Theorem genpss 11028

Description: The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
genp.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)

Assertion

Ref	Expression
genpss	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ⊆ Q)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genpss

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		genp.1	. . . 4 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
2		genp.2	. . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
3	1, 2	genpelv 11024	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)))
4		elprnq 11015	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
5	4	ex 412	. . . . . . 7 ⊢ (𝐴 ∈ P → (𝑔 ∈ 𝐴 → 𝑔 ∈ Q))
6		elprnq 11015	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
7	6	ex 412	. . . . . . 7 ⊢ (𝐵 ∈ P → (ℎ ∈ 𝐵 → ℎ ∈ Q))
8	5, 7	im2anan9 619	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑔 ∈ Q ∧ ℎ ∈ Q)))
9	2	caovcl 7615	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔𝐺ℎ) ∈ Q)
10	8, 9	syl6 35	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑔𝐺ℎ) ∈ Q))
11		eleq1a 2824	. . . . 5 ⊢ ((𝑔𝐺ℎ) ∈ Q → (𝑓 = (𝑔𝐺ℎ) → 𝑓 ∈ Q))
12	10, 11	syl6 35	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑓 = (𝑔𝐺ℎ) → 𝑓 ∈ Q)))
13	12	rexlimdvv 3207	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ) → 𝑓 ∈ Q))
14	3, 13	sylbid 239	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → 𝑓 ∈ Q))
15	14	ssrdv 3986	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ⊆ Q)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2705  ∃wrex 3067   ⊆ wss 3947  (class class class)co 7420   ∈ cmpo 7422  Qcnq 10876  Pcnp 10883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9665

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-ni 10896  df-nq 10936  df-np 11005

This theorem is referenced by:  genpcl  11032