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Definition df-plpq 6328
Description: Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition  +Q (df-plqqs 6333) works with the equivalence classes of these ordered pairs determined by the equivalence relation  ~Q (df-enq 6331). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.)
Assertion
Ref Expression
df-plpq  +pQ  N.  X.  N. ,  N. 
X.  N.  |->  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.
Distinct variable group:   ,

Detailed syntax breakdown of Definition df-plpq
StepHypRef Expression
1 cplpq 6260 . 2  +pQ
2 vx . . 3  setvar
3 vy . . 3  setvar
4 cnpi 6256 . . . 4  N.
54, 4cxp 4286 . . 3  N. 
X.  N.
62cv 1241 . . . . . . 7
7 c1st 5707 . . . . . . 7  1st
86, 7cfv 4845 . . . . . 6  1st `
93cv 1241 . . . . . . 7
10 c2nd 5708 . . . . . . 7  2nd
119, 10cfv 4845 . . . . . 6  2nd `
12 cmi 6258 . . . . . 6  .N
138, 11, 12co 5455 . . . . 5  1st `  .N  2nd `
149, 7cfv 4845 . . . . . 6  1st `
156, 10cfv 4845 . . . . . 6  2nd `
1614, 15, 12co 5455 . . . . 5  1st `  .N  2nd `
17 cpli 6257 . . . . 5  +N
1813, 16, 17co 5455 . . . 4  1st `  .N  2nd `  +N  1st `  .N  2nd `
1915, 11, 12co 5455 . . . 4  2nd `  .N  2nd `
2018, 19cop 3370 . . 3  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.
212, 3, 5, 5, 20cmpt2 5457 . 2  N.  X.  N. ,  N.  X.  N.  |->  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.
221, 21wceq 1242 1  +pQ  N.  X.  N. ,  N. 
X.  N.  |->  <. 1st `  .N  2nd `  +N  1st `  .N  2nd `  ,  2nd `  .N  2nd `  >.
Colors of variables: wff set class
This definition is referenced by:  dfplpq2  6338
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