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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 Q𝑟 Q 𝑠 Q (𝑟 (1stw) 𝑠 (1stv) 𝑞 = (𝑟𝐺𝑠))}, {𝑞 Q𝑟 Q 𝑠 Q (𝑟 (2ndw) 𝑠 (2ndv) 𝑞 = (𝑟𝐺𝑠))}⟩)
Assertion
Ref Expression
genpdf 𝐹 = (w P, v P ↦ ⟨{𝑞 Q𝑟 (1stw)𝑠 (1stv)𝑞 = (𝑟𝐺𝑠)}, {𝑞 Q𝑟 (2ndw)𝑠 (2ndv)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
StepHypRef Expression
1 genpdf.1 . 2 𝐹 = (w P, v P ↦ ⟨{𝑞 Q𝑟 Q 𝑠 Q (𝑟 (1stw) 𝑠 (1stv) 𝑞 = (𝑟𝐺𝑠))}, {𝑞 Q𝑟 Q 𝑠 Q (𝑟 (2ndw) 𝑠 (2ndv) 𝑞 = (𝑟𝐺𝑠))}⟩)
2 prop 6458 . . . . . . 7 (w P → ⟨(1stw), (2ndw)⟩ P)
3 elprnql 6464 . . . . . . 7 ((⟨(1stw), (2ndw)⟩ P 𝑟 (1stw)) → 𝑟 Q)
42, 3sylan 267 . . . . . 6 ((w P 𝑟 (1stw)) → 𝑟 Q)
54adantlr 446 . . . . 5 (((w P v P) 𝑟 (1stw)) → 𝑟 Q)
6 prop 6458 . . . . . . 7 (v P → ⟨(1stv), (2ndv)⟩ P)
7 elprnql 6464 . . . . . . 7 ((⟨(1stv), (2ndv)⟩ P 𝑠 (1stv)) → 𝑠 Q)
86, 7sylan 267 . . . . . 6 ((v P 𝑠 (1stv)) → 𝑠 Q)
98adantll 445 . . . . 5 (((w P v P) 𝑠 (1stv)) → 𝑠 Q)
105, 9genpdflem 6490 . . . 4 ((w P v P) → {𝑞 Q𝑟 Q 𝑠 Q (𝑟 (1stw) 𝑠 (1stv) 𝑞 = (𝑟𝐺𝑠))} = {𝑞 Q𝑟 (1stw)𝑠 (1stv)𝑞 = (𝑟𝐺𝑠)})
11 elprnqu 6465 . . . . . . 7 ((⟨(1stw), (2ndw)⟩ P 𝑟 (2ndw)) → 𝑟 Q)
122, 11sylan 267 . . . . . 6 ((w P 𝑟 (2ndw)) → 𝑟 Q)
1312adantlr 446 . . . . 5 (((w P v P) 𝑟 (2ndw)) → 𝑟 Q)
14 elprnqu 6465 . . . . . . 7 ((⟨(1stv), (2ndv)⟩ P 𝑠 (2ndv)) → 𝑠 Q)
156, 14sylan 267 . . . . . 6 ((v P 𝑠 (2ndv)) → 𝑠 Q)
1615adantll 445 . . . . 5 (((w P v P) 𝑠 (2ndv)) → 𝑠 Q)
1713, 16genpdflem 6490 . . . 4 ((w P v P) → {𝑞 Q𝑟 Q 𝑠 Q (𝑟 (2ndw) 𝑠 (2ndv) 𝑞 = (𝑟𝐺𝑠))} = {𝑞 Q𝑟 (2ndw)𝑠 (2ndv)𝑞 = (𝑟𝐺𝑠)})
1810, 17opeq12d 3548 . . 3 ((w P v P) → ⟨{𝑞 Q𝑟 Q 𝑠 Q (𝑟 (1stw) 𝑠 (1stv) 𝑞 = (𝑟𝐺𝑠))}, {𝑞 Q𝑟 Q 𝑠 Q (𝑟 (2ndw) 𝑠 (2ndv) 𝑞 = (𝑟𝐺𝑠))}⟩ = ⟨{𝑞 Q𝑟 (1stw)𝑠 (1stv)𝑞 = (𝑟𝐺𝑠)}, {𝑞 Q𝑟 (2ndw)𝑠 (2ndv)𝑞 = (𝑟𝐺𝑠)}⟩)
1918mpt2eq3ia 5512 . 2 (w P, v P ↦ ⟨{𝑞 Q𝑟 Q 𝑠 Q (𝑟 (1stw) 𝑠 (1stv) 𝑞 = (𝑟𝐺𝑠))}, {𝑞 Q𝑟 Q 𝑠 Q (𝑟 (2ndw) 𝑠 (2ndv) 𝑞 = (𝑟𝐺𝑠))}⟩) = (w P, v P ↦ ⟨{𝑞 Q𝑟 (1stw)𝑠 (1stv)𝑞 = (𝑟𝐺𝑠)}, {𝑞 Q𝑟 (2ndw)𝑠 (2ndv)𝑞 = (𝑟𝐺𝑠)}⟩)
201, 19eqtri 2057 1 𝐹 = (w P, v P ↦ ⟨{𝑞 Q𝑟 (1stw)𝑠 (1stv)𝑞 = (𝑟𝐺𝑠)}, {𝑞 Q𝑟 (2ndw)𝑠 (2ndv)𝑞 = (𝑟𝐺𝑠)}⟩)
Colors of variables: wff set class
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  1st c1st 5707  2nd 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
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