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Theorem ltexprlemopu 6567
Description: The upper cut of our constructed difference is open. Lemma for ltexpri 6577. (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 6563 . . . 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 1783 . . . . . . . 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 1783 . . . . . . . . 9 (y(𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB))) ↔ (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB))))
65anbi2i 430 . . . . . . . 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 6480 . . . . . . . . . . . . . . 15 <P ⊆ (P × P)
98brel 4334 . . . . . . . . . . . . . 14 (A<P B → (A P B P))
109simprd 107 . . . . . . . . . . . . 13 (A<P BB P)
11 prop 6450 . . . . . . . . . . . . 13 (B P → ⟨(1stB), (2ndB)⟩ P)
1210, 11syl 14 . . . . . . . . . . . 12 (A<P B → ⟨(1stB), (2ndB)⟩ P)
13 prnminu 6464 . . . . . . . . . . . 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 447 . . . . . . . . . 10 ((A<P B (y (1stA) (y +Q 𝑟) (2ndB))) → 𝑠 (2ndB)𝑠 <Q (y +Q 𝑟))
1615adantrl 447 . . . . . . . . 9 ((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) → 𝑠 (2ndB)𝑠 <Q (y +Q 𝑟))
17 ltdfpr 6481 . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . 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 6450 . . . . . . . . . . . . . . 15 (A P → ⟨(1stA), (2ndA)⟩ P)
29 prltlu 6462 . . . . . . . . . . . . . . 15 ((⟨(1stA), (2ndA)⟩ P y (1stA) 𝑡 (2ndA)) → y <Q 𝑡)
3028, 29syl3an1 1167 . . . . . . . . . . . . . 14 ((A P y (1stA) 𝑡 (2ndA)) → y <Q 𝑡)
3123, 26, 27, 30syl3anc 1134 . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . 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 6462 . . . . . . . . . . . . . . 15 ((⟨(1stB), (2ndB)⟩ P 𝑡 (1stB) 𝑠 (2ndB)) → 𝑡 <Q 𝑠)
3612, 35syl3an1 1167 . . . . . . . . . . . . . 14 ((A<P B 𝑡 (1stB) 𝑠 (2ndB)) → 𝑡 <Q 𝑠)
3732, 33, 34, 36syl3anc 1134 . . . . . . . . . . . . 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 6375 . . . . . . . . . . . . . 14 <Q Or Q
39 ltrelnq 6342 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4038, 39sotri 4662 . . . . . . . . . . . . 13 ((y <Q 𝑡 𝑡 <Q 𝑠) → y <Q 𝑠)
4131, 37, 40syl2anc 391 . . . . . . . . . . . 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 2431 . . . . . . . . . . 11 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → y <Q 𝑠)
43 elprnql 6456 . . . . . . . . . . . . . 14 ((⟨(1stA), (2ndA)⟩ P y (1stA)) → y Q)
4428, 43sylan 267 . . . . . . . . . . . . 13 ((A P y (1stA)) → y Q)
4522, 25, 44syl2anc 391 . . . . . . . . . . . 12 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → y Q)
46 elprnqu 6457 . . . . . . . . . . . . . 14 ((⟨(1stB), (2ndB)⟩ P 𝑠 (2ndB)) → 𝑠 Q)
4712, 46sylan 267 . . . . . . . . . . . . 13 ((A<P B 𝑠 (2ndB)) → 𝑠 Q)
4847ad2ant2r 478 . . . . . . . . . . . 12 (((A<P B (𝑟 Q (y (1stA) (y +Q 𝑟) (2ndB)))) (𝑠 (2ndB) 𝑠 <Q (y +Q 𝑟))) → 𝑠 Q)
49 ltexnqq 6384 . . . . . . . . . . . 12 ((y Q 𝑠 Q) → (y <Q 𝑠𝑞 Q (y +Q 𝑞) = 𝑠))
5045, 48, 49syl2anc 391 . . . . . . . . . . 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 484 . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . 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 3774 . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . 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 6377 . . . . . . . . . . . . . . 15 ((𝑞 Q 𝑟 Q y Q) → (𝑞 <Q 𝑟 ↔ (y +Q 𝑞) <Q (y +Q 𝑟)))
6055, 57, 58, 59syl3anc 1134 . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . 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 2111 . . . . . . . . . . . . . 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 2415 . . . . . . . . . 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 2431 . . . . . . . 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 1488 . . . . . . 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 2571 . . . . . 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 1783 . . . . . . 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 1783 . . . . . . . 8 (y(𝑞 Q (y (1stA) (y +Q 𝑞) (2ndB))) ↔ (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB))))
7776anbi2i 430 . . . . . . 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 2325 . . . . 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 6563 . . . . . 6 (𝑞 (2nd𝐶) ↔ (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB))))
8281anbi2i 430 . . . . 5 ((𝑞 <Q 𝑟 𝑞 (2nd𝐶)) ↔ (𝑞 <Q 𝑟 (𝑞 Q y(y (1stA) (y +Q 𝑞) (2ndB)))))
8382rexbii 2325 . . . 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 385 . 2 ((A<P B (𝑟 Q 𝑟 (2nd𝐶))) → 𝑞 Q (𝑞 <Q 𝑟 𝑞 (2nd𝐶)))
86853impb 1099 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 884   = wceq 1242  wex 1378   wcel 1390  wrex 2301  {crab 2304  cop 3369   class class class wbr 3754  cfv 4844  (class class class)co 5452  1st c1st 5704  2nd c2nd 5705  Qcnq 6257   +Q cplq 6259   <Q cltq 6262  Pcnp 6268  <P cltp 6272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-eprel 4016  df-id 4020  df-po 4023  df-iso 4024  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-1o 5933  df-oadd 5937  df-omul 5938  df-er 6035  df-ec 6037  df-qs 6041  df-ni 6281  df-pli 6282  df-mi 6283  df-lti 6284  df-plpq 6321  df-mpq 6322  df-enq 6324  df-nqqs 6325  df-plqqs 6326  df-mqqs 6327  df-1nqqs 6328  df-ltnqqs 6330  df-inp 6441  df-iltp 6445
This theorem is referenced by:  ltexprlemrnd  6569
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