Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltexprlemopu Structured version   GIF version

Theorem ltexprlemopu 6434
 Description: The upper cut of our constructed difference is open. Lemma for ltexpri 6444. (Contributed by Jim Kingdon, 21-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
ltexprlemopu ((A<P B 𝑟 Q 𝑟 (2nd𝐶)) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))
Distinct variable groups:   x,y,𝑞,𝑟,A   x,B,y,𝑞,𝑟   x,𝐶,y,𝑞,𝑟

Proof of Theorem ltexprlemopu
Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexprlem.1 . . . . 5 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
21ltexprlemelu 6430 . . . 4 (𝑟 (2nd𝐶) ↔ (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB))))
32simprbi 260 . . 3 (𝑟 (2nd𝐶) → y(y (1stA) (y +Q 𝑟) (2ndB)))
4 19.42v 1764 . . . . . . . 8 (y(A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) ↔ (A<P B y(𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))))
5 19.42v 1764 . . . . . . . . 9 (y(𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB))) ↔ (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB))))
65anbi2i 433 . . . . . . . 8 ((A<P B y(𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) ↔ (A<P B (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB)))))
74, 6bitri 173 . . . . . . 7 (y(A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) ↔ (A<P B (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB)))))
8 ltrelpr 6353 . . . . . . . . . . . . . . 15 <P ⊆ (P × P)
98brel 4315 . . . . . . . . . . . . . 14 (A<P B → (A P B P))
109simprd 107 . . . . . . . . . . . . 13 (A<P BB P)
11 prop 6323 . . . . . . . . . . . . 13 (B P → ⟨(1stB), (2ndB)⟩ P)
1210, 11syl 14 . . . . . . . . . . . 12 (A<P B → ⟨(1stB), (2ndB)⟩ P)
13 prnminu 6337 . . . . . . . . . . . 12 ((⟨(1stB), (2ndB)⟩ P (y +Q 𝑟) (2ndB)) → 𝑠 (2ndB)𝑠 <Q (y +Q 𝑟))
1412, 13sylan 267 . . . . . . . . . . 11 ((A<P B (y +Q 𝑟) (2ndB)) → 𝑠 (2ndB)𝑠 <Q (y +Q 𝑟))
1514adantrl 450 . . . . . . . . . 10 ((A<P B (y (1stA) (y +Q 𝑟) (2ndB))) → 𝑠 (2ndB)𝑠 <Q (y +Q 𝑟))
1615adantrl 450 . . . . . . . . 9 ((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) → 𝑠 (2ndB)𝑠 <Q (y +Q 𝑟))
17 ltdfpr 6354 . . . . . . . . . . . . . . 15 ((A P B P) → (A<P B𝑡 Q (𝑡 (2ndA) 𝑡 (1stB))))
1817biimpd 132 . . . . . . . . . . . . . 14 ((A P B P) → (A<P B𝑡 Q (𝑡 (2ndA) 𝑡 (1stB))))
199, 18mpcom 32 . . . . . . . . . . . . 13 (A<P B𝑡 Q (𝑡 (2ndA) 𝑡 (1stB)))
2019ad2antrr 460 . . . . . . . . . . . 12 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → 𝑡 Q (𝑡 (2ndA) 𝑡 (1stB)))
219simpld 105 . . . . . . . . . . . . . . . 16 (A<P BA P)
2221ad2antrr 460 . . . . . . . . . . . . . . 15 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → A P)
2322adantr 261 . . . . . . . . . . . . . 14 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑡 Q (𝑡 (2ndA) 𝑡 (1stB)))) → A P)
24 simplrr 476 . . . . . . . . . . . . . . . 16 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → (y (1stA) (y +Q 𝑟) (2ndB)))
2524simpld 105 . . . . . . . . . . . . . . 15 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → y (1stA))
2625adantr 261 . . . . . . . . . . . . . 14 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑡 Q (𝑡 (2ndA) 𝑡 (1stB)))) → y (1stA))
27 simprrl 479 . . . . . . . . . . . . . 14 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑡 Q (𝑡 (2ndA) 𝑡 (1stB)))) → 𝑡 (2ndA))
28 prop 6323 . . . . . . . . . . . . . . 15 (A P → ⟨(1stA), (2ndA)⟩ P)
29 prltlu 6335 . . . . . . . . . . . . . . 15 ((⟨(1stA), (2ndA)⟩ P y (1stA) 𝑡 (2ndA)) → y <Q 𝑡)
3028, 29syl3an1 1152 . . . . . . . . . . . . . 14 ((A P y (1stA) 𝑡 (2ndA)) → y <Q 𝑡)
3123, 26, 27, 30syl3anc 1119 . . . . . . . . . . . . 13 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑡 Q (𝑡 (2ndA) 𝑡 (1stB)))) → y <Q 𝑡)
32 simplll 473 . . . . . . . . . . . . . 14 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑡 Q (𝑡 (2ndA) 𝑡 (1stB)))) → A<P B)
33 simprrr 480 . . . . . . . . . . . . . 14 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑡 Q (𝑡 (2ndA) 𝑡 (1stB)))) → 𝑡 (1stB))
34 simplrl 475 . . . . . . . . . . . . . 14 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑡 Q (𝑡 (2ndA) 𝑡 (1stB)))) → 𝑠 (2ndB))
35 prltlu 6335 . . . . . . . . . . . . . . 15 ((⟨(1stB), (2ndB)⟩ P 𝑡 (1stB) 𝑠 (2ndB)) → 𝑡 <Q 𝑠)
3612, 35syl3an1 1152 . . . . . . . . . . . . . 14 ((A<P B 𝑡 (1stB) 𝑠 (2ndB)) → 𝑡 <Q 𝑠)
3732, 33, 34, 36syl3anc 1119 . . . . . . . . . . . . 13 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑡 Q (𝑡 (2ndA) 𝑡 (1stB)))) → 𝑡 <Q 𝑠)
38 ltsonq 6251 . . . . . . . . . . . . . 14 <Q Or Q
39 ltrelnq 6218 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4038, 39sotri 4643 . . . . . . . . . . . . 13 ((y <Q 𝑡 𝑡 <Q 𝑠) → y <Q 𝑠)
4131, 37, 40syl2anc 393 . . . . . . . . . . . 12 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑡 Q (𝑡 (2ndA) 𝑡 (1stB)))) → y <Q 𝑠)
4220, 41rexlimddv 2411 . . . . . . . . . . 11 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → y <Q 𝑠)
43 elprnql 6329 . . . . . . . . . . . . . 14 ((⟨(1stA), (2ndA)⟩ P y (1stA)) → y Q)
4428, 43sylan 267 . . . . . . . . . . . . 13 ((A P y (1stA)) → y Q)
4522, 25, 44syl2anc 393 . . . . . . . . . . . 12 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → y Q)
46 elprnqu 6330 . . . . . . . . . . . . . 14 ((⟨(1stB), (2ndB)⟩ P 𝑠 (2ndB)) → 𝑠 Q)
4712, 46sylan 267 . . . . . . . . . . . . 13 ((A<P B 𝑠 (2ndB)) → 𝑠 Q)
4847ad2ant2r 466 . . . . . . . . . . . 12 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → 𝑠 Q)
49 ltexnqq 6260 . . . . . . . . . . . 12 ((y Q 𝑠 Q) → (y <Q 𝑠𝑞 Q (y +Q 𝑞) = 𝑠))
5045, 48, 49syl2anc 393 . . . . . . . . . . 11 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → (y <Q 𝑠𝑞 Q (y +Q 𝑞) = 𝑠))
5142, 50mpbid 135 . . . . . . . . . 10 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → 𝑞 Q (y +Q 𝑞) = 𝑠)
52 simprr 472 . . . . . . . . . . . . . . 15 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → (y +Q 𝑞) = 𝑠)
53 simplrr 476 . . . . . . . . . . . . . . 15 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → 𝑠 <Q (y +Q 𝑟))
5452, 53eqbrtrd 3754 . . . . . . . . . . . . . 14 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → (y +Q 𝑞) <Q (y +Q 𝑟))
55 simprl 471 . . . . . . . . . . . . . . 15 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → 𝑞 Q)
56 simplrl 475 . . . . . . . . . . . . . . . 16 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → 𝑟 Q)
5756adantr 261 . . . . . . . . . . . . . . 15 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → 𝑟 Q)
5845adantr 261 . . . . . . . . . . . . . . 15 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → y Q)
59 ltanqg 6253 . . . . . . . . . . . . . . 15 ((𝑞 Q 𝑟 Q y Q) → (𝑞 <Q 𝑟 ↔ (y +Q 𝑞) <Q (y +Q 𝑟)))
6055, 57, 58, 59syl3anc 1119 . . . . . . . . . . . . . 14 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → (𝑞 <Q 𝑟 ↔ (y +Q 𝑞) <Q (y +Q 𝑟)))
6154, 60mpbird 156 . . . . . . . . . . . . 13 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → 𝑞 <Q 𝑟)
6225adantr 261 . . . . . . . . . . . . . 14 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → y (1stA))
63 simplrl 475 . . . . . . . . . . . . . . 15 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → 𝑠 (2ndB))
6452, 63eqeltrd 2092 . . . . . . . . . . . . . 14 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → (y +Q 𝑞) (2ndB))
6562, 64jca 290 . . . . . . . . . . . . 13 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → (y (1stA) (y +Q 𝑞) (2ndB)))
6661, 55, 65jca32 293 . . . . . . . . . . . 12 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) (𝑞 Q (y +Q 𝑞) = 𝑠)) → (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))))
6766expr 357 . . . . . . . . . . 11 ((((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) 𝑞 Q) → ((y +Q 𝑞) = 𝑠 → (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))))
6867reximdva 2395 . . . . . . . . . 10 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → (𝑞 Q (y +Q 𝑞) = 𝑠𝑞 Q (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))))))
6951, 68mpd 13 . . . . . . . . 9 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → 𝑞 Q (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))))
7016, 69rexlimddv 2411 . . . . . . . 8 ((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) → 𝑞 Q (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))))
7170eximi 1469 . . . . . . 7 (y(A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) → y𝑞 Q (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))))
727, 71sylbir 125 . . . . . 6 ((A<P B (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB)))) → y𝑞 Q (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))))
73 rexcom4 2550 . . . . . 6 (𝑞 Q y(𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))) ↔ y𝑞 Q (𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))))
7472, 73sylibr 137 . . . . 5 ((A<P B (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB)))) → 𝑞 Q y(𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))))
75 19.42v 1764 . . . . . . 7 (y(𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))) ↔ (𝑞 <Q 𝑟 y(𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))))
76 19.42v 1764 . . . . . . . 8 (y(𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))) ↔ (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB))))
7776anbi2i 433 . . . . . . 7 ((𝑞 <Q 𝑟 y(𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))) ↔ (𝑞 <Q 𝑟 (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB)))))
7875, 77bitri 173 . . . . . 6 (y(𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))) ↔ (𝑞 <Q 𝑟 (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB)))))
7978rexbii 2305 . . . . 5 (𝑞 Q y(𝑞 <Q 𝑟 (𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB)))) ↔ 𝑞 Q (𝑞 <Q 𝑟 (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB)))))
8074, 79sylib 127 . . . 4 ((A<P B (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB)))) → 𝑞 Q (𝑞 <Q 𝑟 (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB)))))
811ltexprlemelu 6430 . . . . . 6 (𝑞 (2nd𝐶) ↔ (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB))))
8281anbi2i 433 . . . . 5 ((𝑞 <Q 𝑟 𝑞 (2nd𝐶)) ↔ (𝑞 <Q 𝑟 (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB)))))
8382rexbii 2305 . . . 4 (𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)) ↔ 𝑞 Q (𝑞 <Q 𝑟 (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB)))))
8480, 83sylibr 137 . . 3 ((A<P B (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB)))) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))
853, 84sylanr2 387 . 2 ((A<P B (𝑟 Q 𝑟 (2nd𝐶))) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))
86853impb 1084 1 ((A<P B 𝑟 Q 𝑟 (2nd𝐶)) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 871   = wceq 1226  ∃wex 1358   ∈ wcel 1370  ∃wrex 2281  {crab 2284  ⟨cop 3349   class class class wbr 3734  ‘cfv 4825  (class class class)co 5432  1st c1st 5684  2nd c2nd 5685  Qcnq 6134   +Q cplq 6136
 Copyright terms: Public domain W3C validator