Theorem lnatexN 36930

Description: There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnatex.b	⊢ 𝐵 = (Base‘𝐾)
lnatex.l	⊢ ≤ = (le‘𝐾)
lnatex.a	⊢ 𝐴 = (Atoms‘𝐾)
lnatex.n	⊢ 𝑁 = (Lines‘𝐾)
lnatex.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
lnatexN	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))

Distinct variable groups:   𝐴,𝑞   ≤ ,𝑞   𝑃,𝑞   𝑋,𝑞

Allowed substitution hints:   𝐵(𝑞)   𝐾(𝑞)   𝑀(𝑞)   𝑁(𝑞)

Proof of Theorem lnatexN

Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnatex.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2821	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
3		lnatex.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4		lnatex.n	. . . 4 ⊢ 𝑁 = (Lines‘𝐾)
5		lnatex.m	. . . 4 ⊢ 𝑀 = (pmap‘𝐾)
6	1, 2, 3, 4, 5	isline3 36927	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))))
7	6	biimp3a 1465	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠)))
8		simpl2r 1223	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠 ∈ 𝐴)
9		simpl3l 1224	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑟 ≠ 𝑠)
10	9	necomd 3071	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠 ≠ 𝑟)
11		simpr 487	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑟 = 𝑃)
12	10, 11	neeqtrd 3085	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠 ≠ 𝑃)
13		simpl11 1244	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝐾 ∈ HL)
14		simpl2l 1222	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑟 ∈ 𝐴)
15		lnatex.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
16	15, 2, 3	hlatlej2 36527	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑠 ≤ (𝑟(join‘𝐾)𝑠))
17	13, 14, 8, 16	syl3anc 1367	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠 ≤ (𝑟(join‘𝐾)𝑠))
18		simpl3r 1225	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑋 = (𝑟(join‘𝐾)𝑠))
19	17, 18	breqtrrd 5094	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠 ≤ 𝑋)
20		neeq1 3078	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → (𝑞 ≠ 𝑃 ↔ 𝑠 ≠ 𝑃))
21		breq1 5069	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → (𝑞 ≤ 𝑋 ↔ 𝑠 ≤ 𝑋))
22	20, 21	anbi12d 632	. . . . . . 7 ⊢ (𝑞 = 𝑠 → ((𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋) ↔ (𝑠 ≠ 𝑃 ∧ 𝑠 ≤ 𝑋)))
23	22	rspcev 3623	. . . . . 6 ⊢ ((𝑠 ∈ 𝐴 ∧ (𝑠 ≠ 𝑃 ∧ 𝑠 ≤ 𝑋)) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))
24	8, 12, 19, 23	syl12anc 834	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))
25		simpl2l 1222	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 ≠ 𝑃) → 𝑟 ∈ 𝐴)
26		simpr 487	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 ≠ 𝑃) → 𝑟 ≠ 𝑃)
27		simpl11 1244	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 ≠ 𝑃) → 𝐾 ∈ HL)
28		simpl2r 1223	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 ≠ 𝑃) → 𝑠 ∈ 𝐴)
29	15, 2, 3	hlatlej1 36526	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑟 ≤ (𝑟(join‘𝐾)𝑠))
30	27, 25, 28, 29	syl3anc 1367	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 ≠ 𝑃) → 𝑟 ≤ (𝑟(join‘𝐾)𝑠))
31		simpl3r 1225	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 ≠ 𝑃) → 𝑋 = (𝑟(join‘𝐾)𝑠))
32	30, 31	breqtrrd 5094	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 ≠ 𝑃) → 𝑟 ≤ 𝑋)
33		neeq1 3078	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝑞 ≠ 𝑃 ↔ 𝑟 ≠ 𝑃))
34		breq1 5069	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝑞 ≤ 𝑋 ↔ 𝑟 ≤ 𝑋))
35	33, 34	anbi12d 632	. . . . . . 7 ⊢ (𝑞 = 𝑟 → ((𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋) ↔ (𝑟 ≠ 𝑃 ∧ 𝑟 ≤ 𝑋)))
36	35	rspcev 3623	. . . . . 6 ⊢ ((𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≤ 𝑋)) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))
37	25, 26, 32, 36	syl12anc 834	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 ≠ 𝑃) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))
38	24, 37	pm2.61dane 3104	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))
39	38	3exp 1115	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) → ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → ((𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠)) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))))
40	39	rexlimdvv 3293	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) → (∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠)) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋)))
41	7, 40	mpd 15	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∃wrex 3139   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  lecple 16572  joincjn 17554  Atomscatm 36414  HLchlt 36501  Linesclines 36645  pmapcpmap 36648

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-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-lat 17656  df-clat 17718  df-oposet 36327  df-ol 36329  df-oml 36330  df-covers 36417  df-ats 36418  df-atl 36449  df-cvlat 36473  df-hlat 36502  df-lines 36652  df-pmap 36655

This theorem is referenced by:  lnjatN  36931