Theorem cofcut1d 27388

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	⊢ (𝜑 → 𝐴 <<s 𝐵)
cofcut1d.2	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦)
cofcut1d.3	⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)
cofcut1d.4	⊢ (𝜑 → 𝐶 <<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

Step	Hyp	Ref	Expression
1		cofcut1d.1	. 2 ⊢ (𝜑 → 𝐴 <<s 𝐵)
2		cofcut1d.2	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦)
3		cofcut1d.3	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)
4		cofcut1d.4	. 2 ⊢ (𝜑 → 𝐶 <<s {(𝐴 |s 𝐵)})
5		cofcut1d.5	. 2 ⊢ (𝜑 → {(𝐴 |s 𝐵)} <<s 𝐷)
6		cofcut1 27387	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
7	1, 2, 3, 4, 5, 6	syl122anc 1380	1 ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))

Syntax hints:   → wi 4   = wceq 1542  ∀wral 3062  ∃wrex 3071  {csn 4627   class class class wbr 5147  (class class class)co 7404   ≤s csle 27227   <<s csslt 27262   |s cscut 27264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1o 8461  df-2o 8462  df-no 27126  df-slt 27127  df-bday 27128  df-sle 27228  df-sslt 27263  df-scut 27265

This theorem is referenced by:  addsuniflem  27464  negsunif  27509  mulsuniflem  27584