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Theorem cofcut2 27971
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 27861 . . . . 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 27967 . . 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 27968 . . 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 27969 . 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 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  (class class class)co 7431   No csur 27699   ≤s csle 27804   <<s csslt 27840   |s cscut 27842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843
This theorem is referenced by:  cofcut2d  27972
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