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Theorem prslem 16852
Description: Lemma for prsref 16853 and prstr 16854. (Contributed by Mario Carneiro, 1-Feb-2015.)
Hypotheses
Ref Expression
isprs.b 𝐵 = (Base‘𝐾)
isprs.l = (le‘𝐾)
Assertion
Ref Expression
prslem ((𝐾 ∈ Preset ∧ (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 16851 . . 3 (𝐾 ∈ Preset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
43simprbi 480 . 2 (𝐾 ∈ Preset → ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
5 breq12 4618 . . . . 5 ((𝑥 = 𝑋𝑥 = 𝑋) → (𝑥 𝑥𝑋 𝑋))
65anidms 676 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑥𝑋 𝑋))
7 breq1 4616 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
87anbi1d 740 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑧) ↔ (𝑋 𝑦𝑦 𝑧)))
9 breq1 4616 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑧𝑋 𝑧))
108, 9imbi12d 334 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧) ↔ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧)))
116, 10anbi12d 746 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧))))
12 breq2 4617 . . . . . 6 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
13 breq1 4616 . . . . . 6 (𝑦 = 𝑌 → (𝑦 𝑧𝑌 𝑧))
1412, 13anbi12d 746 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑧) ↔ (𝑋 𝑌𝑌 𝑧)))
1514imbi1d 331 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧)))
1615anbi2d 739 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑋 ∧ ((𝑋 𝑦𝑦 𝑧) → 𝑋 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧))))
17 breq2 4617 . . . . . 6 (𝑧 = 𝑍 → (𝑌 𝑧𝑌 𝑍))
1817anbi2d 739 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌𝑌 𝑧) ↔ (𝑋 𝑌𝑌 𝑍)))
19 breq2 4617 . . . . 5 (𝑧 = 𝑍 → (𝑋 𝑧𝑋 𝑍))
2018, 19imbi12d 334 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧) ↔ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
2120anbi2d 739 . . 3 (𝑧 = 𝑍 → ((𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑧) → 𝑋 𝑧)) ↔ (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))))
2211, 16, 21rspc3v 3309 . 2 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))))
234, 22mpan9 486 1 ((𝐾 ∈ Preset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186   class class class wbr 4613  cfv 5847  Basecbs 15781  lecple 15869   Preset cpreset 16847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-preset 16849
This theorem is referenced by:  prsref  16853  prstr  16854
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