Theorem cofcut1d 27886

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 27885	. 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 1381	1 ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))

Syntax hints:   → wi 4   = wceq 1540  ∀wral 3052  ∃wrex 3061  {csn 4606   class class class wbr 5124  (class class class)co 7410   ≤s csle 27713   <<s csslt 27749   |s cscut 27751

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1o 8485  df-2o 8486  df-no 27611  df-slt 27612  df-bday 27613  df-sle 27714  df-sslt 27750  df-scut 27752

This theorem is referenced by:  cutmax  27899  cutmin  27900  addsuniflem  27965  negsunif  28018  mulsuniflem  28109  n0sfincut  28303  zscut  28352  halfcut  28390  addhalfcut  28391