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Theorem elpcliN 34197
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 1054 . . . . . 6 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝐾𝑉)
2 simp2 1055 . . . . . . 7 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑋𝑌)
3 eqid 2610 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
4 elpcli.s . . . . . . . . 9 𝑆 = (PSubSp‘𝐾)
53, 4psubssat 34058 . . . . . . . 8 ((𝐾𝑉𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
653adant2 1073 . . . . . . 7 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
72, 6sstrd 3578 . . . . . 6 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
8 elpcli.c . . . . . . 7 𝑈 = (PCl‘𝐾)
93, 4, 8pclvalN 34194 . . . . . 6 ((𝐾𝑉𝑋 ⊆ (Atoms‘𝐾)) → (𝑈𝑋) = {𝑧𝑆𝑋𝑧})
101, 7, 9syl2anc 691 . . . . 5 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑈𝑋) = {𝑧𝑆𝑋𝑧})
1110eleq2d 2673 . . . 4 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) ↔ 𝑄 {𝑧𝑆𝑋𝑧}))
12 elintrabg 4424 . . . . 5 (𝑄 {𝑧𝑆𝑋𝑧} → (𝑄 {𝑧𝑆𝑋𝑧} ↔ ∀𝑧𝑆 (𝑋𝑧𝑄𝑧)))
1312ibi 255 . . . 4 (𝑄 {𝑧𝑆𝑋𝑧} → ∀𝑧𝑆 (𝑋𝑧𝑄𝑧))
1411, 13syl6bi 242 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) → ∀𝑧𝑆 (𝑋𝑧𝑄𝑧)))
15 sseq2 3590 . . . . . . . 8 (𝑧 = 𝑌 → (𝑋𝑧𝑋𝑌))
16 eleq2 2677 . . . . . . . 8 (𝑧 = 𝑌 → (𝑄𝑧𝑄𝑌))
1715, 16imbi12d 333 . . . . . . 7 (𝑧 = 𝑌 → ((𝑋𝑧𝑄𝑧) ↔ (𝑋𝑌𝑄𝑌)))
1817rspccv 3279 . . . . . 6 (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → (𝑌𝑆 → (𝑋𝑌𝑄𝑌)))
1918com13 86 . . . . 5 (𝑋𝑌 → (𝑌𝑆 → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌)))
2019imp 444 . . . 4 ((𝑋𝑌𝑌𝑆) → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌))
21203adant1 1072 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌))
2214, 21syld 46 . 2 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) → 𝑄𝑌))
2322imp 444 1 (((𝐾𝑉𝑋𝑌𝑌𝑆) ∧ 𝑄 ∈ (𝑈𝑋)) → 𝑄𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  {crab 2900  wss 3540   cint 4410  cfv 5804  Atomscatm 33568  PSubSpcpsubsp 33800  PClcpclN 34191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-psubsp 33807  df-pclN 34192
This theorem is referenced by:  pclfinclN  34254
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