Theorem genpelvu 7663

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 7659	. . . . . 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 5604	. . . . 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 7513	. . . . . . 7 ⊢ Q ∈ V
6	5	rabex 4205	. . . . . 6 ⊢ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1st ‘𝐴)∃ℎ ∈ (1st ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ∈ V
7	5	rabex 4205	. . . . . 6 ⊢ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ∈ V
8	6, 7	op2nd 6258	. . . . 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 2256	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (2nd ‘(𝐴𝐹𝐵)) = {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)})
10	9	eleq2d 2277	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝐶 ∈ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)}))
11		elrabi 2934	. . 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 7625	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
14		elprnqu 7632	. . . . . . 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 7625	. . . . . . 7 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
17		elprnqu 7632	. . . . . . 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 6126	. . . . . 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 588	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ (2nd ‘𝐴) ∧ ℎ ∈ (2nd ‘𝐵))) → (𝑔𝐺ℎ) ∈ Q)
22		eleq1 2270	. . . 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 2634	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ Q))
25		eqeq1 2214	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺ℎ) ↔ 𝐶 = (𝑔𝐺ℎ)))
26	25	2rexbidv 2533	. . . . 5 ⊢ (𝑓 = 𝐶 → (∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ)))
27	26	elrab3 2938	. . . 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 462	. . 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 707	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2178  ∃wrex 2487  {crab 2490  ⟨cop 3647  ‘cfv 5291  (class class class)co 5969   ∈ cmpo 5971  1^st c1st 6249  2^nd c2nd 6250  Qcnq 7430  Pcnp 7441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4176  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-iinf 4655

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-iun 3944  df-br 4061  df-opab 4123  df-mpt 4124  df-id 4359  df-iom 4658  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-f1o 5298  df-fv 5299  df-ov 5972  df-oprab 5973  df-mpo 5974  df-1st 6251  df-2nd 6252  df-qs 6651  df-ni 7454  df-nqqs 7498  df-inp 7616

This theorem is referenced by:  genppreclu  7665  genpcuu  7670  genprndu  7672  genpdisj  7673  genpassu  7675  addnqprlemru  7708  mulnqprlemru  7724  distrlem1pru  7733  distrlem5pru  7737  1idpru  7741  ltexprlemfu  7761  recexprlem1ssu  7784  recexprlemss1u  7786  cauappcvgprlemladdfu  7804  caucvgprlemladdfu  7827