Theorem genpdf 7575

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 7542	. . . . . . 7 ⊢ (𝑤 ∈ P → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ P)
3		elprnql 7548	. . . . . . 7 ⊢ ((⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ P ∧ 𝑟 ∈ (1st ‘𝑤)) → 𝑟 ∈ Q)
4	2, 3	sylan 283	. . . . . 6 ⊢ ((𝑤 ∈ P ∧ 𝑟 ∈ (1st ‘𝑤)) → 𝑟 ∈ Q)
5	4	adantlr 477	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑟 ∈ (1st ‘𝑤)) → 𝑟 ∈ Q)
6		prop 7542	. . . . . . 7 ⊢ (𝑣 ∈ P → ⟨(1st ‘𝑣), (2nd ‘𝑣)⟩ ∈ P)
7		elprnql 7548	. . . . . . 7 ⊢ ((⟨(1st ‘𝑣), (2nd ‘𝑣)⟩ ∈ P ∧ 𝑠 ∈ (1st ‘𝑣)) → 𝑠 ∈ Q)
8	6, 7	sylan 283	. . . . . 6 ⊢ ((𝑣 ∈ P ∧ 𝑠 ∈ (1st ‘𝑣)) → 𝑠 ∈ Q)
9	8	adantll 476	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑠 ∈ (1st ‘𝑣)) → 𝑠 ∈ Q)
10	5, 9	genpdflem 7574	. . . 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 7549	. . . . . . 7 ⊢ ((⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘𝑤)) → 𝑟 ∈ Q)
12	2, 11	sylan 283	. . . . . 6 ⊢ ((𝑤 ∈ P ∧ 𝑟 ∈ (2nd ‘𝑤)) → 𝑟 ∈ Q)
13	12	adantlr 477	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑟 ∈ (2nd ‘𝑤)) → 𝑟 ∈ Q)
14		elprnqu 7549	. . . . . . 7 ⊢ ((⟨(1st ‘𝑣), (2nd ‘𝑣)⟩ ∈ P ∧ 𝑠 ∈ (2nd ‘𝑣)) → 𝑠 ∈ Q)
15	6, 14	sylan 283	. . . . . 6 ⊢ ((𝑣 ∈ P ∧ 𝑠 ∈ (2nd ‘𝑣)) → 𝑠 ∈ Q)
16	15	adantll 476	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑠 ∈ (2nd ‘𝑣)) → 𝑠 ∈ Q)
17	13, 16	genpdflem 7574	. . . 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 3816	. . 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 5987	. 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 2217	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 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∃wrex 2476  {crab 2479  ⟨cop 3625  ‘cfv 5258  (class class class)co 5922   ∈ cmpo 5924  1^st c1st 6196  2^nd c2nd 6197  Qcnq 7347  Pcnp 7358

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-qs 6598  df-ni 7371  df-nqqs 7415  df-inp 7533

This theorem is referenced by:  genipv  7576