Theorem genpelv 10756

Description: Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (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
genpelv	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑔,ℎ,𝐴   𝑥,𝐵,𝑦,𝑧,𝑔,ℎ   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧,𝑔,ℎ   𝑔,𝐹   𝐶,𝑔,ℎ

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

Proof of Theorem genpelv

Dummy variable 𝑓 is 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	genpv 10755	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)})
4	3	eleq2d 2824	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ 𝐶 ∈ {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)}))
5		id 22	. . . . . 6 ⊢ (𝐶 = (𝑔𝐺ℎ) → 𝐶 = (𝑔𝐺ℎ))
6		ovex 7308	. . . . . 6 ⊢ (𝑔𝐺ℎ) ∈ V
7	5, 6	eqeltrdi 2847	. . . . 5 ⊢ (𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ V)
8	7	rexlimivw 3211	. . . 4 ⊢ (∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ V)
9	8	rexlimivw 3211	. . 3 ⊢ (∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ V)
10		eqeq1 2742	. . . 4 ⊢ (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺ℎ) ↔ 𝐶 = (𝑔𝐺ℎ)))
11	10	2rexbidv 3229	. . 3 ⊢ (𝑓 = 𝐶 → (∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ)))
12	9, 11	elab3 3617	. 2 ⊢ (𝐶 ∈ {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)} ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ))
13	4, 12	bitrdi 287	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065  Vcvv 3432  (class class class)co 7275   ∈ cmpo 7277  Qcnq 10608  Pcnp 10615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-ni 10628  df-nq 10668  df-np 10737

This theorem is referenced by:  genpprecl  10757  genpss  10760  genpnnp  10761  genpcd  10762  genpnmax  10763  genpass  10765  distrlem1pr  10781  distrlem5pr  10783  1idpr  10785  ltexprlem6  10797  reclem3pr  10805  reclem4pr  10806