Theorem cofcut2d 27867

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 27866	. 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  𝒫 cpw 4547   class class class wbr 5089  (class class class)co 7346   No csur 27578   ≤s csle 27683   <<s csslt 27720   |s cscut 27722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1o 8385  df-2o 8386  df-no 27581  df-slt 27582  df-bday 27583  df-sle 27684  df-sslt 27721  df-scut 27723

This theorem is referenced by:  cutlt  27876