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Theorem cofcut2 34091
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 1202 . 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 1136 . 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 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)) → 𝐶 ∈ 𝒫 No )
4 simp3l 1200 . . 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 33995 . . . . 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 1142 . . 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 34088 . . 3 ((𝐶 ∈ 𝒫 No ∧ ∀𝑡𝐶𝑢𝐴 𝑡 ≤s 𝑢𝐴 <<s {(𝐴 |s 𝐵)}) → 𝐶 <<s {(𝐴 |s 𝐵)})
93, 4, 7, 8syl3anc 1370 . 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 1204 . . 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 1201 . . 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 1143 . . 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 34089 . . 3 ((𝐷 ∈ 𝒫 No ∧ ∀𝑟𝐷𝑠𝐵 𝑠 ≤s 𝑟 ∧ {(𝐴 |s 𝐵)} <<s 𝐵) → {(𝐴 |s 𝐵)} <<s 𝐷)
1410, 11, 12, 13syl3anc 1370 . 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 34090 . 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 1373 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 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065  𝒫 cpw 4533  {csn 4561   class class class wbr 5074  (class class class)co 7275   No csur 33843   ≤s csle 33947   <<s csslt 33975   |s cscut 33977
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sle 33948  df-sslt 33976  df-scut 33978
This theorem is referenced by: (None)
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