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Theorem recexprlemopl 6597
 Description: The lower cut of B is open. Lemma for recexpr 6610. (Contributed by Jim Kingdon, 28-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩
Assertion
Ref Expression
recexprlemopl ((A P 𝑞 Q 𝑞 (1stB)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)))
Distinct variable groups:   𝑟,𝑞,x,y,A   B,𝑞,𝑟,x,y

Proof of Theorem recexprlemopl
StepHypRef Expression
1 recexpr.1 . . . 4 B = ⟨{xy(x <Q y (*Qy) (2ndA))}, {xy(y <Q x (*Qy) (1stA))}⟩
21recexprlemell 6594 . . 3 (𝑞 (1stB) ↔ y(𝑞 <Q y (*Qy) (2ndA)))
3 ltbtwnnqq 6398 . . . . . 6 (𝑞 <Q y𝑟 Q (𝑞 <Q 𝑟 𝑟 <Q y))
43biimpi 113 . . . . 5 (𝑞 <Q y𝑟 Q (𝑞 <Q 𝑟 𝑟 <Q y))
5 simpll 481 . . . . . . . 8 (((𝑞 <Q 𝑟 𝑟 <Q y) (*Qy) (2ndA)) → 𝑞 <Q 𝑟)
6 19.8a 1479 . . . . . . . . . 10 ((𝑟 <Q y (*Qy) (2ndA)) → y(𝑟 <Q y (*Qy) (2ndA)))
71recexprlemell 6594 . . . . . . . . . 10 (𝑟 (1stB) ↔ y(𝑟 <Q y (*Qy) (2ndA)))
86, 7sylibr 137 . . . . . . . . 9 ((𝑟 <Q y (*Qy) (2ndA)) → 𝑟 (1stB))
98adantll 445 . . . . . . . 8 (((𝑞 <Q 𝑟 𝑟 <Q y) (*Qy) (2ndA)) → 𝑟 (1stB))
105, 9jca 290 . . . . . . 7 (((𝑞 <Q 𝑟 𝑟 <Q y) (*Qy) (2ndA)) → (𝑞 <Q 𝑟 𝑟 (1stB)))
1110expcom 109 . . . . . 6 ((*Qy) (2ndA) → ((𝑞 <Q 𝑟 𝑟 <Q y) → (𝑞 <Q 𝑟 𝑟 (1stB))))
1211reximdv 2414 . . . . 5 ((*Qy) (2ndA) → (𝑟 Q (𝑞 <Q 𝑟 𝑟 <Q y) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB))))
134, 12mpan9 265 . . . 4 ((𝑞 <Q y (*Qy) (2ndA)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)))
1413exlimiv 1486 . . 3 (y(𝑞 <Q y (*Qy) (2ndA)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)))
152, 14sylbi 114 . 2 (𝑞 (1stB) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)))
16153ad2ant3 926 1 ((A P 𝑞 Q 𝑞 (1stB)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1stB)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∃wrex 2301  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  *Qcrq 6268
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