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Theorem cvlsupr3 36585
Description: Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄, which can replace the superposition part of ishlat1 36593, (𝑥𝑦 → ∃𝑧𝐴(𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)) ), with the simpler 𝑧𝐴(𝑥 𝑧) = (𝑦 𝑧) as shown in ishlat3N 36595. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
cvlsupr2.a 𝐴 = (Atoms‘𝐾)
cvlsupr2.l = (le‘𝐾)
cvlsupr2.j = (join‘𝐾)
Assertion
Ref Expression
cvlsupr3 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃𝑄 → (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))))

Proof of Theorem cvlsupr3
StepHypRef Expression
1 df-ne 3015 . . . 4 (𝑃𝑄 ↔ ¬ 𝑃 = 𝑄)
21imbi1i 353 . . 3 ((𝑃𝑄 → (𝑃 𝑅) = (𝑄 𝑅)) ↔ (¬ 𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅)))
3 oveq1 7156 . . . 4 (𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅))
43biantrur 534 . . 3 ((¬ 𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅)) ↔ ((𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅)) ∧ (¬ 𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅))))
5 pm4.83 1022 . . 3 (((𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅)) ∧ (¬ 𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅))) ↔ (𝑃 𝑅) = (𝑄 𝑅))
62, 4, 53bitrri 301 . 2 ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃𝑄 → (𝑃 𝑅) = (𝑄 𝑅)))
7 cvlsupr2.a . . . . 5 𝐴 = (Atoms‘𝐾)
8 cvlsupr2.l . . . . 5 = (le‘𝐾)
9 cvlsupr2.j . . . . 5 = (join‘𝐾)
107, 8, 9cvlsupr2 36584 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
11103expia 1118 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃𝑄 → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))))
1211pm5.74d 276 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃𝑄 → (𝑃 𝑅) = (𝑄 𝑅)) ↔ (𝑃𝑄 → (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))))
136, 12syl5bb 286 1 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃𝑄 → (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  (class class class)co 7149  lecple 16572  joincjn 17554  Atomscatm 36504  CvLatclc 36506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-lat 17656  df-covers 36507  df-ats 36508  df-atl 36539  df-cvlat 36563
This theorem is referenced by:  ishlat3N  36595  hlsupr2  36628
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