Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  resspos Structured version   Visualization version   GIF version

Theorem resspos 30404
Description: The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Assertion
Ref Expression
resspos ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Poset)

Proof of Theorem resspos
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovexd 7008 . 2 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ V)
2 eqid 2771 . . . . . . 7 (𝐹s 𝐴) = (𝐹s 𝐴)
3 eqid 2771 . . . . . . 7 (Base‘𝐹) = (Base‘𝐹)
42, 3ressbas 16408 . . . . . 6 (𝐴𝑉 → (𝐴 ∩ (Base‘𝐹)) = (Base‘(𝐹s 𝐴)))
5 inss2 4087 . . . . . 6 (𝐴 ∩ (Base‘𝐹)) ⊆ (Base‘𝐹)
64, 5syl6eqssr 3905 . . . . 5 (𝐴𝑉 → (Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹))
76adantl 474 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹))
8 eqid 2771 . . . . . . 7 (le‘𝐹) = (le‘𝐹)
93, 8ispos 17427 . . . . . 6 (𝐹 ∈ Poset ↔ (𝐹 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
109simprbi 489 . . . . 5 (𝐹 ∈ Poset → ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)))
1110adantr 473 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)))
12 ssralv 3916 . . . . . . . 8 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
1312ralimdv 3121 . . . . . . 7 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
14 ssralv 3916 . . . . . . 7 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
1513, 14syld 47 . . . . . 6 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
1615ralimdv 3121 . . . . 5 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
17 ssralv 3916 . . . . 5 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
1816, 17syld 47 . . . 4 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
197, 11, 18sylc 65 . . 3 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)))
202, 8ressle 16526 . . . . 5 (𝐴𝑉 → (le‘𝐹) = (le‘(𝐹s 𝐴)))
2120adantl 474 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (le‘𝐹) = (le‘(𝐹s 𝐴)))
22 breq 4927 . . . . . . 7 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑥𝑥(le‘(𝐹s 𝐴))𝑥))
23 breq 4927 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑦𝑥(le‘(𝐹s 𝐴))𝑦))
24 breq 4927 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑦(le‘𝐹)𝑥𝑦(le‘(𝐹s 𝐴))𝑥))
2523, 24anbi12d 622 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) ↔ (𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥)))
2625imbi1d 334 . . . . . . 7 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ↔ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦)))
27 breq 4927 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑦(le‘𝐹)𝑧𝑦(le‘(𝐹s 𝐴))𝑧))
2823, 27anbi12d 622 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) ↔ (𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧)))
29 breq 4927 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑧𝑥(le‘(𝐹s 𝐴))𝑧))
3028, 29imbi12d 337 . . . . . . 7 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧) ↔ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧)))
3122, 26, 303anbi123d 1416 . . . . . 6 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → ((𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ (𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧))))
3231ralbidv 3140 . . . . 5 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ ∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧))))
33322ralbidv 3142 . . . 4 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧))))
3421, 33syl 17 . . 3 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧))))
3519, 34mpbid 224 . 2 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧)))
36 eqid 2771 . . 3 (Base‘(𝐹s 𝐴)) = (Base‘(𝐹s 𝐴))
37 eqid 2771 . . 3 (le‘(𝐹s 𝐴)) = (le‘(𝐹s 𝐴))
3836, 37ispos 17427 . 2 ((𝐹s 𝐴) ∈ Poset ↔ ((𝐹s 𝐴) ∈ V ∧ ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧))))
391, 35, 38sylanbrc 575 1 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069   = wceq 1508  wcel 2051  wral 3081  Vcvv 3408  cin 3821  wss 3822   class class class wbr 4925  cfv 6185  (class class class)co 6974  Basecbs 16337  s cress 16338  lecple 16426  Posetcpo 17420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-nn 11438  df-2 11501  df-3 11502  df-4 11503  df-5 11504  df-6 11505  df-7 11506  df-8 11507  df-9 11508  df-dec 11910  df-ndx 16340  df-slot 16341  df-base 16343  df-sets 16344  df-ress 16345  df-ple 16439  df-poset 17426
This theorem is referenced by:  resstos  30405
  Copyright terms: Public domain W3C validator