Theorem genpdf 7482

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 7449	. . . . . . 7 ⊢ (𝑤 ∈ P → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ P)
3		elprnql 7455	. . . . . . 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 7449	. . . . . . 7 ⊢ (𝑣 ∈ P → ⟨(1st ‘𝑣), (2nd ‘𝑣)⟩ ∈ P)
7		elprnql 7455	. . . . . . 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 7481	. . . 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 7456	. . . . . . 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 7456	. . . . . . 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 7481	. . . 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 3782	. . 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 5930	. 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 2196	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 978   = wceq 1353   ∈ wcel 2146  ∃wrex 2454  {crab 2457  ⟨cop 3592  ‘cfv 5208  (class class class)co 5865   ∈ cmpo 5867  1^st c1st 6129  2^nd c2nd 6130  Qcnq 7254  Pcnp 7265

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-iinf 4581

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-qs 6531  df-ni 7278  df-nqqs 7322  df-inp 7440

This theorem is referenced by:  genipv  7483