Theorem prcdnq 11062

Description: A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
prcdnq	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → (𝐶 <Q 𝐵 → 𝐶 ∈ 𝐴))

Proof of Theorem prcdnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 10995	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
2		relxp 5718	. . . . . . 7 ⊢ Rel (Q × Q)
3		relss 5805	. . . . . . 7 ⊢ ( <Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel <Q ))
4	1, 2, 3	mp2 9	. . . . . 6 ⊢ Rel <Q
5	4	brrelex1i 5756	. . . . 5 ⊢ (𝐶 <Q 𝐵 → 𝐶 ∈ V)
6		eleq1 2832	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
7	6	anbi2d 629	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ 𝐴)))
8		breq2 5170	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑦 <Q 𝑥 ↔ 𝑦 <Q 𝐵))
9	7, 8	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 <Q 𝑥) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 <Q 𝐵)))
10	9	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦 ∈ 𝐴) ↔ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦 ∈ 𝐴)))
11		breq1 5169	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝑦 <Q 𝐵 ↔ 𝐶 <Q 𝐵))
12	11	anbi2d 629	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 <Q 𝐵) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵)))
13		eleq1 2832	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
14	12, 13	imbi12d 344	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦 ∈ 𝐴) ↔ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴)))
15		elnpi 11057	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
16	15	simprbi 496	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
17	16	r19.21bi 3257	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
18	17	simpld 494	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴))
19	18	19.21bi 2190	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → (𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴))
20	19	imp 406	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦 ∈ 𝐴)
21	10, 14, 20	vtocl2g 3586	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V) → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴))
22	5, 21	sylan2 592	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 <Q 𝐵) → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴))
23	22	adantll 713	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴))
24	23	pm2.43i 52	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴)
25	24	ex 412	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → (𝐶 <Q 𝐵 → 𝐶 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1535   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976   ⊊ wpss 3977  ∅c0 4352   class class class wbr 5166   × cxp 5698  Rel wrel 5705  Qcnq 10921   <_Q cltq 10927  Pcnp 10928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-ltnq 10987  df-np 11050

This theorem is referenced by:  prub  11063  addclprlem1  11085  mulclprlem  11088  distrlem4pr  11095  1idpr  11098  psslinpr  11100  prlem934  11102  ltaddpr  11103  ltexprlem2  11106  ltexprlem3  11107  ltexprlem6  11110  prlem936  11116  reclem2pr  11117  suplem1pr  11121