Theorem genpdf 6491

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 6458	. . . . . . 7 ⊢ (w ∈ P → ⟨(1st ‘w), (2nd ‘w)⟩ ∈ P)
3		elprnql 6464	. . . . . . 7 ⊢ ((⟨(1st ‘w), (2nd ‘w)⟩ ∈ P ∧ 𝑟 ∈ (1st ‘w)) → 𝑟 ∈ Q)
4	2, 3	sylan 267	. . . . . 6 ⊢ ((w ∈ P ∧ 𝑟 ∈ (1st ‘w)) → 𝑟 ∈ Q)
5	4	adantlr 446	. . . . 5 ⊢ (((w ∈ P ∧ v ∈ P) ∧ 𝑟 ∈ (1st ‘w)) → 𝑟 ∈ Q)
6		prop 6458	. . . . . . 7 ⊢ (v ∈ P → ⟨(1st ‘v), (2nd ‘v)⟩ ∈ P)
7		elprnql 6464	. . . . . . 7 ⊢ ((⟨(1st ‘v), (2nd ‘v)⟩ ∈ P ∧ 𝑠 ∈ (1st ‘v)) → 𝑠 ∈ Q)
8	6, 7	sylan 267	. . . . . 6 ⊢ ((v ∈ P ∧ 𝑠 ∈ (1st ‘v)) → 𝑠 ∈ Q)
9	8	adantll 445	. . . . 5 ⊢ (((w ∈ P ∧ v ∈ P) ∧ 𝑠 ∈ (1st ‘v)) → 𝑠 ∈ Q)
10	5, 9	genpdflem 6490	. . . 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 6465	. . . . . . 7 ⊢ ((⟨(1st ‘w), (2nd ‘w)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘w)) → 𝑟 ∈ Q)
12	2, 11	sylan 267	. . . . . 6 ⊢ ((w ∈ P ∧ 𝑟 ∈ (2nd ‘w)) → 𝑟 ∈ Q)
13	12	adantlr 446	. . . . 5 ⊢ (((w ∈ P ∧ v ∈ P) ∧ 𝑟 ∈ (2nd ‘w)) → 𝑟 ∈ Q)
14		elprnqu 6465	. . . . . . 7 ⊢ ((⟨(1st ‘v), (2nd ‘v)⟩ ∈ P ∧ 𝑠 ∈ (2nd ‘v)) → 𝑠 ∈ Q)
15	6, 14	sylan 267	. . . . . 6 ⊢ ((v ∈ P ∧ 𝑠 ∈ (2nd ‘v)) → 𝑠 ∈ Q)
16	15	adantll 445	. . . . 5 ⊢ (((w ∈ P ∧ v ∈ P) ∧ 𝑠 ∈ (2nd ‘v)) → 𝑠 ∈ Q)
17	13, 16	genpdflem 6490	. . . 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 3548	. . 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	mpt2eq3ia 5512	. 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 2057	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 97   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  {crab 2304  ⟨cop 3370  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  1^st c1st 5707  2^nd c2nd 5708  Qcnq 6264  Pcnp 6275

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-qs 6048  df-ni 6288  df-nqqs 6332  df-inp 6449

This theorem is referenced by:  genipv  6492