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Theorem elpcliN 36909
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 1128 . . . . . 6 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝐾𝑉)
2 simp2 1129 . . . . . . 7 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑋𝑌)
3 eqid 2818 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
4 elpcli.s . . . . . . . . 9 𝑆 = (PSubSp‘𝐾)
53, 4psubssat 36770 . . . . . . . 8 ((𝐾𝑉𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
653adant2 1123 . . . . . . 7 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
72, 6sstrd 3974 . . . . . 6 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
8 elpcli.c . . . . . . 7 𝑈 = (PCl‘𝐾)
93, 4, 8pclvalN 36906 . . . . . 6 ((𝐾𝑉𝑋 ⊆ (Atoms‘𝐾)) → (𝑈𝑋) = {𝑧𝑆𝑋𝑧})
101, 7, 9syl2anc 584 . . . . 5 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑈𝑋) = {𝑧𝑆𝑋𝑧})
1110eleq2d 2895 . . . 4 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) ↔ 𝑄 {𝑧𝑆𝑋𝑧}))
12 elintrabg 4880 . . . . 5 (𝑄 {𝑧𝑆𝑋𝑧} → (𝑄 {𝑧𝑆𝑋𝑧} ↔ ∀𝑧𝑆 (𝑋𝑧𝑄𝑧)))
1312ibi 268 . . . 4 (𝑄 {𝑧𝑆𝑋𝑧} → ∀𝑧𝑆 (𝑋𝑧𝑄𝑧))
1411, 13syl6bi 254 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) → ∀𝑧𝑆 (𝑋𝑧𝑄𝑧)))
15 sseq2 3990 . . . . . . . 8 (𝑧 = 𝑌 → (𝑋𝑧𝑋𝑌))
16 eleq2 2898 . . . . . . . 8 (𝑧 = 𝑌 → (𝑄𝑧𝑄𝑌))
1715, 16imbi12d 346 . . . . . . 7 (𝑧 = 𝑌 → ((𝑋𝑧𝑄𝑧) ↔ (𝑋𝑌𝑄𝑌)))
1817rspccv 3617 . . . . . 6 (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → (𝑌𝑆 → (𝑋𝑌𝑄𝑌)))
1918com13 88 . . . . 5 (𝑋𝑌 → (𝑌𝑆 → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌)))
2019imp 407 . . . 4 ((𝑋𝑌𝑌𝑆) → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌))
21203adant1 1122 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌))
2214, 21syld 47 . 2 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) → 𝑄𝑌))
2322imp 407 1 (((𝐾𝑉𝑋𝑌𝑌𝑆) ∧ 𝑄 ∈ (𝑈𝑋)) → 𝑄𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {crab 3139  wss 3933   cint 4867  cfv 6348  Atomscatm 36279  PSubSpcpsubsp 36512  PClcpclN 36903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-psubsp 36519  df-pclN 36904
This theorem is referenced by:  pclfinclN  36966
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