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

Theorem oviec 6148
 Description: Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.)
Hypotheses
Ref Expression
oviec.1 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → 𝐻 (𝑆 × 𝑆))
oviec.2 (((𝑎 𝑆 𝑏 𝑆) (g 𝑆 𝑆)) → 𝐾 (𝑆 × 𝑆))
oviec.3 (((𝑐 𝑆 𝑑 𝑆) (𝑡 𝑆 𝑠 𝑆)) → 𝐿 (𝑆 × 𝑆))
oviec.4 V
oviec.5 Er (𝑆 × 𝑆)
oviec.7 = {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) φ))}
oviec.8 (((z = 𝑎 w = 𝑏) (v = 𝑐 u = 𝑑)) → (φψ))
oviec.9 (((z = g w = ) (v = 𝑡 u = 𝑠)) → (φχ))
oviec.10 + = {⟨⟨x, y⟩, z⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝐽))}
oviec.11 (((w = 𝑎 v = 𝑏) (u = g f = )) → 𝐽 = 𝐾)
oviec.12 (((w = 𝑐 v = 𝑑) (u = 𝑡 f = 𝑠)) → 𝐽 = 𝐿)
oviec.13 (((w = A v = B) (u = 𝐶 f = 𝐷)) → 𝐽 = 𝐻)
oviec.14 = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑄 y 𝑄) 𝑎𝑏𝑐𝑑((x = [⟨𝑎, 𝑏⟩] y = [⟨𝑐, 𝑑⟩] ) z = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] ))}
oviec.15 𝑄 = ((𝑆 × 𝑆) / )
oviec.16 ((((𝑎 𝑆 𝑏 𝑆) (𝑐 𝑆 𝑑 𝑆)) ((g 𝑆 𝑆) (𝑡 𝑆 𝑠 𝑆))) → ((ψ χ) → 𝐾 𝐿))
Assertion
Ref Expression
oviec (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → ([⟨A, B⟩] [⟨𝐶, 𝐷⟩] ) = [𝐻] )
Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,f,u,v,w,x,y,z,𝐶   𝐷,𝑎,𝑏,𝑐,𝑑,f,u,v,w,x,y,z   x,𝐽,y,z   g,𝑎,,A,𝑏,𝑐,𝑑,f,u,v,w,x,y,z   χ,u,v,w,z   f,𝐻,u,v,w,x,y,z   B,𝑎,𝑏,𝑐,𝑑,f,g,,u,v,w,x,y,z   f,𝐾,u,v,w,x,y,z   ψ,u,v,w,z   f,𝐿,u,v,w,x,y,z   φ,x,y   𝑠,𝑎,𝑡,𝑆,𝑏,𝑐,𝑑,f,g,,u,v,w,x,y,z   + ,𝑎,𝑏,𝑐,𝑑,g,,𝑠,𝑡,x,y,z   ,𝑎,𝑏,𝑐,𝑑,g,,𝑠,𝑡,x,y,z
Allowed substitution hints:   φ(z,w,v,u,𝑡,f,g,,𝑠,𝑎,𝑏,𝑐,𝑑)   ψ(x,y,𝑡,f,g,,𝑠,𝑎,𝑏,𝑐,𝑑)   χ(x,y,𝑡,f,g,,𝑠,𝑎,𝑏,𝑐,𝑑)   A(𝑡,𝑠)   B(𝑡,𝑠)   𝐶(𝑡,g,,𝑠)   𝐷(𝑡,g,,𝑠)   + (w,v,u,f)   (x,y,z,w,v,u,𝑡,f,g,,𝑠,𝑎,𝑏,𝑐,𝑑)   𝑄(x,y,z,w,v,u,𝑡,f,g,,𝑠,𝑎,𝑏,𝑐,𝑑)   (w,v,u,f)   𝐻(𝑡,g,,𝑠,𝑎,𝑏,𝑐,𝑑)   𝐽(w,v,u,𝑡,f,g,,𝑠,𝑎,𝑏,𝑐,𝑑)   𝐾(𝑡,g,,𝑠,𝑎,𝑏,𝑐,𝑑)   𝐿(𝑡,g,,𝑠,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem oviec
StepHypRef Expression
1 oviec.4 . . 3 V
2 oviec.5 . . 3 Er (𝑆 × 𝑆)
3 oviec.16 . . . 4 ((((𝑎 𝑆 𝑏 𝑆) (𝑐 𝑆 𝑑 𝑆)) ((g 𝑆 𝑆) (𝑡 𝑆 𝑠 𝑆))) → ((ψ χ) → 𝐾 𝐿))
4 oviec.8 . . . . . 6 (((z = 𝑎 w = 𝑏) (v = 𝑐 u = 𝑑)) → (φψ))
5 oviec.7 . . . . . 6 = {⟨x, y⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) zwvu((x = ⟨z, w y = ⟨v, u⟩) φ))}
64, 5opbrop 4362 . . . . 5 (((𝑎 𝑆 𝑏 𝑆) (𝑐 𝑆 𝑑 𝑆)) → (⟨𝑎, 𝑏𝑐, 𝑑⟩ ↔ ψ))
7 oviec.9 . . . . . 6 (((z = g w = ) (v = 𝑡 u = 𝑠)) → (φχ))
87, 5opbrop 4362 . . . . 5 (((g 𝑆 𝑆) (𝑡 𝑆 𝑠 𝑆)) → (⟨g, 𝑡, 𝑠⟩ ↔ χ))
96, 8bi2anan9 538 . . . 4 ((((𝑎 𝑆 𝑏 𝑆) (𝑐 𝑆 𝑑 𝑆)) ((g 𝑆 𝑆) (𝑡 𝑆 𝑠 𝑆))) → ((⟨𝑎, 𝑏𝑐, 𝑑g, 𝑡, 𝑠⟩) ↔ (ψ χ)))
10 oviec.2 . . . . . . 7 (((𝑎 𝑆 𝑏 𝑆) (g 𝑆 𝑆)) → 𝐾 (𝑆 × 𝑆))
11 oviec.11 . . . . . . 7 (((w = 𝑎 v = 𝑏) (u = g f = )) → 𝐽 = 𝐾)
12 oviec.10 . . . . . . 7 + = {⟨⟨x, y⟩, z⟩ ∣ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = 𝐽))}
1310, 11, 12ovi3 5579 . . . . . 6 (((𝑎 𝑆 𝑏 𝑆) (g 𝑆 𝑆)) → (⟨𝑎, 𝑏+g, ⟩) = 𝐾)
14 oviec.3 . . . . . . 7 (((𝑐 𝑆 𝑑 𝑆) (𝑡 𝑆 𝑠 𝑆)) → 𝐿 (𝑆 × 𝑆))
15 oviec.12 . . . . . . 7 (((w = 𝑐 v = 𝑑) (u = 𝑡 f = 𝑠)) → 𝐽 = 𝐿)
1614, 15, 12ovi3 5579 . . . . . 6 (((𝑐 𝑆 𝑑 𝑆) (𝑡 𝑆 𝑠 𝑆)) → (⟨𝑐, 𝑑+𝑡, 𝑠⟩) = 𝐿)
1713, 16breqan12d 3770 . . . . 5 ((((𝑎 𝑆 𝑏 𝑆) (g 𝑆 𝑆)) ((𝑐 𝑆 𝑑 𝑆) (𝑡 𝑆 𝑠 𝑆))) → ((⟨𝑎, 𝑏+g, ⟩) (⟨𝑐, 𝑑+𝑡, 𝑠⟩) ↔ 𝐾 𝐿))
1817an4s 522 . . . 4 ((((𝑎 𝑆 𝑏 𝑆) (𝑐 𝑆 𝑑 𝑆)) ((g 𝑆 𝑆) (𝑡 𝑆 𝑠 𝑆))) → ((⟨𝑎, 𝑏+g, ⟩) (⟨𝑐, 𝑑+𝑡, 𝑠⟩) ↔ 𝐾 𝐿))
193, 9, 183imtr4d 192 . . 3 ((((𝑎 𝑆 𝑏 𝑆) (𝑐 𝑆 𝑑 𝑆)) ((g 𝑆 𝑆) (𝑡 𝑆 𝑠 𝑆))) → ((⟨𝑎, 𝑏𝑐, 𝑑g, 𝑡, 𝑠⟩) → (⟨𝑎, 𝑏+g, ⟩) (⟨𝑐, 𝑑+𝑡, 𝑠⟩)))
20 oviec.14 . . . 4 = {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑄 y 𝑄) 𝑎𝑏𝑐𝑑((x = [⟨𝑎, 𝑏⟩] y = [⟨𝑐, 𝑑⟩] ) z = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] ))}
21 oviec.15 . . . . . . . 8 𝑄 = ((𝑆 × 𝑆) / )
2221eleq2i 2101 . . . . . . 7 (x 𝑄x ((𝑆 × 𝑆) / ))
2321eleq2i 2101 . . . . . . 7 (y 𝑄y ((𝑆 × 𝑆) / ))
2422, 23anbi12i 433 . . . . . 6 ((x 𝑄 y 𝑄) ↔ (x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )))
2524anbi1i 431 . . . . 5 (((x 𝑄 y 𝑄) 𝑎𝑏𝑐𝑑((x = [⟨𝑎, 𝑏⟩] y = [⟨𝑐, 𝑑⟩] ) z = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] )) ↔ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) 𝑎𝑏𝑐𝑑((x = [⟨𝑎, 𝑏⟩] y = [⟨𝑐, 𝑑⟩] ) z = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] )))
2625oprabbii 5502 . . . 4 {⟨⟨x, y⟩, z⟩ ∣ ((x 𝑄 y 𝑄) 𝑎𝑏𝑐𝑑((x = [⟨𝑎, 𝑏⟩] y = [⟨𝑐, 𝑑⟩] ) z = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] ))} = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) 𝑎𝑏𝑐𝑑((x = [⟨𝑎, 𝑏⟩] y = [⟨𝑐, 𝑑⟩] ) z = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] ))}
2720, 26eqtri 2057 . . 3 = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) 𝑎𝑏𝑐𝑑((x = [⟨𝑎, 𝑏⟩] y = [⟨𝑐, 𝑑⟩] ) z = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] ))}
281, 2, 19, 27th3q 6147 . 2 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → ([⟨A, B⟩] [⟨𝐶, 𝐷⟩] ) = [(⟨A, B+𝐶, 𝐷⟩)] )
29 oviec.1 . . . 4 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → 𝐻 (𝑆 × 𝑆))
30 oviec.13 . . . 4 (((w = A v = B) (u = 𝐶 f = 𝐷)) → 𝐽 = 𝐻)
3129, 30, 12ovi3 5579 . . 3 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → (⟨A, B+𝐶, 𝐷⟩) = 𝐻)
3231eceq1d 6078 . 2 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → [(⟨A, B+𝐶, 𝐷⟩)] = [𝐻] )
3328, 32eqtrd 2069 1 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → ([⟨A, B⟩] [⟨𝐶, 𝐷⟩] ) = [𝐻] )
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   class class class wbr 3755  {copab 3808   × cxp 4286  (class class class)co 5455  {coprab 5456   Er wer 6039  [cec 6040   / cqs 6041 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-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220 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-v 2553  df-sbc 2759  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-br 3756  df-opab 3810  df-id 4021  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-fv 4853  df-ov 5458  df-oprab 5459  df-er 6042  df-ec 6044  df-qs 6048 This theorem is referenced by:  addpipqqs  6354  mulpipqqs  6357
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