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Theorem ltexprlemopl 6432
 Description: The lower 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
ltexprlemopl ((A<P B 𝑞 Q 𝑞 (1st𝐶)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)))
Distinct variable groups:   x,y,𝑞,𝑟,A   x,B,y,𝑞,𝑟   x,𝐶,y,𝑞,𝑟

Proof of Theorem ltexprlemopl
Dummy variable 𝑠 is 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))}⟩
21ltexprlemell 6429 . . . 4 (𝑞 (1st𝐶) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
32simprbi 260 . . 3 (𝑞 (1st𝐶) → y(y (2ndA) (y +Q 𝑞) (1stB)))
4 19.42v 1764 . . . . . . . 8 (y(A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) ↔ (A<P B y(𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))))
5 19.42v 1764 . . . . . . . . 9 (y(𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB))) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
65anbi2i 433 . . . . . . . 8 ((A<P B y(𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) ↔ (A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))))
74, 6bitri 173 . . . . . . 7 (y(A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) ↔ (A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))))
8 ltrelpr 6353 . . . . . . . . . . . . . 14 <P ⊆ (P × P)
98brel 4315 . . . . . . . . . . . . 13 (A<P B → (A P B P))
109simprd 107 . . . . . . . . . . . 12 (A<P BB P)
11 prop 6323 . . . . . . . . . . . . 13 (B P → ⟨(1stB), (2ndB)⟩ P)
12 prnmaxl 6336 . . . . . . . . . . . . 13 ((⟨(1stB), (2ndB)⟩ P (y +Q 𝑞) (1stB)) → 𝑠 (1stB)(y +Q 𝑞) <Q 𝑠)
1311, 12sylan 267 . . . . . . . . . . . 12 ((B P (y +Q 𝑞) (1stB)) → 𝑠 (1stB)(y +Q 𝑞) <Q 𝑠)
1410, 13sylan 267 . . . . . . . . . . 11 ((A<P B (y +Q 𝑞) (1stB)) → 𝑠 (1stB)(y +Q 𝑞) <Q 𝑠)
1514adantrl 450 . . . . . . . . . 10 ((A<P B (y (2ndA) (y +Q 𝑞) (1stB))) → 𝑠 (1stB)(y +Q 𝑞) <Q 𝑠)
1615adantrl 450 . . . . . . . . 9 ((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) → 𝑠 (1stB)(y +Q 𝑞) <Q 𝑠)
179simpld 105 . . . . . . . . . . . . . . 15 (A<P BA P)
1817ad2antrr 460 . . . . . . . . . . . . . 14 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → A P)
19 simplrr 476 . . . . . . . . . . . . . . 15 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → (y (2ndA) (y +Q 𝑞) (1stB)))
2019simpld 105 . . . . . . . . . . . . . 14 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → y (2ndA))
21 prop 6323 . . . . . . . . . . . . . . 15 (A P → ⟨(1stA), (2ndA)⟩ P)
22 elprnqu 6330 . . . . . . . . . . . . . . 15 ((⟨(1stA), (2ndA)⟩ P y (2ndA)) → y Q)
2321, 22sylan 267 . . . . . . . . . . . . . 14 ((A P y (2ndA)) → y Q)
2418, 20, 23syl2anc 393 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → y Q)
25 simplrl 475 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑞 Q)
26 ltaddnq 6259 . . . . . . . . . . . . 13 ((y Q 𝑞 Q) → y <Q (y +Q 𝑞))
2724, 25, 26syl2anc 393 . . . . . . . . . . . 12 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → y <Q (y +Q 𝑞))
28 simprr 472 . . . . . . . . . . . 12 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → (y +Q 𝑞) <Q 𝑠)
29 ltsonq 6251 . . . . . . . . . . . . 13 <Q Or Q
30 ltrelnq 6218 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
3129, 30sotri 4643 . . . . . . . . . . . 12 ((y <Q (y +Q 𝑞) (y +Q 𝑞) <Q 𝑠) → y <Q 𝑠)
3227, 28, 31syl2anc 393 . . . . . . . . . . 11 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → y <Q 𝑠)
3310ad2antrr 460 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → B P)
34 simprl 471 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑠 (1stB))
35 elprnql 6329 . . . . . . . . . . . . . 14 ((⟨(1stB), (2ndB)⟩ P 𝑠 (1stB)) → 𝑠 Q)
3611, 35sylan 267 . . . . . . . . . . . . 13 ((B P 𝑠 (1stB)) → 𝑠 Q)
3733, 34, 36syl2anc 393 . . . . . . . . . . . 12 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑠 Q)
38 ltexnqq 6260 . . . . . . . . . . . 12 ((y Q 𝑠 Q) → (y <Q 𝑠𝑟 Q (y +Q 𝑟) = 𝑠))
3924, 37, 38syl2anc 393 . . . . . . . . . . 11 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → (y <Q 𝑠𝑟 Q (y +Q 𝑟) = 𝑠))
4032, 39mpbid 135 . . . . . . . . . 10 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑟 Q (y +Q 𝑟) = 𝑠)
41 simplrr 476 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (y +Q 𝑞) <Q 𝑠)
42 simprr 472 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (y +Q 𝑟) = 𝑠)
4341, 42breqtrrd 3760 . . . . . . . . . . . . . 14 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (y +Q 𝑞) <Q (y +Q 𝑟))
4425adantr 261 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → 𝑞 Q)
45 simprl 471 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → 𝑟 Q)
4624adantr 261 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → y Q)
47 ltanqg 6253 . . . . . . . . . . . . . . 15 ((𝑞 Q 𝑟 Q y Q) → (𝑞 <Q 𝑟 ↔ (y +Q 𝑞) <Q (y +Q 𝑟)))
4844, 45, 46, 47syl3anc 1119 . . . . . . . . . . . . . 14 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (𝑞 <Q 𝑟 ↔ (y +Q 𝑞) <Q (y +Q 𝑟)))
4943, 48mpbird 156 . . . . . . . . . . . . 13 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → 𝑞 <Q 𝑟)
5020adantr 261 . . . . . . . . . . . . . 14 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → y (2ndA))
51 simplrl 475 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → 𝑠 (1stB))
5242, 51eqeltrd 2092 . . . . . . . . . . . . . 14 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (y +Q 𝑟) (1stB))
5350, 52jca 290 . . . . . . . . . . . . 13 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (y (2ndA) (y +Q 𝑟) (1stB)))
5449, 45, 53jca32 293 . . . . . . . . . . . 12 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
5554expr 357 . . . . . . . . . . 11 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) 𝑟 Q) → ((y +Q 𝑟) = 𝑠 → (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))))
5655reximdva 2395 . . . . . . . . . 10 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → (𝑟 Q (y +Q 𝑟) = 𝑠𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))))
5740, 56mpd 13 . . . . . . . . 9 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
5816, 57rexlimddv 2411 . . . . . . . 8 ((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) → 𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
5958eximi 1469 . . . . . . 7 (y(A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) → y𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
607, 59sylbir 125 . . . . . 6 ((A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))) → y𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
61 rexcom4 2550 . . . . . 6 (𝑟 Q y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))) ↔ y𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
6260, 61sylibr 137 . . . . 5 ((A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))) → 𝑟 Q y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
63 19.42v 1764 . . . . . . 7 (y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))) ↔ (𝑞 <Q 𝑟 y(𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
64 19.42v 1764 . . . . . . . 8 (y(𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))) ↔ (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB))))
6564anbi2i 433 . . . . . . 7 ((𝑞 <Q 𝑟 y(𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))) ↔ (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
6663, 65bitri 173 . . . . . 6 (y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))) ↔ (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
6766rexbii 2305 . . . . 5 (𝑟 Q y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))) ↔ 𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
6862, 67sylib 127 . . . 4 ((A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))) → 𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
691ltexprlemell 6429 . . . . . 6 (𝑟 (1st𝐶) ↔ (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB))))
7069anbi2i 433 . . . . 5 ((𝑞 <Q 𝑟 𝑟 (1st𝐶)) ↔ (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
7170rexbii 2305 . . . 4 (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)) ↔ 𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
7268, 71sylibr 137 . . 3 ((A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)))
733, 72sylanr2 387 . 2 ((A<P B (𝑞 Q 𝑞 (1st𝐶))) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)))
74733impb 1084 1 ((A<P B 𝑞 Q 𝑞 (1st𝐶)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)))
 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
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