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Theorem elpcliN 35682
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 1131 . . . . . 6 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝐾𝑉)
2 simp2 1132 . . . . . . 7 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑋𝑌)
3 eqid 2760 . . . . . . . . 9 (Atoms‘𝐾) = (Atoms‘𝐾)
4 elpcli.s . . . . . . . . 9 𝑆 = (PSubSp‘𝐾)
53, 4psubssat 35543 . . . . . . . 8 ((𝐾𝑉𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
653adant2 1126 . . . . . . 7 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
72, 6sstrd 3754 . . . . . 6 ((𝐾𝑉𝑋𝑌𝑌𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
8 elpcli.c . . . . . . 7 𝑈 = (PCl‘𝐾)
93, 4, 8pclvalN 35679 . . . . . 6 ((𝐾𝑉𝑋 ⊆ (Atoms‘𝐾)) → (𝑈𝑋) = {𝑧𝑆𝑋𝑧})
101, 7, 9syl2anc 696 . . . . 5 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑈𝑋) = {𝑧𝑆𝑋𝑧})
1110eleq2d 2825 . . . 4 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) ↔ 𝑄 {𝑧𝑆𝑋𝑧}))
12 elintrabg 4641 . . . . 5 (𝑄 {𝑧𝑆𝑋𝑧} → (𝑄 {𝑧𝑆𝑋𝑧} ↔ ∀𝑧𝑆 (𝑋𝑧𝑄𝑧)))
1312ibi 256 . . . 4 (𝑄 {𝑧𝑆𝑋𝑧} → ∀𝑧𝑆 (𝑋𝑧𝑄𝑧))
1411, 13syl6bi 243 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (𝑄 ∈ (𝑈𝑋) → ∀𝑧𝑆 (𝑋𝑧𝑄𝑧)))
15 sseq2 3768 . . . . . . . 8 (𝑧 = 𝑌 → (𝑋𝑧𝑋𝑌))
16 eleq2 2828 . . . . . . . 8 (𝑧 = 𝑌 → (𝑄𝑧𝑄𝑌))
1715, 16imbi12d 333 . . . . . . 7 (𝑧 = 𝑌 → ((𝑋𝑧𝑄𝑧) ↔ (𝑋𝑌𝑄𝑌)))
1817rspccv 3446 . . . . . 6 (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → (𝑌𝑆 → (𝑋𝑌𝑄𝑌)))
1918com13 88 . . . . 5 (𝑋𝑌 → (𝑌𝑆 → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌)))
2019imp 444 . . . 4 ((𝑋𝑌𝑌𝑆) → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌))
21203adant1 1125 . . 3 ((𝐾𝑉𝑋𝑌𝑌𝑆) → (∀𝑧𝑆 (𝑋𝑧𝑄𝑧) → 𝑄𝑌))
2214, 21syld 47 . 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 1072   = wceq 1632  wcel 2139  wral 3050  {crab 3054  wss 3715   cint 4627  cfv 6049  Atomscatm 35053  PSubSpcpsubsp 35285  PClcpclN 35676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-psubsp 35292  df-pclN 35677
This theorem is referenced by:  pclfinclN  35739
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