Theorem cofcut1d 28218

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 28217	. 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 1406	1 ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∀wral 3076  ∃wrex 3086  {csn 4584   class class class wbr 5103  (class class class)co 7416   ≤s cles 28012   <<s cslts 28054   |s ccuts 28056

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6362  df-on 6363  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1o 8462  df-2o 8463  df-no 27911  df-lts 27912  df-bday 27913  df-les 28013  df-slts 28055  df-cuts 28057

This theorem is used by:  cutmax  28231  cutmin  28232  addsuniflem  28298  negsunif  28352  mulsuniflem  28446  n0fincut  28652  zcuts  28704  halfcut  28755  addhalfcut  28756  elreno2  28792