Theorem genpdf 7316

Description: Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)

Hypothesis

Ref	Expression
genpdf.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑤) ∧ 𝑠 ∈ (1st ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑤) ∧ 𝑠 ∈ (2nd ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩)

Assertion

Ref	Expression
genpdf	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝑤)∃𝑠 ∈ (1st ‘𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝑤)∃𝑠 ∈ (2nd ‘𝑣)𝑞 = (𝑟𝐺𝑠)}⟩)

Distinct variable group:   𝑟,𝑞,𝑠,𝑣,𝑤

Allowed substitution hints:   𝐹(𝑤,𝑣,𝑠,𝑟,𝑞)   𝐺(𝑤,𝑣,𝑠,𝑟,𝑞)

Proof of Theorem genpdf

Step	Hyp	Ref	Expression
1		genpdf.1	. 2 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑤) ∧ 𝑠 ∈ (1st ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑤) ∧ 𝑠 ∈ (2nd ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩)
2		prop 7283	. . . . . . 7 ⊢ (𝑤 ∈ P → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ P)
3		elprnql 7289	. . . . . . 7 ⊢ ((⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ P ∧ 𝑟 ∈ (1st ‘𝑤)) → 𝑟 ∈ Q)
4	2, 3	sylan 281	. . . . . 6 ⊢ ((𝑤 ∈ P ∧ 𝑟 ∈ (1st ‘𝑤)) → 𝑟 ∈ Q)
5	4	adantlr 468	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑟 ∈ (1st ‘𝑤)) → 𝑟 ∈ Q)
6		prop 7283	. . . . . . 7 ⊢ (𝑣 ∈ P → ⟨(1st ‘𝑣), (2nd ‘𝑣)⟩ ∈ P)
7		elprnql 7289	. . . . . . 7 ⊢ ((⟨(1st ‘𝑣), (2nd ‘𝑣)⟩ ∈ P ∧ 𝑠 ∈ (1st ‘𝑣)) → 𝑠 ∈ Q)
8	6, 7	sylan 281	. . . . . 6 ⊢ ((𝑣 ∈ P ∧ 𝑠 ∈ (1st ‘𝑣)) → 𝑠 ∈ Q)
9	8	adantll 467	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑠 ∈ (1st ‘𝑣)) → 𝑠 ∈ Q)
10	5, 9	genpdflem 7315	. . . 4 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑤) ∧ 𝑠 ∈ (1st ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝑤)∃𝑠 ∈ (1st ‘𝑣)𝑞 = (𝑟𝐺𝑠)})
11		elprnqu 7290	. . . . . . 7 ⊢ ((⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘𝑤)) → 𝑟 ∈ Q)
12	2, 11	sylan 281	. . . . . 6 ⊢ ((𝑤 ∈ P ∧ 𝑟 ∈ (2nd ‘𝑤)) → 𝑟 ∈ Q)
13	12	adantlr 468	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑟 ∈ (2nd ‘𝑤)) → 𝑟 ∈ Q)
14		elprnqu 7290	. . . . . . 7 ⊢ ((⟨(1st ‘𝑣), (2nd ‘𝑣)⟩ ∈ P ∧ 𝑠 ∈ (2nd ‘𝑣)) → 𝑠 ∈ Q)
15	6, 14	sylan 281	. . . . . 6 ⊢ ((𝑣 ∈ P ∧ 𝑠 ∈ (2nd ‘𝑣)) → 𝑠 ∈ Q)
16	15	adantll 467	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑠 ∈ (2nd ‘𝑣)) → 𝑠 ∈ Q)
17	13, 16	genpdflem 7315	. . . 4 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑤) ∧ 𝑠 ∈ (2nd ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))} = {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝑤)∃𝑠 ∈ (2nd ‘𝑣)𝑞 = (𝑟𝐺𝑠)})
18	10, 17	opeq12d 3713	. . 3 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑤) ∧ 𝑠 ∈ (1st ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑤) ∧ 𝑠 ∈ (2nd ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩ = ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝑤)∃𝑠 ∈ (1st ‘𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝑤)∃𝑠 ∈ (2nd ‘𝑣)𝑞 = (𝑟𝐺𝑠)}⟩)
19	18	mpoeq3ia 5836	. 2 ⊢ (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑤) ∧ 𝑠 ∈ (1st ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑤) ∧ 𝑠 ∈ (2nd ‘𝑣) ∧ 𝑞 = (𝑟𝐺𝑠))}⟩) = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝑤)∃𝑠 ∈ (1st ‘𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝑤)∃𝑠 ∈ (2nd ‘𝑣)𝑞 = (𝑟𝐺𝑠)}⟩)
20	1, 19	eqtri 2160	1 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ (1st ‘𝑤)∃𝑠 ∈ (1st ‘𝑣)𝑞 = (𝑟𝐺𝑠)}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ (2nd ‘𝑤)∃𝑠 ∈ (2nd ‘𝑣)𝑞 = (𝑟𝐺𝑠)}⟩)

Syntax hints:   ∧ wa 103   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∃wrex 2417  {crab 2420  ⟨cop 3530  ‘cfv 5123  (class class class)co 5774   ∈ cmpo 5776  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088  Pcnp 7099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7112  df-nqqs 7156  df-inp 7274

This theorem is referenced by:  genipv  7317