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Theorem cvlsupr3 35300
Description: Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄, which can replace the superposition part of ishlat1 35308, (𝑥𝑦 → ∃𝑧𝐴(𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)) ), with the simpler 𝑧𝐴(𝑥 𝑧) = (𝑦 𝑧) as shown in ishlat3N 35310. (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 2938 . . . 4 (𝑃𝑄 ↔ ¬ 𝑃 = 𝑄)
21imbi1i 340 . . 3 ((𝑃𝑄 → (𝑃 𝑅) = (𝑄 𝑅)) ↔ (¬ 𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅)))
3 oveq1 6849 . . . 4 (𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅))
43biantrur 526 . . 3 ((¬ 𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅)) ↔ ((𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅)) ∧ (¬ 𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅))))
5 pm4.83 1048 . . 3 (((𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅)) ∧ (¬ 𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅))) ↔ (𝑃 𝑅) = (𝑄 𝑅))
62, 4, 53bitrri 289 . 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 35299 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))
11103expia 1150 . . 3 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃𝑄 → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))))
1211pm5.74d 264 . 2 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃𝑄 → (𝑃 𝑅) = (𝑄 𝑅)) ↔ (𝑃𝑄 → (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))))
136, 12syl5bb 274 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937   class class class wbr 4809  cfv 6068  (class class class)co 6842  lecple 16221  joincjn 17210  Atomscatm 35219  CvLatclc 35221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-proset 17194  df-poset 17212  df-plt 17224  df-lub 17240  df-glb 17241  df-join 17242  df-meet 17243  df-p0 17305  df-lat 17312  df-covers 35222  df-ats 35223  df-atl 35254  df-cvlat 35278
This theorem is referenced by:  ishlat3N  35310  hlsupr2  35343
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