Theorem genpss 10984

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 10980	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)))
4		elprnq 10971	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
5	4	ex 417	. . . . . . 7 ⊢ (𝐴 ∈ P → (𝑔 ∈ 𝐴 → 𝑔 ∈ Q))
6		elprnq 10971	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
7	6	ex 417	. . . . . . 7 ⊢ (𝐵 ∈ P → (ℎ ∈ 𝐵 → ℎ ∈ Q))
8	5, 7	im2anan9 631	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑔 ∈ Q ∧ ℎ ∈ Q)))
9	2	caovcl 7604	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔𝐺ℎ) ∈ Q)
10	8, 9	syl6 36	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑔𝐺ℎ) ∈ Q))
11		eleq1a 2858	. . . . 5 ⊢ ((𝑔𝐺ℎ) ∈ Q → (𝑓 = (𝑔𝐺ℎ) → 𝑓 ∈ Q))
12	10, 11	syl6 36	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑓 = (𝑔𝐺ℎ) → 𝑓 ∈ Q)))
13	12	rexlimdvv 3221	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ) → 𝑓 ∈ Q))
14	3, 13	sylbid 243	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → 𝑓 ∈ Q))
15	14	ssrdv 3943	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ⊆ Q)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  ∃wrex 3089   ⊆ wss 3905  (class class class)co 7410   ∈ cmpo 7412  Qcnq 10832  Pcnp 10839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-ni 10852  df-nq 10892  df-np 10961

This theorem is referenced by:  genpcl  10988