Theorem genpelvu 7512

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 7508	. . . . . 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 5520	. . . . 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 7362	. . . . . . 7 ⊢ Q ∈ V
6	5	rabex 4148	. . . . . 6 ⊢ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1st ‘𝐴)∃ℎ ∈ (1st ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ∈ V
7	5	rabex 4148	. . . . . 6 ⊢ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ∈ V
8	6, 7	op2nd 6148	. . . . 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 2226	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (2nd ‘(𝐴𝐹𝐵)) = {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)})
10	9	eleq2d 2247	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ 𝐶 ∈ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)}))
11		elrabi 2891	. . 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 7474	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
14		elprnqu 7481	. . . . . . 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 7474	. . . . . . 7 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
17		elprnqu 7481	. . . . . . 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 6029	. . . . . 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 2240	. . . 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 2602	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ Q))
25		eqeq1 2184	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺ℎ) ↔ 𝐶 = (𝑔𝐺ℎ)))
26	25	2rexbidv 2502	. . . . 5 ⊢ (𝑓 = 𝐶 → (∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝐶 = (𝑔𝐺ℎ)))
27	26	elrab3 2895	. . . 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 705	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 978   = wceq 1353   ∈ wcel 2148  ∃wrex 2456  {crab 2459  ⟨cop 3596  ‘cfv 5217  (class class class)co 5875   ∈ cmpo 5877  1^st c1st 6139  2^nd c2nd 6140  Qcnq 7279  Pcnp 7290

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-qs 6541  df-ni 7303  df-nqqs 7347  df-inp 7465

This theorem is referenced by:  genppreclu  7514  genpcuu  7519  genprndu  7521  genpdisj  7522  genpassu  7524  addnqprlemru  7557  mulnqprlemru  7573  distrlem1pru  7582  distrlem5pru  7586  1idpru  7590  ltexprlemfu  7610  recexprlem1ssu  7633  recexprlemss1u  7635  cauappcvgprlemladdfu  7653  caucvgprlemladdfu  7676