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Theorem dfmpq2 6339
Description: Alternative definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.)
Assertion
Ref Expression
dfmpq2 ·pQ = {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩))}
Distinct variable group:   x,y,z,w,v,u,f

Proof of Theorem dfmpq2
StepHypRef Expression
1 df-mpt2 5460 . 2 (x (N × N), y (N × N) ↦ ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩) = {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) z = ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩)}
2 df-mpq 6329 . 2 ·pQ = (x (N × N), y (N × N) ↦ ⟨((1stx) ·N (1sty)), ((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 u), (v ·N f)⟩) ↔ ((⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩)))
98bicomd 129 . . . . . 6 ((x (N × N) y (N × N)) → (((⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩) ↔ ((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩)))
1094exbidv 1747 . . . . 5 ((x (N × N) y (N × N)) → (wvuf((⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩) ↔ wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩)))
11 xp1st 5734 . . . . . . 7 (x (N × N) → (1stx) N)
12 xp2nd 5735 . . . . . . 7 (x (N × N) → (2ndx) N)
1311, 12jca 290 . . . . . 6 (x (N × N) → ((1stx) N (2ndx) N))
14 xp1st 5734 . . . . . . 7 (y (N × N) → (1sty) N)
15 xp2nd 5735 . . . . . . 7 (y (N × N) → (2ndy) N)
1614, 15jca 290 . . . . . 6 (y (N × N) → ((1sty) N (2ndy) N))
17 simpll 481 . . . . . . . . . 10 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → w = (1stx))
18 simprl 483 . . . . . . . . . 10 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → u = (1sty))
1917, 18oveq12d 5473 . . . . . . . . 9 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → (w ·N u) = ((1stx) ·N (1sty)))
20 simplr 482 . . . . . . . . . 10 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → v = (2ndx))
21 simprr 484 . . . . . . . . . 10 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → f = (2ndy))
2220, 21oveq12d 5473 . . . . . . . . 9 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → (v ·N f) = ((2ndx) ·N (2ndy)))
2319, 22opeq12d 3548 . . . . . . . 8 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → ⟨(w ·N u), (v ·N f)⟩ = ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩)
2423eqeq2d 2048 . . . . . . 7 (((w = (1stx) v = (2ndx)) (u = (1sty) f = (2ndy))) → (z = ⟨(w ·N u), (v ·N f)⟩ ↔ z = ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩))
2524copsex4g 3975 . . . . . 6 ((((1stx) N (2ndx) N) ((1sty) N (2ndy) N)) → (wvuf((⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩) ↔ z = ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩))
2613, 16, 25syl2an 273 . . . . 5 ((x (N × N) y (N × N)) → (wvuf((⟨(1stx), (2ndx)⟩ = ⟨w, v ⟨(1sty), (2ndy)⟩ = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩) ↔ z = ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩))
2710, 26bitr3d 179 . . . 4 ((x (N × N) y (N × N)) → (wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩) ↔ z = ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩))
2827pm5.32i 427 . . 3 (((x (N × N) y (N × N)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩)) ↔ ((x (N × N) y (N × N)) z = ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩))
2928oprabbii 5502 . 2 {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w ·N u), (v ·N f)⟩))} = {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) z = ⟨((1stx) ·N (1sty)), ((2ndx) ·N (2ndy))⟩)}
301, 2, 293eqtr4i 2067 1 ·pQ = {⟨⟨x, y⟩, z⟩ ∣ ((x (N × N) y (N × N)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w ·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 cmi 6258   ·pQ cmpq 6261
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-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-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
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-v 2553  df-sbc 2759  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-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-mpq 6329
This theorem is referenced by:  mulpipqqs  6357
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