Theorem genpelvu 7880

Description: Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.)

Hypotheses

Ref	Expression
genpelvl.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)

Assertion

Ref	Expression
genpelvu	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ)))

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

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

Proof of Theorem genpelvu

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		genpelvl.1	. . . . . . 7 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
2		genpelvl.2	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
3	1, 2	genipv 7876	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ (1st ‘𝐴)∃ℎ ∈ (1st ‘𝐵)𝑓 = (𝑔𝐺ℎ)}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)}⟩)
4	3	fveq2d 5699	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (2nd ‘(𝐴𝐹𝐵)) = (2nd ‘⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ (1st ‘𝐴)∃ℎ ∈ (1st ‘𝐵)𝑓 = (𝑔𝐺ℎ)}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)}⟩))
5		nqex 7730	. . . . . . 7 ⊢ Q ∈ V
6	5	rabex 4280	. . . . . 6 ⊢ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1st ‘𝐴)∃ℎ ∈ (1st ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ∈ V
7	5	rabex 4280	. . . . . 6 ⊢ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ∈ V
8	6, 7	op2nd 6381	. . . . 5 ⊢ (2nd ‘⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ (1st ‘𝐴)∃ℎ ∈ (1st ‘𝐵)𝑓 = (𝑔𝐺ℎ)}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)}⟩) = {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)}
9	4, 8	eqtrdi 2287	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (2nd ‘(𝐴𝐹𝐵)) = {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)})
10	9	eleq2d 2308	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝐶 ∈ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)}))
11		elrabi 2979	. . 3 ⊢ (𝐶 ∈ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)} → 𝐶 ∈ Q)
12	10, 11	biimtrdi 163	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) → 𝐶 ∈ Q))
13		prop 7842	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
14		elprnqu 7849	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) → 𝑔 ∈ Q)
15	13, 14	sylan 283	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) → 𝑔 ∈ Q)
16		prop 7842	. . . . . . 7 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
17		elprnqu 7849	. . . . . . 7 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ ℎ ∈ (2nd ‘𝐵)) → ℎ ∈ Q)
18	16, 17	sylan 283	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵)) → ℎ ∈ Q)
19	2	caovcl 6244	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔𝐺ℎ) ∈ Q)
20	15, 18, 19	syl2an 289	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) → (𝑔𝐺ℎ) ∈ Q)
21	20	an4s 596	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ (2nd ‘𝐴) ∧ ℎ ∈ (2nd ‘𝐵))) → (𝑔𝐺ℎ) ∈ Q)
22		eleq1 2301	. . . 4 ⊢ (𝐶 = (𝑔𝐺ℎ) → (𝐶 ∈ Q ↔ (𝑔𝐺ℎ) ∈ Q))
23	21, 22	syl5ibrcom 157	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ (2nd ‘𝐴) ∧ ℎ ∈ (2nd ‘𝐵))) → (𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ Q))
24	23	rexlimdvva 2676	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ Q))
25		eqeq1 2245	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺ℎ) ↔ 𝐶 = (𝑔𝐺ℎ)))
26	25	2rexbidv 2575	. . . . 5 ⊢ (𝑓 = 𝐶 → (∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ)))
27	26	elrab3 2983	. . . 4 ⊢ (𝐶 ∈ Q → (𝐶 ∈ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ)))
28	10, 27	sylan9bb 466	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐶 ∈ Q) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ)))
29	28	ex 115	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ Q → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ))))
30	12, 24, 29	pm5.21ndd 717	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  {crab 2532  ⟨cop 3712  ‘cfv 5377  (class class class)co 6085   ∈ cmpo 6087  1^st c1st 6372  2^nd c2nd 6373  Qcnq 7647  Pcnp 7658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-qs 6813  df-ni 7671  df-nqqs 7715  df-inp 7833

This theorem is used by:  genppreclu  7882  genpcuu  7887  genprndu  7889  genpdisj  7890  genpassu  7892  addnqprlemru  7925  mulnqprlemru  7941  distrlem1pru  7950  distrlem5pru  7954  1idpru  7958  ltexprlemfu  7978  recexprlem1ssu  8001  recexprlemss1u  8003  cauappcvgprlemladdfu  8021  caucvgprlemladdfu  8044