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Theorem oviec 6212
 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 4419 . . . . 5 (((𝑎𝑆𝑏𝑆) ∧ (𝑐𝑆𝑑𝑆)) → (⟨𝑎, 𝑏𝑐, 𝑑⟩ ↔ 𝜓))
7 oviec.9 . . . . . 6 (((𝑧 = 𝑔𝑤 = ) ∧ (𝑣 = 𝑡𝑢 = 𝑠)) → (𝜑𝜒))
87, 5opbrop 4419 . . . . 5 (((𝑔𝑆𝑆) ∧ (𝑡𝑆𝑠𝑆)) → (⟨𝑔, 𝑡, 𝑠⟩ ↔ 𝜒))
96, 8bi2anan9 538 . . . 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 5637 . . . . . 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 5637 . . . . . 6 (((𝑐𝑆𝑑𝑆) ∧ (𝑡𝑆𝑠𝑆)) → (⟨𝑐, 𝑑+𝑡, 𝑠⟩) = 𝐿)
1713, 16breqan12d 3779 . . . . 5 ((((𝑎𝑆𝑏𝑆) ∧ (𝑔𝑆𝑆)) ∧ ((𝑐𝑆𝑑𝑆) ∧ (𝑡𝑆𝑠𝑆))) → ((⟨𝑎, 𝑏+𝑔, ⟩) (⟨𝑐, 𝑑+𝑡, 𝑠⟩) ↔ 𝐾 𝐿))
1817an4s 522 . . . 4 ((((𝑎𝑆𝑏𝑆) ∧ (𝑐𝑆𝑑𝑆)) ∧ ((𝑔𝑆𝑆) ∧ (𝑡𝑆𝑠𝑆))) → ((⟨𝑎, 𝑏+𝑔, ⟩) (⟨𝑐, 𝑑+𝑡, 𝑠⟩) ↔ 𝐾 𝐿))
193, 9, 183imtr4d 192 . . 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 2104 . . . . . . 7 (𝑥𝑄𝑥 ∈ ((𝑆 × 𝑆) / ))
2321eleq2i 2104 . . . . . . 7 (𝑦𝑄𝑦 ∈ ((𝑆 × 𝑆) / ))
2422, 23anbi12i 433 . . . . . 6 ((𝑥𝑄𝑦𝑄) ↔ (𝑥 ∈ ((𝑆 × 𝑆) / ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / )))
2524anbi1i 431 . . . . 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 5560 . . . 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 2060 . . 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 6211 . 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 5637 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵+𝐶, 𝐷⟩) = 𝐻)
3231eceq1d 6142 . 2 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → [(⟨𝐴, 𝐵+𝐶, 𝐷⟩)] = [𝐻] )
3328, 32eqtrd 2072 1 (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → ([⟨𝐴, 𝐵⟩] [⟨𝐶, 𝐷⟩] ) = [𝐻] )
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  ⟨cop 3378   class class class wbr 3764  {copab 3817   × cxp 4343  (class class class)co 5512  {coprab 5513   Er wer 6103  [cec 6104   / cqs 6105 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-er 6106  df-ec 6108  df-qs 6112 This theorem is referenced by:  addpipqqs  6468  mulpipqqs  6471
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