Theorem genpdf 7865

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 7832	. . . . . . 7 ⊢ (𝑤 ∈ P → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ P)
3		elprnql 7838	. . . . . . 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 481	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑟 ∈ (1st ‘𝑤)) → 𝑟 ∈ Q)
6		prop 7832	. . . . . . 7 ⊢ (𝑣 ∈ P → ⟨(1st ‘𝑣), (2nd ‘𝑣)⟩ ∈ P)
7		elprnql 7838	. . . . . . 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 480	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑠 ∈ (1st ‘𝑣)) → 𝑠 ∈ Q)
10	5, 9	genpdflem 7864	. . . 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 7839	. . . . . . 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 481	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑟 ∈ (2nd ‘𝑤)) → 𝑟 ∈ Q)
14		elprnqu 7839	. . . . . . 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 480	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑠 ∈ (2nd ‘𝑣)) → 𝑠 ∈ Q)
17	13, 16	genpdflem 7864	. . . 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 3907	. . 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 6143	. 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 2259	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 1009   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  {crab 2532  ⟨cop 3708  ‘cfv 5372  (class class class)co 6075   ∈ cmpo 6077  1^st c1st 6362  2^nd c2nd 6363  Qcnq 7637  Pcnp 7648

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-qs 6803  df-ni 7661  df-nqqs 7705  df-inp 7823

This theorem is referenced by:  genipv  7866