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Theorem cofcut2 33746
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, 25-Sep-2024.)
Assertion
Ref Expression
cofcut2 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → (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:   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 cofcut2
StepHypRef Expression
1 simp11 1204 . 2 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → 𝐴 <<s 𝐵)
2 simp2 1138 . 2 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧))
3 simp12 1205 . . 3 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → 𝐶 ∈ 𝒫 No )
4 simp3l 1202 . . 3 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → ∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢)
5 scutcut 33650 . . . . 5 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
61, 5syl 17 . . . 4 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
76simp2d 1144 . . 3 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → 𝐴 <<s {(𝐴 |s 𝐵)})
8 cofsslt 33743 . . 3 ((𝐶 ∈ 𝒫 No ∧ ∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢𝐴 <<s {(𝐴 |s 𝐵)}) → 𝐶 <<s {(𝐴 |s 𝐵)})
93, 4, 7, 8syl3anc 1372 . 2 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → 𝐶 <<s {(𝐴 |s 𝐵)})
10 simp13 1206 . . 3 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → 𝐷 ∈ 𝒫 No )
11 simp3r 1203 . . 3 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)
126simp3d 1145 . . 3 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → {(𝐴 |s 𝐵)} <<s 𝐵)
13 coinitsslt 33744 . . 3 ((𝐷 ∈ 𝒫 No ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟 ∧ {(𝐴 |s 𝐵)} <<s 𝐵) → {(𝐴 |s 𝐵)} <<s 𝐷)
1410, 11, 12, 13syl3anc 1372 . 2 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → {(𝐴 |s 𝐵)} <<s 𝐷)
15 cofcut1 33745 . 2 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
161, 2, 9, 14, 15syl112anc 1375 1 (((𝐴 <<s 𝐵𝐶 ∈ 𝒫 No 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  wrex 3055  𝒫 cpw 4498  {csn 4526   class class class wbr 5040  (class class class)co 7182   No csur 33498   ≤s csle 33602   <<s csslt 33630   |s cscut 33632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-ord 6185  df-on 6186  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-1o 8143  df-2o 8144  df-no 33501  df-slt 33502  df-bday 33503  df-sle 33603  df-sslt 33631  df-scut 33633
This theorem is referenced by: (None)
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