Theorem lnatexN 40039

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 2736	. . . 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 40036	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))))
7	6	biimp3a 1471	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠)))
8		simpl2r 1228	. . . . . 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 1229	. . . . . . . 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 2987	. . . . . . 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 484	. . . . . . 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 3001	. . . . . 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 1249	. . . . . . . 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 1227	. . . . . . . 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 39636	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑠 ≤ (𝑟(join‘𝐾)𝑠))
17	13, 14, 8, 16	syl3anc 1373	. . . . . . 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 1230	. . . . . . 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 5126	. . . . . 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 2994	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → (𝑞 ≠ 𝑃 ↔ 𝑠 ≠ 𝑃))
21		breq1 5101	. . . . . . . 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 3576	. . . . . 6 ⊢ ((𝑠 ∈ 𝐴 ∧ (𝑠 ≠ 𝑃 ∧ 𝑠 ≤ 𝑋)) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))
24	8, 12, 19, 23	syl12anc 836	. . . . 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 1227	. . . . . 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 484	. . . . . 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 1249	. . . . . . . 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 1228	. . . . . . . 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 39635	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑟 ≤ (𝑟(join‘𝐾)𝑠))
30	27, 25, 28, 29	syl3anc 1373	. . . . . . 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 1230	. . . . . . 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 5126	. . . . . 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 2994	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝑞 ≠ 𝑃 ↔ 𝑟 ≠ 𝑃))
34		breq1 5101	. . . . . . . 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 3576	. . . . . 6 ⊢ ((𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≤ 𝑋)) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))
37	25, 26, 32, 36	syl12anc 836	. . . . 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 3019	. . . 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 1119	. . 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 3192	. 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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  joincjn 18234  Atomscatm 39523  HLchlt 39610  Linesclines 39754  pmapcpmap 39757

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-proset 18217  df-poset 18236  df-plt 18251  df-lub 18267  df-glb 18268  df-join 18269  df-meet 18270  df-p0 18346  df-lat 18355  df-clat 18422  df-oposet 39436  df-ol 39438  df-oml 39439  df-covers 39526  df-ats 39527  df-atl 39558  df-cvlat 39582  df-hlat 39611  df-lines 39761  df-pmap 39764

This theorem is referenced by:  lnjatN  40040