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Theorem genipv 6491
Description: Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)
Hypotheses
Ref Expression
genp.1 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
genp.2 ((y Q z Q) → (y𝐺z) Q)
Assertion
Ref Expression
genipv ((A P B P) → (A𝐹B) = ⟨{𝑞 Q𝑟 (1stA)𝑠 (1stB)𝑞 = (𝑟𝐺𝑠)}, {𝑞 Q𝑟 (2ndA)𝑠 (2ndB)𝑞 = (𝑟𝐺𝑠)}⟩)
Distinct variable groups:   x,y,z,𝑞,𝑟,𝑠,A   x,B,y,z,𝑞,𝑟,𝑠   x,w,v,𝐺,y,z,𝑞,𝑟,𝑠
Allowed substitution hints:   A(w,v)   B(w,v)   𝐹(x,y,z,w,v,𝑠,𝑟,𝑞)

Proof of Theorem genipv
Dummy variables f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5462 . . . 4 (f = A → (f𝐹g) = (A𝐹g))
2 fveq2 5121 . . . . . . 7 (f = A → (1stf) = (1stA))
32rexeqdv 2506 . . . . . 6 (f = A → (y (1stf)z (1stg)x = (y𝐺z) ↔ y (1stA)z (1stg)x = (y𝐺z)))
43rabbidv 2543 . . . . 5 (f = A → {x Qy (1stf)z (1stg)x = (y𝐺z)} = {x Qy (1stA)z (1stg)x = (y𝐺z)})
5 fveq2 5121 . . . . . . 7 (f = A → (2ndf) = (2ndA))
65rexeqdv 2506 . . . . . 6 (f = A → (y (2ndf)z (2ndg)x = (y𝐺z) ↔ y (2ndA)z (2ndg)x = (y𝐺z)))
76rabbidv 2543 . . . . 5 (f = A → {x Qy (2ndf)z (2ndg)x = (y𝐺z)} = {x Qy (2ndA)z (2ndg)x = (y𝐺z)})
84, 7opeq12d 3548 . . . 4 (f = A → ⟨{x Qy (1stf)z (1stg)x = (y𝐺z)}, {x Qy (2ndf)z (2ndg)x = (y𝐺z)}⟩ = ⟨{x Qy (1stA)z (1stg)x = (y𝐺z)}, {x Qy (2ndA)z (2ndg)x = (y𝐺z)}⟩)
91, 8eqeq12d 2051 . . 3 (f = A → ((f𝐹g) = ⟨{x Qy (1stf)z (1stg)x = (y𝐺z)}, {x Qy (2ndf)z (2ndg)x = (y𝐺z)}⟩ ↔ (A𝐹g) = ⟨{x Qy (1stA)z (1stg)x = (y𝐺z)}, {x Qy (2ndA)z (2ndg)x = (y𝐺z)}⟩))
10 oveq2 5463 . . . 4 (g = B → (A𝐹g) = (A𝐹B))
11 fveq2 5121 . . . . . . . 8 (g = B → (1stg) = (1stB))
1211rexeqdv 2506 . . . . . . 7 (g = B → (z (1stg)x = (y𝐺z) ↔ z (1stB)x = (y𝐺z)))
1312rexbidv 2321 . . . . . 6 (g = B → (y (1stA)z (1stg)x = (y𝐺z) ↔ y (1stA)z (1stB)x = (y𝐺z)))
1413rabbidv 2543 . . . . 5 (g = B → {x Qy (1stA)z (1stg)x = (y𝐺z)} = {x Qy (1stA)z (1stB)x = (y𝐺z)})
15 fveq2 5121 . . . . . . . 8 (g = B → (2ndg) = (2ndB))
1615rexeqdv 2506 . . . . . . 7 (g = B → (z (2ndg)x = (y𝐺z) ↔ z (2ndB)x = (y𝐺z)))
1716rexbidv 2321 . . . . . 6 (g = B → (y (2ndA)z (2ndg)x = (y𝐺z) ↔ y (2ndA)z (2ndB)x = (y𝐺z)))
1817rabbidv 2543 . . . . 5 (g = B → {x Qy (2ndA)z (2ndg)x = (y𝐺z)} = {x Qy (2ndA)z (2ndB)x = (y𝐺z)})
1914, 18opeq12d 3548 . . . 4 (g = B → ⟨{x Qy (1stA)z (1stg)x = (y𝐺z)}, {x Qy (2ndA)z (2ndg)x = (y𝐺z)}⟩ = ⟨{x Qy (1stA)z (1stB)x = (y𝐺z)}, {x Qy (2ndA)z (2ndB)x = (y𝐺z)}⟩)
2010, 19eqeq12d 2051 . . 3 (g = B → ((A𝐹g) = ⟨{x Qy (1stA)z (1stg)x = (y𝐺z)}, {x Qy (2ndA)z (2ndg)x = (y𝐺z)}⟩ ↔ (A𝐹B) = ⟨{x Qy (1stA)z (1stB)x = (y𝐺z)}, {x Qy (2ndA)z (2ndB)x = (y𝐺z)}⟩))
21 nqex 6347 . . . . . . 7 Q V
2221a1i 9 . . . . . 6 ((f P g P) → Q V)
23 rabssab 3021 . . . . . . 7 {x Qy (1stf)z (1stg)x = (y𝐺z)} ⊆ {xy (1stf)z (1stg)x = (y𝐺z)}
24 prop 6457 . . . . . . . . . . . 12 (f P → ⟨(1stf), (2ndf)⟩ P)
25 elprnql 6463 . . . . . . . . . . . 12 ((⟨(1stf), (2ndf)⟩ P y (1stf)) → y Q)
2624, 25sylan 267 . . . . . . . . . . 11 ((f P y (1stf)) → y Q)
27 prop 6457 . . . . . . . . . . . 12 (g P → ⟨(1stg), (2ndg)⟩ P)
28 elprnql 6463 . . . . . . . . . . . 12 ((⟨(1stg), (2ndg)⟩ P z (1stg)) → z Q)
2927, 28sylan 267 . . . . . . . . . . 11 ((g P z (1stg)) → z Q)
30 genp.2 . . . . . . . . . . . 12 ((y Q z Q) → (y𝐺z) Q)
31 eleq1 2097 . . . . . . . . . . . 12 (x = (y𝐺z) → (x Q ↔ (y𝐺z) Q))
3230, 31syl5ibrcom 146 . . . . . . . . . . 11 ((y Q z Q) → (x = (y𝐺z) → x Q))
3326, 29, 32syl2an 273 . . . . . . . . . 10 (((f P y (1stf)) (g P z (1stg))) → (x = (y𝐺z) → x Q))
3433an4s 522 . . . . . . . . 9 (((f P g P) (y (1stf) z (1stg))) → (x = (y𝐺z) → x Q))
3534rexlimdvva 2434 . . . . . . . 8 ((f P g P) → (y (1stf)z (1stg)x = (y𝐺z) → x Q))
3635abssdv 3008 . . . . . . 7 ((f P g P) → {xy (1stf)z (1stg)x = (y𝐺z)} ⊆ Q)
3723, 36syl5ss 2950 . . . . . 6 ((f P g P) → {x Qy (1stf)z (1stg)x = (y𝐺z)} ⊆ Q)
3822, 37ssexd 3888 . . . . 5 ((f P g P) → {x Qy (1stf)z (1stg)x = (y𝐺z)} V)
39 rabssab 3021 . . . . . . 7 {x Qy (2ndf)z (2ndg)x = (y𝐺z)} ⊆ {xy (2ndf)z (2ndg)x = (y𝐺z)}
40 elprnqu 6464 . . . . . . . . . . . 12 ((⟨(1stf), (2ndf)⟩ P y (2ndf)) → y Q)
4124, 40sylan 267 . . . . . . . . . . 11 ((f P y (2ndf)) → y Q)
42 elprnqu 6464 . . . . . . . . . . . 12 ((⟨(1stg), (2ndg)⟩ P z (2ndg)) → z Q)
4327, 42sylan 267 . . . . . . . . . . 11 ((g P z (2ndg)) → z Q)
4441, 43, 32syl2an 273 . . . . . . . . . 10 (((f P y (2ndf)) (g P z (2ndg))) → (x = (y𝐺z) → x Q))
4544an4s 522 . . . . . . . . 9 (((f P g P) (y (2ndf) z (2ndg))) → (x = (y𝐺z) → x Q))
4645rexlimdvva 2434 . . . . . . . 8 ((f P g P) → (y (2ndf)z (2ndg)x = (y𝐺z) → x Q))
4746abssdv 3008 . . . . . . 7 ((f P g P) → {xy (2ndf)z (2ndg)x = (y𝐺z)} ⊆ Q)
4839, 47syl5ss 2950 . . . . . 6 ((f P g P) → {x Qy (2ndf)z (2ndg)x = (y𝐺z)} ⊆ Q)
4922, 48ssexd 3888 . . . . 5 ((f P g P) → {x Qy (2ndf)z (2ndg)x = (y𝐺z)} V)
50 opelxp 4317 . . . . 5 (⟨{x Qy (1stf)z (1stg)x = (y𝐺z)}, {x Qy (2ndf)z (2ndg)x = (y𝐺z)}⟩ (V × V) ↔ ({x Qy (1stf)z (1stg)x = (y𝐺z)} V {x Qy (2ndf)z (2ndg)x = (y𝐺z)} V))
5138, 49, 50sylanbrc 394 . . . 4 ((f P g P) → ⟨{x Qy (1stf)z (1stg)x = (y𝐺z)}, {x Qy (2ndf)z (2ndg)x = (y𝐺z)}⟩ (V × V))
52 fveq2 5121 . . . . . . . 8 (w = f → (1stw) = (1stf))
5352rexeqdv 2506 . . . . . . 7 (w = f → (y (1stw)z (1stv)x = (y𝐺z) ↔ y (1stf)z (1stv)x = (y𝐺z)))
5453rabbidv 2543 . . . . . 6 (w = f → {x Qy (1stw)z (1stv)x = (y𝐺z)} = {x Qy (1stf)z (1stv)x = (y𝐺z)})
55 fveq2 5121 . . . . . . . 8 (w = f → (2ndw) = (2ndf))
5655rexeqdv 2506 . . . . . . 7 (w = f → (y (2ndw)z (2ndv)x = (y𝐺z) ↔ y (2ndf)z (2ndv)x = (y𝐺z)))
5756rabbidv 2543 . . . . . 6 (w = f → {x Qy (2ndw)z (2ndv)x = (y𝐺z)} = {x Qy (2ndf)z (2ndv)x = (y𝐺z)})
5854, 57opeq12d 3548 . . . . 5 (w = f → ⟨{x Qy (1stw)z (1stv)x = (y𝐺z)}, {x Qy (2ndw)z (2ndv)x = (y𝐺z)}⟩ = ⟨{x Qy (1stf)z (1stv)x = (y𝐺z)}, {x Qy (2ndf)z (2ndv)x = (y𝐺z)}⟩)
59 fveq2 5121 . . . . . . . . 9 (v = g → (1stv) = (1stg))
6059rexeqdv 2506 . . . . . . . 8 (v = g → (z (1stv)x = (y𝐺z) ↔ z (1stg)x = (y𝐺z)))
6160rexbidv 2321 . . . . . . 7 (v = g → (y (1stf)z (1stv)x = (y𝐺z) ↔ y (1stf)z (1stg)x = (y𝐺z)))
6261rabbidv 2543 . . . . . 6 (v = g → {x Qy (1stf)z (1stv)x = (y𝐺z)} = {x Qy (1stf)z (1stg)x = (y𝐺z)})
63 fveq2 5121 . . . . . . . . 9 (v = g → (2ndv) = (2ndg))
6463rexeqdv 2506 . . . . . . . 8 (v = g → (z (2ndv)x = (y𝐺z) ↔ z (2ndg)x = (y𝐺z)))
6564rexbidv 2321 . . . . . . 7 (v = g → (y (2ndf)z (2ndv)x = (y𝐺z) ↔ y (2ndf)z (2ndg)x = (y𝐺z)))
6665rabbidv 2543 . . . . . 6 (v = g → {x Qy (2ndf)z (2ndv)x = (y𝐺z)} = {x Qy (2ndf)z (2ndg)x = (y𝐺z)})
6762, 66opeq12d 3548 . . . . 5 (v = g → ⟨{x Qy (1stf)z (1stv)x = (y𝐺z)}, {x Qy (2ndf)z (2ndv)x = (y𝐺z)}⟩ = ⟨{x Qy (1stf)z (1stg)x = (y𝐺z)}, {x Qy (2ndf)z (2ndg)x = (y𝐺z)}⟩)
68 genp.1 . . . . . 6 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
6968genpdf 6490 . . . . 5 𝐹 = (w P, v P ↦ ⟨{x Qy (1stw)z (1stv)x = (y𝐺z)}, {x Qy (2ndw)z (2ndv)x = (y𝐺z)}⟩)
7058, 67, 69ovmpt2g 5577 . . . 4 ((f P g P ⟨{x Qy (1stf)z (1stg)x = (y𝐺z)}, {x Qy (2ndf)z (2ndg)x = (y𝐺z)}⟩ (V × V)) → (f𝐹g) = ⟨{x Qy (1stf)z (1stg)x = (y𝐺z)}, {x Qy (2ndf)z (2ndg)x = (y𝐺z)}⟩)
7151, 70mpd3an3 1232 . . 3 ((f P g P) → (f𝐹g) = ⟨{x Qy (1stf)z (1stg)x = (y𝐺z)}, {x Qy (2ndf)z (2ndg)x = (y𝐺z)}⟩)
729, 20, 71vtocl2ga 2615 . 2 ((A P B P) → (A𝐹B) = ⟨{x Qy (1stA)z (1stB)x = (y𝐺z)}, {x Qy (2ndA)z (2ndB)x = (y𝐺z)}⟩)
73 eqeq1 2043 . . . . . 6 (x = 𝑞 → (x = (y𝐺z) ↔ 𝑞 = (y𝐺z)))
74732rexbidv 2343 . . . . 5 (x = 𝑞 → (y (1stA)z (1stB)x = (y𝐺z) ↔ y (1stA)z (1stB)𝑞 = (y𝐺z)))
75 oveq1 5462 . . . . . . 7 (y = 𝑟 → (y𝐺z) = (𝑟𝐺z))
7675eqeq2d 2048 . . . . . 6 (y = 𝑟 → (𝑞 = (y𝐺z) ↔ 𝑞 = (𝑟𝐺z)))
77 oveq2 5463 . . . . . . 7 (z = 𝑠 → (𝑟𝐺z) = (𝑟𝐺𝑠))
7877eqeq2d 2048 . . . . . 6 (z = 𝑠 → (𝑞 = (𝑟𝐺z) ↔ 𝑞 = (𝑟𝐺𝑠)))
7976, 78cbvrex2v 2536 . . . . 5 (y (1stA)z (1stB)𝑞 = (y𝐺z) ↔ 𝑟 (1stA)𝑠 (1stB)𝑞 = (𝑟𝐺𝑠))
8074, 79syl6bb 185 . . . 4 (x = 𝑞 → (y (1stA)z (1stB)x = (y𝐺z) ↔ 𝑟 (1stA)𝑠 (1stB)𝑞 = (𝑟𝐺𝑠)))
8180cbvrabv 2550 . . 3 {x Qy (1stA)z (1stB)x = (y𝐺z)} = {𝑞 Q𝑟 (1stA)𝑠 (1stB)𝑞 = (𝑟𝐺𝑠)}
82732rexbidv 2343 . . . . 5 (x = 𝑞 → (y (2ndA)z (2ndB)x = (y𝐺z) ↔ y (2ndA)z (2ndB)𝑞 = (y𝐺z)))
8376, 78cbvrex2v 2536 . . . . 5 (y (2ndA)z (2ndB)𝑞 = (y𝐺z) ↔ 𝑟 (2ndA)𝑠 (2ndB)𝑞 = (𝑟𝐺𝑠))
8482, 83syl6bb 185 . . . 4 (x = 𝑞 → (y (2ndA)z (2ndB)x = (y𝐺z) ↔ 𝑟 (2ndA)𝑠 (2ndB)𝑞 = (𝑟𝐺𝑠)))
8584cbvrabv 2550 . . 3 {x Qy (2ndA)z (2ndB)x = (y𝐺z)} = {𝑞 Q𝑟 (2ndA)𝑠 (2ndB)𝑞 = (𝑟𝐺𝑠)}
8681, 85opeq12i 3545 . 2 ⟨{x Qy (1stA)z (1stB)x = (y𝐺z)}, {x Qy (2ndA)z (2ndB)x = (y𝐺z)}⟩ = ⟨{𝑞 Q𝑟 (1stA)𝑠 (1stB)𝑞 = (𝑟𝐺𝑠)}, {𝑞 Q𝑟 (2ndA)𝑠 (2ndB)𝑞 = (𝑟𝐺𝑠)}⟩
8772, 86syl6eq 2085 1 ((A P B P) → (A𝐹B) = ⟨{𝑞 Q𝑟 (1stA)𝑠 (1stB)𝑞 = (𝑟𝐺𝑠)}, {𝑞 Q𝑟 (2ndA)𝑠 (2ndB)𝑞 = (𝑟𝐺𝑠)}⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  {cab 2023  wrex 2301  {crab 2304  Vcvv 2551  cop 3370   × cxp 4286  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-bnd 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-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  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-ne 2203  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-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-qs 6048  df-ni 6288  df-nqqs 6332  df-inp 6448
This theorem is referenced by:  genpelvl  6494  genpelvu  6495  plpvlu  6520  mpvlu  6521
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