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Theorem cofcut1d 27956
Description: If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷, then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.)
Hypotheses
Ref Expression
cofcut1d.1 (𝜑𝐴 <<s 𝐵)
cofcut1d.2 (𝜑 → ∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦)
cofcut1d.3 (𝜑 → ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧)
cofcut1d.4 (𝜑𝐶 <<s {(𝐴 |s 𝐵)})
cofcut1d.5 (𝜑 → {(𝐴 |s 𝐵)} <<s 𝐷)
Assertion
Ref Expression
cofcut1d (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Distinct variable groups:   𝑥,𝐴   𝑧,𝐵   𝑥,𝐶,𝑦   𝑤,𝐷,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑦,𝑧,𝑤)   𝐵(𝑥,𝑦,𝑤)   𝐶(𝑧,𝑤)   𝐷(𝑥,𝑦)

Proof of Theorem cofcut1d
StepHypRef Expression
1 cofcut1d.1 . 2 (𝜑𝐴 <<s 𝐵)
2 cofcut1d.2 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦)
3 cofcut1d.3 . 2 (𝜑 → ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧)
4 cofcut1d.4 . 2 (𝜑𝐶 <<s {(𝐴 |s 𝐵)})
5 cofcut1d.5 . 2 (𝜑 → {(𝐴 |s 𝐵)} <<s 𝐷)
6 cofcut1 27955 . 2 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
71, 2, 3, 4, 5, 6syl122anc 1380 1 (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wral 3060  wrex 3069  {csn 4625   class class class wbr 5142  (class class class)co 7432   ≤s csle 27790   <<s csslt 27826   |s cscut 27828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1o 8507  df-2o 8508  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829
This theorem is referenced by:  cutmax  27969  cutmin  27970  addsuniflem  28035  negsunif  28088  mulsuniflem  28176  zscut  28394  halfcut  28417  addhalfcut  28420
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