ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfplpq2 Structured version   GIF version

Theorem dfplpq2 6338
Description: Alternative definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.)
Assertion
Ref Expression
dfplpq2 +pQ = {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩))}
Distinct variable group:   x,y,z,w,v,u,f

Proof of Theorem dfplpq2
StepHypRef Expression
1 df-mpt2 5460 . 2 (x (N × N), y (N × N) ↦ ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩) = {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) z = ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩)}
2 df-plpq 6328 . 2 +pQ = (x (N × N), y (N × N) ↦ ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩)
3 1st2nd2 5743 . . . . . . . . . 10 (x (N × N) → x = ⟨(1stx), (2ndx)⟩)
43eqeq1d 2045 . . . . . . . . 9 (x (N × N) → (x = ⟨w, v⟩ ↔ ⟨(1stx), (2ndx)⟩ = ⟨w, v⟩))
5 1st2nd2 5743 . . . . . . . . . 10 (y (N × N) → y = ⟨(1sty), (2ndy)⟩)
65eqeq1d 2045 . . . . . . . . 9 (y (N × N) → (y = ⟨u, f⟩ ↔ ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩))
74, 6bi2anan9 538 . . . . . . . 8 ((x (N × N) y (N × N)) → ((x = ⟨w, v y = ⟨u, f⟩) ↔ (⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩)))
87anbi1d 438 . . . . . . 7 ((x (N × N) y (N × N)) → (((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩) ↔ ((⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩) z = ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩)))
9 xp1st 5734 . . . . . . . . . . . . . 14 (y (N × N) → (1sty) N)
109ad2antlr 458 . . . . . . . . . . . . 13 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → (1sty) N)
117biimpa 280 . . . . . . . . . . . . . . 15 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → (⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩))
1211simprd 107 . . . . . . . . . . . . . 14 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩)
13 vex 2554 . . . . . . . . . . . . . . . . 17 u V
14 vex 2554 . . . . . . . . . . . . . . . . 17 f V
1513, 14opth2 3968 . . . . . . . . . . . . . . . 16 (⟨(1sty), (2ndy)⟩ = ⟨u, f⟩ ↔ ((1sty) = u (2ndy) = f))
1615simplbi 259 . . . . . . . . . . . . . . 15 (⟨(1sty), (2ndy)⟩ = ⟨u, f⟩ → (1sty) = u)
1716eleq1d 2103 . . . . . . . . . . . . . 14 (⟨(1sty), (2ndy)⟩ = ⟨u, f⟩ → ((1sty) Nu N))
1812, 17syl 14 . . . . . . . . . . . . 13 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → ((1sty) Nu N))
1910, 18mpbid 135 . . . . . . . . . . . 12 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → u N)
20 xp2nd 5735 . . . . . . . . . . . . . 14 (x (N × N) → (2ndx) N)
2120ad2antrr 457 . . . . . . . . . . . . 13 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → (2ndx) N)
2211simpld 105 . . . . . . . . . . . . . 14 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → ⟨(1stx), (2ndx)⟩ = ⟨w, v⟩)
23 vex 2554 . . . . . . . . . . . . . . . . 17 w V
24 vex 2554 . . . . . . . . . . . . . . . . 17 v V
2523, 24opth2 3968 . . . . . . . . . . . . . . . 16 (⟨(1stx), (2ndx)⟩ = ⟨w, v⟩ ↔ ((1stx) = w (2ndx) = v))
2625simprbi 260 . . . . . . . . . . . . . . 15 (⟨(1stx), (2ndx)⟩ = ⟨w, v⟩ → (2ndx) = v)
2726eleq1d 2103 . . . . . . . . . . . . . 14 (⟨(1stx), (2ndx)⟩ = ⟨w, v⟩ → ((2ndx) Nv N))
2822, 27syl 14 . . . . . . . . . . . . 13 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → ((2ndx) Nv N))
2921, 28mpbid 135 . . . . . . . . . . . 12 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → v N)
30 mulcompig 6315 . . . . . . . . . . . 12 ((u N v N) → (u ·N v) = (v ·N u))
3119, 29, 30syl2anc 391 . . . . . . . . . . 11 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → (u ·N v) = (v ·N u))
3231oveq2d 5471 . . . . . . . . . 10 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → ((w ·N f) +N (u ·N v)) = ((w ·N f) +N (v ·N u)))
3332opeq1d 3546 . . . . . . . . 9 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩ = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩)
3433eqeq2d 2048 . . . . . . . 8 (((x (N × N) y (N × N)) (x = ⟨w, v y = ⟨u, f⟩)) → (z = ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩ ↔ z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩))
3534pm5.32da 425 . . . . . . 7 ((x (N × N) y (N × N)) → (((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩) ↔ ((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩)))
368, 35bitr3d 179 . . . . . 6 ((x (N × N) y (N × N)) → (((⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩) z = ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩) ↔ ((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩)))
37364exbidv 1747 . . . . 5 ((x (N × N) y (N × N)) → (wvuf((⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩) z = ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩) ↔ wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩)))
38 xp1st 5734 . . . . . . 7 (x (N × N) → (1stx) N)
3938, 20jca 290 . . . . . 6 (x (N × N) → ((1stx) N (2ndx) N))
40 xp2nd 5735 . . . . . . 7 (y (N × N) → (2ndy) N)
419, 40jca 290 . . . . . 6 (y (N × N) → ((1sty) N (2ndy) N))
42 simpll 481 . . . . . . . . . . 11 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → w = (1stx))
43 simprr 484 . . . . . . . . . . 11 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → f = (2ndy))
4442, 43oveq12d 5473 . . . . . . . . . 10 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → (w ·N f) = ((1stx) ·N (2ndy)))
45 simprl 483 . . . . . . . . . . 11 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → u = (1sty))
46 simplr 482 . . . . . . . . . . 11 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → v = (2ndx))
4745, 46oveq12d 5473 . . . . . . . . . 10 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → (u ·N v) = ((1sty) ·N (2ndx)))
4844, 47oveq12d 5473 . . . . . . . . 9 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → ((w ·N f) +N (u ·N v)) = (((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))))
4946, 43oveq12d 5473 . . . . . . . . 9 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → (v ·N f) = ((2ndx) ·N (2ndy)))
5048, 49opeq12d 3548 . . . . . . . 8 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩ = ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩)
5150eqeq2d 2048 . . . . . . 7 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → (z = ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩ ↔ z = ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩))
5251copsex4g 3975 . . . . . 6 ((((1stx) N (2ndx) N) ((1sty) N (2ndy) N)) → (wvuf((⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩) z = ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩) ↔ z = ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩))
5339, 41, 52syl2an 273 . . . . 5 ((x (N × N) y (N × N)) → (wvuf((⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩) z = ⟨((w ·N f) +N (u ·N v)), (v ·N f)⟩) ↔ z = ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩))
5437, 53bitr3d 179 . . . 4 ((x (N × N) y (N × N)) → (wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩) ↔ z = ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩))
5554pm5.32i 427 . . 3 (((x (N × N) y (N × N)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩)) ↔ ((x (N × N) y (N × N)) z = ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩))
5655oprabbii 5502 . 2 {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩))} = {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) z = ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩)}
571, 2, 563eqtr4i 2067 1 +pQ = {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩))}
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  cop 3370   × cxp 4286  cfv 4845  (class class class)co 5455  {coprab 5456  cmpt2 5457  1st c1st 5707  2nd c2nd 5708  Ncnpi 6256   +N cpli 6257   ·N cmi 6258   +pQ cplpq 6260
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-nul 3874  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-nul 3219  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-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  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-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-ni 6288  df-mi 6290  df-plpq 6328
This theorem is referenced by:  addpipqqs  6354
  Copyright terms: Public domain W3C validator