Theorem lnatexN 40408

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 2763	. . . 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 40405	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠))))
7	6	biimp3a 1491	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟(join‘𝐾)𝑠)))
8		simpl2r 1242	. . . . . 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 1243	. . . . . . . 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 3013	. . . . . . 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 488	. . . . . . 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 3027	. . . . . 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 1263	. . . . . . . 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 1241	. . . . . . . 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 40005	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑠 ≤ (𝑟(join‘𝐾)𝑠))
17	13, 14, 8, 16	syl3anc 1392	. . . . . . 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 1244	. . . . . . 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 5129	. . . . . 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 3020	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → (𝑞 ≠ 𝑃 ↔ 𝑠 ≠ 𝑃))
21		breq1 5104	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → (𝑞 ≤ 𝑋 ↔ 𝑠 ≤ 𝑋))
22	20, 21	anbi12d 641	. . . . . . 7 ⊢ (𝑞 = 𝑠 → ((𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋) ↔ (𝑠 ≠ 𝑃 ∧ 𝑠 ≤ 𝑋)))
23	22	rspcev 3582	. . . . . 6 ⊢ ((𝑠 ∈ 𝐴 ∧ (𝑠 ≠ 𝑃 ∧ 𝑠 ≤ 𝑋)) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))
24	8, 12, 19, 23	syl12anc 847	. . . . 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 1241	. . . . . 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 488	. . . . . 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 1263	. . . . . . . 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 1242	. . . . . . . 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 40004	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑟 ≤ (𝑟(join‘𝐾)𝑠))
30	27, 25, 28, 29	syl3anc 1392	. . . . . . 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 1244	. . . . . . 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 5129	. . . . . 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 3020	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝑞 ≠ 𝑃 ↔ 𝑟 ≠ 𝑃))
34		breq1 5104	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝑞 ≤ 𝑋 ↔ 𝑟 ≤ 𝑋))
35	33, 34	anbi12d 641	. . . . . . 7 ⊢ (𝑞 = 𝑟 → ((𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋) ↔ (𝑟 ≠ 𝑃 ∧ 𝑟 ≤ 𝑋)))
36	35	rspcev 3582	. . . . . 6 ⊢ ((𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≤ 𝑋)) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))
37	25, 26, 32, 36	syl12anc 847	. . . . 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 3045	. . . 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 1133	. . 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 3219	. 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 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3087   class class class wbr 5101  ‘cfv 6521  (class class class)co 7396  Basecbs 17255  lecple 17303  joincjn 18353  Atomscatm 39892  HLchlt 39979  Linesclines 40123  pmapcpmap 40126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-proset 18336  df-poset 18355  df-plt 18370  df-lub 18386  df-glb 18387  df-join 18388  df-meet 18389  df-p0 18465  df-lat 18474  df-clat 18541  df-oposet 39805  df-ol 39807  df-oml 39808  df-covers 39895  df-ats 39896  df-atl 39927  df-cvlat 39951  df-hlat 39980  df-lines 40130  df-pmap 40133

This theorem is referenced by:  lnjatN  40409