝑦))) → 𝑠 ≠ 𝑧)

Proof of Theorem paddasslem5

Step	Hyp	Ref	Expression
1		breq1 5069	. . . . . . . . 9 ⊢ (𝑠 = 𝑧 → (𝑠 ≤ (𝑥 ∨ 𝑦) ↔ 𝑧 ≤ (𝑥 ∨ 𝑦)))
2	1	biimpac 481	. . . . . . . 8 ⊢ ((𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 = 𝑧) → 𝑧 ≤ (𝑥 ∨ 𝑦))
3		eqid 2821	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		paddasslem.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
5		simpll1 1208	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝐾 ∈ HL)
6	5	hllatd 36515	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝐾 ∈ Lat)
7		simpll2 1209	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑟 ∈ 𝐴)
8		paddasslem.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
9	3, 8	atbase 36440	. . . . . . . . . . 11 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ (Base‘𝐾))
10	7, 9	syl 17	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑟 ∈ (Base‘𝐾))
11		simp32 1206	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
12	11	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑦 ∈ 𝐴)
13	3, 8	atbase 36440	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ (Base‘𝐾))
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑦 ∈ (Base‘𝐾))
15		simp33 1207	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝐴)
16	15	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑧 ∈ 𝐴)
17	3, 8	atbase 36440	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (Base‘𝐾))
18	16, 17	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑧 ∈ (Base‘𝐾))
19		paddasslem.j	. . . . . . . . . . . 12 ⊢ ∨ = (join‘𝐾)
20	3, 19	latjcl 17661	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑦 ∨ 𝑧) ∈ (Base‘𝐾))
21	6, 14, 18, 20	syl3anc 1367	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → (𝑦 ∨ 𝑧) ∈ (Base‘𝐾))
22		simp31 1205	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
23	22	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑥 ∈ 𝐴)
24	3, 8	atbase 36440	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (Base‘𝐾))
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑥 ∈ (Base‘𝐾))
26	3, 19	latjcl 17661	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥 ∨ 𝑦) ∈ (Base‘𝐾))
27	6, 25, 14, 26	syl3anc 1367	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → (𝑥 ∨ 𝑦) ∈ (Base‘𝐾))
28		simplr 767	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑟 ≤ (𝑦 ∨ 𝑧))
29	4, 19, 8	hlatlej2 36527	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ (𝑥 ∨ 𝑦))
30	5, 23, 12, 29	syl3anc 1367	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑦 ≤ (𝑥 ∨ 𝑦))
31		simpr 487	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑧 ≤ (𝑥 ∨ 𝑦))
32	3, 4, 19	latjle12 17672	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾) ∧ (𝑥 ∨ 𝑦) ∈ (Base‘𝐾))) → ((𝑦 ≤ (𝑥 ∨ 𝑦) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) ↔ (𝑦 ∨ 𝑧) ≤ (𝑥 ∨ 𝑦)))
33	32	biimpd 231	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾) ∧ (𝑥 ∨ 𝑦) ∈ (Base‘𝐾))) → ((𝑦 ≤ (𝑥 ∨ 𝑦) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → (𝑦 ∨ 𝑧) ≤ (𝑥 ∨ 𝑦)))
34	6, 14, 18, 27, 33	syl13anc 1368	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → ((𝑦 ≤ (𝑥 ∨ 𝑦) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → (𝑦 ∨ 𝑧) ≤ (𝑥 ∨ 𝑦)))
35	30, 31, 34	mp2and 697	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → (𝑦 ∨ 𝑧) ≤ (𝑥 ∨ 𝑦))
36	3, 4, 6, 10, 21, 27, 28, 35	lattrd 17668	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) → 𝑟 ≤ (𝑥 ∨ 𝑦))
37	36	ex 415	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) → (𝑧 ≤ (𝑥 ∨ 𝑦) → 𝑟 ≤ (𝑥 ∨ 𝑦)))
38	2, 37	syl5 34	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) → ((𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 = 𝑧) → 𝑟 ≤ (𝑥 ∨ 𝑦)))
39	38	expdimp 455	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑠 ≤ (𝑥 ∨ 𝑦)) → (𝑠 = 𝑧 → 𝑟 ≤ (𝑥 ∨ 𝑦)))
40	39	necon3bd 3030	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ 𝑠 ≤ (𝑥 ∨ 𝑦)) → (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) → 𝑠 ≠ 𝑧))
41	40	exp31 422	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑟 ≤ (𝑦 ∨ 𝑧) → (𝑠 ≤ (𝑥 ∨ 𝑦) → (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) → 𝑠 ≠ 𝑧))))
42	41	com23 86	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑠 ≤ (𝑥 ∨ 𝑦) → (𝑟 ≤ (𝑦 ∨ 𝑧) → (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) → 𝑠 ≠ 𝑧))))
43	42	com24 95	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) → (𝑟 ≤ (𝑦 ∨ 𝑧) → (𝑠 ≤ (𝑥 ∨ 𝑦) → 𝑠 ≠ 𝑧))))
44	43	3imp2 1345	1 ⊢ (((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧) ∧ 𝑠 ≤ (𝑥 ∨ 𝑦))) → 𝑠 ≠ 𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  lecple 16572  joincjn 17554  Latclat 17655  Atomscatm 36414  HLchlt 36501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-poset 17556  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-lat 17656  df-ats 36418  df-atl 36449  df-cvlat 36473  df-hlat 36502

This theorem is referenced by:  paddasslem7  36977