Theorem linepmap 35379

Description: A line described with a projective map. (Contributed by NM, 3-Feb-2012.)

Hypotheses

Ref	Expression
isline2.j	⊢ ∨ = (join‘𝐾)
isline2.a	⊢ 𝐴 = (Atoms‘𝐾)
isline2.n	⊢ 𝑁 = (Lines‘𝐾)
isline2.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
linepmap	⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑀‘(𝑃 ∨ 𝑄)) ∈ 𝑁)

Proof of Theorem linepmap

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1084	. . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ Lat)
2		simpl2 1085	. . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ 𝐴)
3		eqid 2651	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		isline2.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	atbase 34894	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
6	2, 5	syl 17	. . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ (Base‘𝐾))
7		simpl3 1086	. . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ 𝐴)
8	3, 4	atbase 34894	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
9	7, 8	syl 17	. . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ (Base‘𝐾))
10		isline2.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
11	3, 10	latjcl 17098	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
12	1, 6, 9, 11	syl3anc 1366	. . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
13		eqid 2651	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
14		isline2.m	. . . 4 ⊢ 𝑀 = (pmap‘𝐾)
15	3, 13, 4, 14	pmapval 35361	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑀‘(𝑃 ∨ 𝑄)) = {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)})
16	1, 12, 15	syl2anc 694	. 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑀‘(𝑃 ∨ 𝑄)) = {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)})
17		eqid 2651	. . 3 ⊢ {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)} = {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)}
18		isline2.n	. . . 4 ⊢ 𝑁 = (Lines‘𝐾)
19	13, 10, 4, 18	islinei 35344	. . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)} = {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)})) → {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)} ∈ 𝑁)
20	17, 19	mpanr2 720	. 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → {𝑟 ∈ 𝐴 ∣ 𝑟(le‘𝐾)(𝑃 ∨ 𝑄)} ∈ 𝑁)
21	16, 20	eqeltrd 2730	1 ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑀‘(𝑃 ∨ 𝑄)) ∈ 𝑁)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  {crab 2945   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  Latclat 17092  Atomscatm 34868  Linesclines 35098  pmapcpmap 35101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-lat 17093  df-ats 34872  df-lines 35105  df-pmap 35108

This theorem is referenced by:  cdleme3h  35840  cdleme7ga  35853