Theorem cofcut2d 27836

Description: If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.)

Hypotheses

Ref	Expression
cofcut2d.1	⊢ (𝜑 → 𝐴 <<s 𝐵)
cofcut2d.2	⊢ (𝜑 → 𝐶 ∈ 𝒫 No )
cofcut2d.3	⊢ (𝜑 → 𝐷 ∈ 𝒫 No )
cofcut2d.4	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦)
cofcut2d.5	⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)
cofcut2d.6	⊢ (𝜑 → ∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢)
cofcut2d.7	⊢ (𝜑 → ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)

Assertion

Ref	Expression
cofcut2d	⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))

Distinct variable groups:   𝑡,𝐴,𝑢   𝑥,𝐴   𝐵,𝑟,𝑠   𝑧,𝐵   𝑡,𝐶   𝑥,𝐶,𝑦   𝐷,𝑟   𝑤,𝐷,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑢,𝑡,𝑠,𝑟)   𝐴(𝑦,𝑧,𝑤,𝑠,𝑟)   𝐵(𝑥,𝑦,𝑤,𝑢,𝑡)   𝐶(𝑧,𝑤,𝑢,𝑠,𝑟)   𝐷(𝑥,𝑦,𝑢,𝑡,𝑠)

Proof of Theorem cofcut2d

Step	Hyp	Ref	Expression
1		cofcut2d.1	. 2 ⊢ (𝜑 → 𝐴 <<s 𝐵)
2		cofcut2d.2	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝒫 No )
3		cofcut2d.3	. 2 ⊢ (𝜑 → 𝐷 ∈ 𝒫 No )
4		cofcut2d.4	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦)
5		cofcut2d.5	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)
6		cofcut2d.6	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢)
7		cofcut2d.7	. 2 ⊢ (𝜑 → ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)
8		cofcut2 27835	. 2 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 ∈ 𝒫 No ∧ 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
9	1, 2, 3, 4, 5, 6, 7, 8	syl322anc 1400	1 ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  𝒫 cpw 4551   class class class wbr 5092  (class class class)co 7349   No csur 27549   ≤s csle 27654   <<s csslt 27691   |s cscut 27693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1o 8388  df-2o 8389  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694

This theorem is referenced by:  cutlt  27845