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Theorem elpcliN 37644
Description: Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpcli.s 𝑆 = (PSubSp‘𝐾)
elpcli.c 𝑈 = (PCl‘𝐾)
Assertion
Ref Expression
elpcliN (((𝐾𝑉𝑋𝑌𝑌𝑆) ∧ 𝑄 ∈ (𝑈𝑋)) → 𝑄𝑌)

Proof of Theorem elpcliN
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simp1 1138 . . . . . 6 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝐾𝑉)
2 simp2 1139 . . . . . . 7 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑋𝑌)
3 eqid 2737 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
4 elpcli.s . . . . . . . . 9 𝑆 = (PSubSp‘𝐾)
53, 4psubssat 37505 . . . . . . . 8 ((𝐾𝑉𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
653adant2 1133 . . . . . . 7 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
72, 6sstrd 3911 . . . . . 6 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
8 elpcli.c . . . . . . 7 𝑈 = (PCl‘𝐾)
93, 4, 8pclvalN 37641 . . . . . 6 ((𝐾𝑉𝑋 ⊆ (Atoms‘𝐾)) → (𝑈𝑋) = {𝑧𝑆𝑋𝑧})
101, 7, 9syl2anc 587 . . . . 5 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑈𝑋) = {𝑧𝑆𝑋𝑧})
1110eleq2d 2823 . . . 4 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) ↔ 𝑄 {𝑧𝑆𝑋𝑧}))
12 elintrabg 4872 . . . . 5 (𝑄 {𝑧𝑆𝑋𝑧} → (𝑄 {𝑧𝑆𝑋𝑧} ↔ ∀𝑧𝑆 (𝑋𝑧𝑄𝑧)))
1312ibi 270 . . . 4 (𝑄 {𝑧𝑆𝑋𝑧} → ∀𝑧𝑆 (𝑋𝑧𝑄𝑧))
1411, 13syl6bi 256 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) → ∀𝑧𝑆 (𝑋𝑧𝑄𝑧)))
15 sseq2 3927 . . . . . . . 8 (𝑧 = 𝑌 → (𝑋𝑧𝑋𝑌))
16 eleq2 2826 . . . . . . . 8 (𝑧 = 𝑌 → (𝑄𝑧𝑄𝑌))
1715, 16imbi12d 348 . . . . . . 7 (𝑧 = 𝑌 → ((𝑋𝑧𝑄𝑧) ↔ (𝑋𝑌𝑄𝑌)))
1817rspccv 3534 . . . . . 6 (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → (𝑌𝑆 → (𝑋𝑌𝑄𝑌)))
1918com13 88 . . . . 5 (𝑋𝑌 → (𝑌𝑆 → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌)))
2019imp 410 . . . 4 ((𝑋𝑌𝑌𝑆) → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌))
21203adant1 1132 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌))
2214, 21syld 47 . 2 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) → 𝑄𝑌))
2322imp 410 1 (((𝐾𝑉𝑋𝑌𝑌𝑆) ∧ 𝑄 ∈ (𝑈𝑋)) → 𝑄𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  {crab 3065  wss 3866   cint 4859  cfv 6380  Atomscatm 37014  PSubSpcpsubsp 37247  PClcpclN 37638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-psubsp 37254  df-pclN 37639
This theorem is referenced by:  pclfinclN  37701
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