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Theorem ltprordil 7245
 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 7161 . . . 4 <P ⊆ (P × P)
21brel 4519 . . 3 (𝐴<P 𝐵 → (𝐴P𝐵P))
3 ltdfpr 7162 . . . 4 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵))))
43biimpd 143 . . 3 ((𝐴P𝐵P) → (𝐴<P 𝐵 → ∃𝑥Q (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵))))
52, 4mpcom 36 . 2 (𝐴<P 𝐵 → ∃𝑥Q (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))
6 simpll 497 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝐴<P 𝐵)
7 simpr 109 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 ∈ (1st𝐴))
8 simprrl 507 . . . . . . 7 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → 𝑥 ∈ (2nd𝐴))
98adantr 271 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑥 ∈ (2nd𝐴))
102simpld 111 . . . . . . . 8 (𝐴<P 𝐵𝐴P)
11 prop 7131 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1210, 11syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 prltlu 7143 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑦 <Q 𝑥)
1412, 13syl3an1 1214 . . . . . 6 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑦 <Q 𝑥)
156, 7, 9, 14syl3anc 1181 . . . . 5 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 <Q 𝑥)
16 simprrr 508 . . . . . . 7 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → 𝑥 ∈ (1st𝐵))
1716adantr 271 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑥 ∈ (1st𝐵))
182simprd 113 . . . . . . . 8 (𝐴<P 𝐵𝐵P)
19 prop 7131 . . . . . . . 8 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2018, 19syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
21 prcdnql 7140 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (1st𝐵)) → (𝑦 <Q 𝑥𝑦 ∈ (1st𝐵)))
2220, 21sylan 278 . . . . . 6 ((𝐴<P 𝐵𝑥 ∈ (1st𝐵)) → (𝑦 <Q 𝑥𝑦 ∈ (1st𝐵)))
236, 17, 22syl2anc 404 . . . . 5 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → (𝑦 <Q 𝑥𝑦 ∈ (1st𝐵)))
2415, 23mpd 13 . . . 4 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 ∈ (1st𝐵))
2524ex 114 . . 3 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → (𝑦 ∈ (1st𝐴) → 𝑦 ∈ (1st𝐵)))
2625ssrdv 3045 . 2 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → (1st𝐴) ⊆ (1st𝐵))
275, 26rexlimddv 2507 1 (𝐴<P 𝐵 → (1st𝐴) ⊆ (1st𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1445  ∃wrex 2371   ⊆ wss 3013  ⟨cop 3469   class class class wbr 3867  ‘cfv 5049  1st c1st 5947  2nd c2nd 5948  Qcnq 6936
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