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Theorem ltexprlempr 6582
 Description: Our constructed difference is a positive real. Lemma for ltexpri 6587. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
Assertion
Ref Expression
ltexprlempr (A<P B𝐶 P)
Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y

Proof of Theorem ltexprlempr
Dummy variables 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexprlem.1 . . . 4 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
21ltexprlemm 6574 . . 3 (A<P B → (𝑞 Q 𝑞 (1st𝐶) 𝑟 Q 𝑟 (2nd𝐶)))
3 ssrab2 3019 . . . . . 6 {x Qy(y (2ndA) (y +Q x) (1stB))} ⊆ Q
4 nqex 6347 . . . . . . 7 Q V
54elpw2 3902 . . . . . 6 ({x Qy(y (2ndA) (y +Q x) (1stB))} 𝒫 Q ↔ {x Qy(y (2ndA) (y +Q x) (1stB))} ⊆ Q)
63, 5mpbir 134 . . . . 5 {x Qy(y (2ndA) (y +Q x) (1stB))} 𝒫 Q
7 ssrab2 3019 . . . . . 6 {x Qy(y (1stA) (y +Q x) (2ndB))} ⊆ Q
84elpw2 3902 . . . . . 6 ({x Qy(y (1stA) (y +Q x) (2ndB))} 𝒫 Q ↔ {x Qy(y (1stA) (y +Q x) (2ndB))} ⊆ Q)
97, 8mpbir 134 . . . . 5 {x Qy(y (1stA) (y +Q x) (2ndB))} 𝒫 Q
10 opelxpi 4319 . . . . 5 (({x Qy(y (2ndA) (y +Q x) (1stB))} 𝒫 Q {x Qy(y (1stA) (y +Q x) (2ndB))} 𝒫 Q) → ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩ (𝒫 Q × 𝒫 Q))
116, 9, 10mp2an 402 . . . 4 ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩ (𝒫 Q × 𝒫 Q)
121, 11eqeltri 2107 . . 3 𝐶 (𝒫 Q × 𝒫 Q)
132, 12jctil 295 . 2 (A<P B → (𝐶 (𝒫 Q × 𝒫 Q) (𝑞 Q 𝑞 (1st𝐶) 𝑟 Q 𝑟 (2nd𝐶))))
141ltexprlemrnd 6579 . . 3 (A<P B → (𝑞 Q (𝑞 (1st𝐶) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶))) 𝑟 Q (𝑟 (2nd𝐶) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))))
151ltexprlemdisj 6580 . . 3 (A<P B𝑞 Q ¬ (𝑞 (1st𝐶) 𝑞 (2nd𝐶)))
161ltexprlemloc 6581 . . 3 (A<P B𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐶) 𝑟 (2nd𝐶))))
1714, 15, 163jca 1083 . 2 (A<P B → ((𝑞 Q (𝑞 (1st𝐶) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶))) 𝑟 Q (𝑟 (2nd𝐶) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))) 𝑞 Q ¬ (𝑞 (1st𝐶) 𝑞 (2nd𝐶)) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐶) 𝑟 (2nd𝐶)))))
18 elnp1st2nd 6459 . 2 (𝐶 P ↔ ((𝐶 (𝒫 Q × 𝒫 Q) (𝑞 Q 𝑞 (1st𝐶) 𝑟 Q 𝑟 (2nd𝐶))) ((𝑞 Q (𝑞 (1st𝐶) ↔ 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶))) 𝑟 Q (𝑟 (2nd𝐶) ↔ 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))) 𝑞 Q ¬ (𝑞 (1st𝐶) 𝑞 (2nd𝐶)) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐶) 𝑟 (2nd𝐶))))))
1913, 17, 18sylanbrc 394 1 (A<P B𝐶 P)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   ∧ w3a 884   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  {crab 2304   ⊆ wss 2911  𝒫 cpw 3351  ⟨cop 3370   class class class wbr 3755   × cxp 4286  ‘cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +Q cplq 6266
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