MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cofcutrtime Structured version   Visualization version   GIF version

Theorem cofcutrtime 27975
Description: If 𝑋 is the cut of 𝐴 and 𝐵 and all of 𝐴 and 𝐵 are older than 𝑋, then ( L ‘𝑋) is cofinal with 𝐴 and ( R ‘𝑋) is coinitial with 𝐵. Note: we will call a cut where all of the elements of the cut are older than the cut itself a "timely" cut. Part of Theorem 4.02(12) of [Alling] p. 125. (Contributed by Scott Fenton, 27-Sep-2024.)
Assertion
Ref Expression
cofcutrtime (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (∀𝑥𝐴𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧))
Distinct variable groups:   𝑥,𝐴   𝑧,𝐴   𝑥,𝐵   𝑧,𝐵   𝑧,𝑤,𝑋   𝑥,𝑋,𝑦   𝑧,𝑋
Allowed substitution hints:   𝐴(𝑦,𝑤)   𝐵(𝑦,𝑤)

Proof of Theorem cofcutrtime
StepHypRef Expression
1 ssun1 4187 . . . . . . . 8 𝐴 ⊆ (𝐴𝐵)
2 sstr 4003 . . . . . . . 8 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ ( O ‘( bday 𝑋))) → 𝐴 ⊆ ( O ‘( bday 𝑋)))
31, 2mpan 690 . . . . . . 7 ((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) → 𝐴 ⊆ ( O ‘( bday 𝑋)))
433ad2ant1 1132 . . . . . 6 (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → 𝐴 ⊆ ( O ‘( bday 𝑋)))
54sselda 3994 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 ∈ ( O ‘( bday 𝑋)))
6 simpl2 1191 . . . . . . . . 9 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝐴 <<s 𝐵)
7 scutcut 27860 . . . . . . . . 9 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
86, 7syl 17 . . . . . . . 8 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
98simp2d 1142 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝐴 <<s {(𝐴 |s 𝐵)})
10 simpr 484 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥𝐴)
11 ovex 7463 . . . . . . . . 9 (𝐴 |s 𝐵) ∈ V
1211snid 4666 . . . . . . . 8 (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)}
1312a1i 11 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)})
149, 10, 13ssltsepcd 27853 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 <s (𝐴 |s 𝐵))
15 simpl3 1192 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑋 = (𝐴 |s 𝐵))
1614, 15breqtrrd 5175 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 <s 𝑋)
17 leftval 27916 . . . . . . . 8 ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋}
1817a1i 11 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋})
1918eleq2d 2824 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → (𝑥 ∈ ( L ‘𝑋) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋}))
20 rabid 3454 . . . . . 6 (𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋} ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋))
2119, 20bitrdi 287 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋)))
225, 16, 21mpbir2and 713 . . . 4 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 ∈ ( L ‘𝑋))
23 leftssno 27933 . . . . . 6 ( L ‘𝑋) ⊆ No
2423, 22sselid 3992 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 No )
25 slerflex 27822 . . . . 5 (𝑥 No 𝑥 ≤s 𝑥)
2624, 25syl 17 . . . 4 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → 𝑥 ≤s 𝑥)
27 breq2 5151 . . . . 5 (𝑦 = 𝑥 → (𝑥 ≤s 𝑦𝑥 ≤s 𝑥))
2827rspcev 3621 . . . 4 ((𝑥 ∈ ( L ‘𝑋) ∧ 𝑥 ≤s 𝑥) → ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
2922, 26, 28syl2anc 584 . . 3 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥𝐴) → ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
3029ralrimiva 3143 . 2 (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ∀𝑥𝐴𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)
31 ssun2 4188 . . . . . . . 8 𝐵 ⊆ (𝐴𝐵)
32 sstr 4003 . . . . . . . 8 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ ( O ‘( bday 𝑋))) → 𝐵 ⊆ ( O ‘( bday 𝑋)))
3331, 32mpan 690 . . . . . . 7 ((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) → 𝐵 ⊆ ( O ‘( bday 𝑋)))
34333ad2ant1 1132 . . . . . 6 (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → 𝐵 ⊆ ( O ‘( bday 𝑋)))
3534sselda 3994 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑧 ∈ ( O ‘( bday 𝑋)))
36 simpl3 1192 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑋 = (𝐴 |s 𝐵))
37 simpl2 1191 . . . . . . . . 9 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝐴 <<s 𝐵)
3837, 7syl 17 . . . . . . . 8 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
3938simp3d 1143 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → {(𝐴 |s 𝐵)} <<s 𝐵)
4012a1i 11 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)})
41 simpr 484 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑧𝐵)
4239, 40, 41ssltsepcd 27853 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → (𝐴 |s 𝐵) <s 𝑧)
4336, 42eqbrtrd 5169 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑋 <s 𝑧)
44 rightval 27917 . . . . . . . 8 ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}
4544a1i 11 . . . . . . 7 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧})
4645eleq2d 2824 . . . . . 6 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → (𝑧 ∈ ( R ‘𝑋) ↔ 𝑧 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}))
47 rabid 3454 . . . . . 6 (𝑧 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧} ↔ (𝑧 ∈ ( O ‘( bday 𝑋)) ∧ 𝑋 <s 𝑧))
4846, 47bitrdi 287 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → (𝑧 ∈ ( R ‘𝑋) ↔ (𝑧 ∈ ( O ‘( bday 𝑋)) ∧ 𝑋 <s 𝑧)))
4935, 43, 48mpbir2and 713 . . . 4 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑧 ∈ ( R ‘𝑋))
50 rightssno 27934 . . . . . 6 ( R ‘𝑋) ⊆ No
5150, 49sselid 3992 . . . . 5 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑧 No )
52 slerflex 27822 . . . . 5 (𝑧 No 𝑧 ≤s 𝑧)
5351, 52syl 17 . . . 4 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → 𝑧 ≤s 𝑧)
54 breq1 5150 . . . . 5 (𝑤 = 𝑧 → (𝑤 ≤s 𝑧𝑧 ≤s 𝑧))
5554rspcev 3621 . . . 4 ((𝑧 ∈ ( R ‘𝑋) ∧ 𝑧 ≤s 𝑧) → ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
5649, 53, 55syl2anc 584 . . 3 ((((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧𝐵) → ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
5756ralrimiva 3143 . 2 (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ∀𝑧𝐵𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)
5830, 57jca 511 1 (((𝐴𝐵) ⊆ ( O ‘( bday 𝑋)) ∧ 𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (∀𝑥𝐴𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  cun 3960  wss 3962  {csn 4630   class class class wbr 5147  cfv 6562  (class class class)co 7430   No csur 27698   <s cslt 27699   bday cbday 27700   ≤s csle 27803   <<s csslt 27839   |s cscut 27841   O cold 27896   L cleft 27898   R cright 27899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-1o 8504  df-2o 8505  df-no 27701  df-slt 27702  df-bday 27703  df-sle 27804  df-sslt 27840  df-scut 27842  df-made 27900  df-old 27901  df-left 27903  df-right 27904
This theorem is referenced by:  cofcutrtime1d  27976  cofcutrtime2d  27977
  Copyright terms: Public domain W3C validator