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Theorem oviec 6589
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 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → 𝐻 ∈ (𝑆 × 𝑆))
oviec.2 (((𝑎𝑆𝑏𝑆) ∧ (𝑔𝑆𝑆)) → 𝐾 ∈ (𝑆 × 𝑆))
oviec.3 (((𝑐𝑆𝑑𝑆) ∧ (𝑡𝑆𝑠𝑆)) → 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 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → (𝜑𝜓))
oviec.9 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 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 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 𝐽 = 𝐾)
oviec.12 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝐽 = 𝐿)
oviec.13 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝐽 = 𝐻)
oviec.14 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑄𝑦𝑄) ∧ ∃𝑎𝑏𝑐𝑑((𝑥 = [⟨𝑎, 𝑏⟩] 𝑦 = [⟨𝑐, 𝑑⟩] ) ∧ 𝑧 = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] ))}
oviec.15 𝑄 = ((𝑆 × 𝑆) / )
oviec.16 ((((𝑎𝑆𝑏𝑆) ∧ (𝑐𝑆𝑑𝑆)) ∧ ((𝑔𝑆𝑆) ∧ (𝑡𝑆𝑠𝑆))) → ((𝜓𝜒) → 𝐾 𝐿))
Assertion
Ref Expression
oviec (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨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 ((((𝑎𝑆𝑏𝑆) ∧ (𝑐𝑆𝑑𝑆)) ∧ ((𝑔𝑆𝑆) ∧ (𝑡𝑆𝑠𝑆))) → ((𝜓𝜒) → 𝐾 𝐿))
4 oviec.8 . . . . . 6 (((𝑧 = 𝑎𝑤 = 𝑏) ∧ (𝑣 = 𝑐𝑢 = 𝑑)) → (𝜑𝜓))
5 oviec.7 . . . . . 6 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}
64, 5opbrop 4668 . . . . 5 (((𝑎𝑆𝑏𝑆) ∧ (𝑐𝑆𝑑𝑆)) → (⟨𝑎, 𝑏𝑐, 𝑑⟩ ↔ 𝜓))
7 oviec.9 . . . . . 6 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → (𝜑𝜒))
87, 5opbrop 4668 . . . . 5 (((𝑔𝑆𝑆) ∧ (𝑡𝑆𝑠𝑆)) → (⟨𝑔, 𝑡, 𝑠⟩ ↔ 𝜒))
96, 8bi2anan9 596 . . . 4 ((((𝑎𝑆𝑏𝑆) ∧ (𝑐𝑆𝑑𝑆)) ∧ ((𝑔𝑆𝑆) ∧ (𝑡𝑆𝑠𝑆))) → ((⟨𝑎, 𝑏𝑐, 𝑑⟩ ∧ ⟨𝑔, 𝑡, 𝑠⟩) ↔ (𝜓𝜒)))
10 oviec.2 . . . . . . 7 (((𝑎𝑆𝑏𝑆) ∧ (𝑔𝑆𝑆)) → 𝐾 ∈ (𝑆 × 𝑆))
11 oviec.11 . . . . . . 7 (((𝑤 = 𝑎𝑣 = 𝑏) ∧ (𝑢 = 𝑔𝑓 = )) → 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 5960 . . . . . 6 (((𝑎𝑆𝑏𝑆) ∧ (𝑔𝑆𝑆)) → (⟨𝑎, 𝑏+𝑔, ⟩) = 𝐾)
14 oviec.3 . . . . . . 7 (((𝑐𝑆𝑑𝑆) ∧ (𝑡𝑆𝑠𝑆)) → 𝐿 ∈ (𝑆 × 𝑆))
15 oviec.12 . . . . . . 7 (((𝑤 = 𝑐𝑣 = 𝑑) ∧ (𝑢 = 𝑡𝑓 = 𝑠)) → 𝐽 = 𝐿)
1614, 15, 12ovi3 5960 . . . . . 6 (((𝑐𝑆𝑑𝑆) ∧ (𝑡𝑆𝑠𝑆)) → (⟨𝑐, 𝑑+𝑡, 𝑠⟩) = 𝐿)
1713, 16breqan12d 3983 . . . . 5 ((((𝑎𝑆𝑏𝑆) ∧ (𝑔𝑆𝑆)) ∧ ((𝑐𝑆𝑑𝑆) ∧ (𝑡𝑆𝑠𝑆))) → ((⟨𝑎, 𝑏+𝑔, ⟩) (⟨𝑐, 𝑑+𝑡, 𝑠⟩) ↔ 𝐾 𝐿))
1817an4s 578 . . . 4 ((((𝑎𝑆𝑏𝑆) ∧ (𝑐𝑆𝑑𝑆)) ∧ ((𝑔𝑆𝑆) ∧ (𝑡𝑆𝑠𝑆))) → ((⟨𝑎, 𝑏+𝑔, ⟩) (⟨𝑐, 𝑑+𝑡, 𝑠⟩) ↔ 𝐾 𝐿))
193, 9, 183imtr4d 202 . . 3 ((((𝑎𝑆𝑏𝑆) ∧ (𝑐𝑆𝑑𝑆)) ∧ ((𝑔𝑆𝑆) ∧ (𝑡𝑆𝑠𝑆))) → ((⟨𝑎, 𝑏𝑐, 𝑑⟩ ∧ ⟨𝑔, 𝑡, 𝑠⟩) → (⟨𝑎, 𝑏+𝑔, ⟩) (⟨𝑐, 𝑑+𝑡, 𝑠⟩)))
20 oviec.14 . . . 4 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑄𝑦𝑄) ∧ ∃𝑎𝑏𝑐𝑑((𝑥 = [⟨𝑎, 𝑏⟩] 𝑦 = [⟨𝑐, 𝑑⟩] ) ∧ 𝑧 = [(⟨𝑎, 𝑏+𝑐, 𝑑⟩)] ))}
21 oviec.15 . . . . . . . 8 𝑄 = ((𝑆 × 𝑆) / )
2221eleq2i 2224 . . . . . . 7 (𝑥𝑄𝑥 ∈ ((𝑆 × 𝑆) / ))
2321eleq2i 2224 . . . . . . 7 (𝑦𝑄𝑦 ∈ ((𝑆 × 𝑆) / ))
2422, 23anbi12i 456 . . . . . 6 ((𝑥𝑄𝑦𝑄) ↔ (𝑥 ∈ ((𝑆 × 𝑆) / ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / )))
2524anbi1i 454 . . . . 5 (((𝑥𝑄𝑦𝑄) ∧ ∃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 5879 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑄𝑦𝑄) ∧ ∃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 2178 . . 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 6588 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] [⟨𝐶, 𝐷⟩] ) = [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] )
29 oviec.1 . . . 4 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → 𝐻 ∈ (𝑆 × 𝑆))
30 oviec.13 . . . 4 (((𝑤 = 𝐴𝑣 = 𝐵) ∧ (𝑢 = 𝐶𝑓 = 𝐷)) → 𝐽 = 𝐻)
3129, 30, 12ovi3 5960 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵+𝐶, 𝐷⟩) = 𝐻)
3231eceq1d 6519 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [𝐻] )
3328, 32eqtrd 2190 1 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] [⟨𝐶, 𝐷⟩] ) = [𝐻] )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wex 1472  wcel 2128  Vcvv 2712  cop 3564   class class class wbr 3967  {copab 4027   × cxp 4587  (class class class)co 5827  {coprab 5828   Er wer 6480  [cec 6481   / cqs 6482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4029  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fv 5181  df-ov 5830  df-oprab 5831  df-er 6483  df-ec 6485  df-qs 6489
This theorem is referenced by:  addpipqqs  7293  mulpipqqs  7296
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