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Theorem cofcut1d 27745
Description: If ðķ is cofinal with ðī and 𝐷 is coinitial with ðĩ and the cut of ðī and ðĩ lies between ðķ and 𝐷, then the cut of ðķ and 𝐷 is equal to the cut of ðī and ðĩ. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.)
Hypotheses
Ref Expression
cofcut1d.1 (𝜑 → ðī <<s ðĩ)
cofcut1d.2 (𝜑 → ∀ð‘Ĩ ∈ ðī ∃ð‘Ķ ∈ ðķ ð‘Ĩ â‰Īs ð‘Ķ)
cofcut1d.3 (𝜑 → ∀𝑧 ∈ ðĩ ∃ð‘Ī ∈ 𝐷 ð‘Ī â‰Īs 𝑧)
cofcut1d.4 (𝜑 → ðķ <<s {(ðī |s ðĩ)})
cofcut1d.5 (𝜑 → {(ðī |s ðĩ)} <<s 𝐷)
Assertion
Ref Expression
cofcut1d (𝜑 → (ðī |s ðĩ) = (ðķ |s 𝐷))
Distinct variable groups:   ð‘Ĩ,ðī   𝑧,ðĩ   ð‘Ĩ,ðķ,ð‘Ķ   ð‘Ī,𝐷,𝑧
Allowed substitution hints:   𝜑(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ī)   ðī(ð‘Ķ,𝑧,ð‘Ī)   ðĩ(ð‘Ĩ,ð‘Ķ,ð‘Ī)   ðķ(𝑧,ð‘Ī)   𝐷(ð‘Ĩ,ð‘Ķ)

Proof of Theorem cofcut1d
StepHypRef Expression
1 cofcut1d.1 . 2 (𝜑 → ðī <<s ðĩ)
2 cofcut1d.2 . 2 (𝜑 → ∀ð‘Ĩ ∈ ðī ∃ð‘Ķ ∈ ðķ ð‘Ĩ â‰Īs ð‘Ķ)
3 cofcut1d.3 . 2 (𝜑 → ∀𝑧 ∈ ðĩ ∃ð‘Ī ∈ 𝐷 ð‘Ī â‰Īs 𝑧)
4 cofcut1d.4 . 2 (𝜑 → ðķ <<s {(ðī |s ðĩ)})
5 cofcut1d.5 . 2 (𝜑 → {(ðī |s ðĩ)} <<s 𝐷)
6 cofcut1 27744 . 2 ((ðī <<s ðĩ ∧ (∀ð‘Ĩ ∈ ðī ∃ð‘Ķ ∈ ðķ ð‘Ĩ â‰Īs ð‘Ķ ∧ ∀𝑧 ∈ ðĩ ∃ð‘Ī ∈ 𝐷 ð‘Ī â‰Īs 𝑧) ∧ (ðķ <<s {(ðī |s ðĩ)} ∧ {(ðī |s ðĩ)} <<s 𝐷)) → (ðī |s ðĩ) = (ðķ |s 𝐷))
71, 2, 3, 4, 5, 6syl122anc 1376 1 (𝜑 → (ðī |s ðĩ) = (ðķ |s 𝐷))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   = wceq 1533  âˆ€wral 3053  âˆƒwrex 3062  {csn 4620   class class class wbr 5138  (class class class)co 7401   â‰Īs csle 27581   <<s csslt 27617   |s cscut 27619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1o 8461  df-2o 8462  df-no 27480  df-slt 27481  df-bday 27482  df-sle 27582  df-sslt 27618  df-scut 27620
This theorem is referenced by:  addsuniflem  27822  negsunif  27871  mulsuniflem  27953
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