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

Theorem prslem 18323
Description: Lemma for prsref 18324 and prstr 18325. (Contributed by Mario Carneiro, 1-Feb-2015.)
Hypotheses
Ref Expression
isprs.b 𝐵 = (Base‘𝐾)
isprs.l = (le‘𝐾)
Assertion
Ref Expression
prslem ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))

Proof of Theorem prslem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isprs.b . . . 4 𝐵 = (Base‘𝐾)
2 isprs.l . . . 4 = (le‘𝐾)
31, 2isprs 18322 . . 3 (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
43simprbi 495 . 2 (𝐾 ∈ Proset → ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
5 breq12 5158 . . . . 5 ((𝑥 = 𝑋𝑥 = 𝑋) → (𝑥 𝑥𝑋 𝑋))
65anidms 565 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑥𝑋 𝑋))
7 breq1 5156 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
87anbi1d 629 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑧) ↔ (𝑋 𝑦𝑦 𝑧)))
9 breq1 5156 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑧𝑋 𝑧))
108, 9imbi12d 343 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧) ↔ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧)))
116, 10anbi12d 630 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧))))
12 breq2 5157 . . . . . 6 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
13 breq1 5156 . . . . . 6 (𝑦 = 𝑌 → (𝑦 𝑧𝑌 𝑧))
1412, 13anbi12d 630 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑧) ↔ (𝑋 𝑌𝑌 𝑧)))
1514imbi1d 340 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧)))
1615anbi2d 628 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑋 ∧ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧))))
17 breq2 5157 . . . . . 6 (𝑧 = 𝑍 → (𝑌 𝑧𝑌 𝑍))
1817anbi2d 628 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌𝑌 𝑧) ↔ (𝑋 𝑌𝑌 𝑍)))
19 breq2 5157 . . . . 5 (𝑧 = 𝑍 → (𝑋 𝑧𝑋 𝑍))
2018, 19imbi12d 343 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
2120anbi2d 628 . . 3 (𝑧 = 𝑍 → ((𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))))
2211, 16, 21rspc3v 3624 . 2 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))))
234, 22mpan9 505 1 ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  Vcvv 3462   class class class wbr 5153  cfv 6554  Basecbs 17213  lecple 17273   Proset cproset 18318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-iota 6506  df-fv 6562  df-proset 18320
This theorem is referenced by:  prsref  18324  prstr  18325
  Copyright terms: Public domain W3C validator