Theorem genpdf 7121

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 7088	. . . . . . 7 ⊢ (𝑤 ∈ P → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ P)
3		elprnql 7094	. . . . . . 7 ⊢ ((⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ P ∧ 𝑟 ∈ (1st ‘𝑤)) → 𝑟 ∈ Q)
4	2, 3	sylan 278	. . . . . 6 ⊢ ((𝑤 ∈ P ∧ 𝑟 ∈ (1st ‘𝑤)) → 𝑟 ∈ Q)
5	4	adantlr 462	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑟 ∈ (1st ‘𝑤)) → 𝑟 ∈ Q)
6		prop 7088	. . . . . . 7 ⊢ (𝑣 ∈ P → ⟨(1st ‘𝑣), (2nd ‘𝑣)⟩ ∈ P)
7		elprnql 7094	. . . . . . 7 ⊢ ((⟨(1st ‘𝑣), (2nd ‘𝑣)⟩ ∈ P ∧ 𝑠 ∈ (1st ‘𝑣)) → 𝑠 ∈ Q)
8	6, 7	sylan 278	. . . . . 6 ⊢ ((𝑣 ∈ P ∧ 𝑠 ∈ (1st ‘𝑣)) → 𝑠 ∈ Q)
9	8	adantll 461	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑠 ∈ (1st ‘𝑣)) → 𝑠 ∈ Q)
10	5, 9	genpdflem 7120	. . . 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 7095	. . . . . . 7 ⊢ ((⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘𝑤)) → 𝑟 ∈ Q)
12	2, 11	sylan 278	. . . . . 6 ⊢ ((𝑤 ∈ P ∧ 𝑟 ∈ (2nd ‘𝑤)) → 𝑟 ∈ Q)
13	12	adantlr 462	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑟 ∈ (2nd ‘𝑤)) → 𝑟 ∈ Q)
14		elprnqu 7095	. . . . . . 7 ⊢ ((⟨(1st ‘𝑣), (2nd ‘𝑣)⟩ ∈ P ∧ 𝑠 ∈ (2nd ‘𝑣)) → 𝑠 ∈ Q)
15	6, 14	sylan 278	. . . . . 6 ⊢ ((𝑣 ∈ P ∧ 𝑠 ∈ (2nd ‘𝑣)) → 𝑠 ∈ Q)
16	15	adantll 461	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑠 ∈ (2nd ‘𝑣)) → 𝑠 ∈ Q)
17	13, 16	genpdflem 7120	. . . 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 3636	. . 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 5728	. 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 2109	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 925   = wceq 1290   ∈ wcel 1439  ∃wrex 2361  {crab 2364  ⟨cop 3453  ‘cfv 5028  (class class class)co 5666   ↦ cmpt2 5668  1^st c1st 5923  2^nd c2nd 5924  Qcnq 6893  Pcnp 6904

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-qs 6312  df-ni 6917  df-nqqs 6961  df-inp 7079

This theorem is referenced by:  genipv  7122