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Theorem ltprordil 6563
 Description: If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.)
Assertion
Ref Expression
ltprordil (A<P B → (1stA) ⊆ (1stB))

Proof of Theorem ltprordil
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 6487 . . . 4 <P ⊆ (P × P)
21brel 4335 . . 3 (A<P B → (A P B P))
3 ltdfpr 6488 . . . 4 ((A P B P) → (A<P Bx Q (x (2ndA) x (1stB))))
43biimpd 132 . . 3 ((A P B P) → (A<P Bx Q (x (2ndA) x (1stB))))
52, 4mpcom 32 . 2 (A<P Bx Q (x (2ndA) x (1stB)))
6 simpll 481 . . . . . 6 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → A<P B)
7 simpr 103 . . . . . 6 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → y (1stA))
8 simprrl 491 . . . . . . 7 ((A<P B (x Q (x (2ndA) x (1stB)))) → x (2ndA))
98adantr 261 . . . . . 6 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → x (2ndA))
102simpld 105 . . . . . . . 8 (A<P BA P)
11 prop 6457 . . . . . . . 8 (A P → ⟨(1stA), (2ndA)⟩ P)
1210, 11syl 14 . . . . . . 7 (A<P B → ⟨(1stA), (2ndA)⟩ P)
13 prltlu 6469 . . . . . . 7 ((⟨(1stA), (2ndA)⟩ P y (1stA) x (2ndA)) → y <Q x)
1412, 13syl3an1 1167 . . . . . 6 ((A<P B y (1stA) x (2ndA)) → y <Q x)
156, 7, 9, 14syl3anc 1134 . . . . 5 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → y <Q x)
16 simprrr 492 . . . . . . 7 ((A<P B (x Q (x (2ndA) x (1stB)))) → x (1stB))
1716adantr 261 . . . . . 6 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → x (1stB))
182simprd 107 . . . . . . . 8 (A<P BB P)
19 prop 6457 . . . . . . . 8 (B P → ⟨(1stB), (2ndB)⟩ P)
2018, 19syl 14 . . . . . . 7 (A<P B → ⟨(1stB), (2ndB)⟩ P)
21 prcdnql 6466 . . . . . . 7 ((⟨(1stB), (2ndB)⟩ P x (1stB)) → (y <Q xy (1stB)))
2220, 21sylan 267 . . . . . 6 ((A<P B x (1stB)) → (y <Q xy (1stB)))
236, 17, 22syl2anc 391 . . . . 5 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → (y <Q xy (1stB)))
2415, 23mpd 13 . . . 4 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → y (1stB))
2524ex 108 . . 3 ((A<P B (x Q (x (2ndA) x (1stB)))) → (y (1stA) → y (1stB)))
2625ssrdv 2945 . 2 ((A<P B (x Q (x (2ndA) x (1stB)))) → (1stA) ⊆ (1stB))
275, 26rexlimddv 2431 1 (A<P B → (1stA) ⊆ (1stB))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  ∃wrex 2301   ⊆ wss 2911  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264
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