Theorem pinq 9749

Description: The representatives of positive integers as positive fractions. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
pinq	|- ( A e. N. -> <. A , 1o >. e. Q. )

Proof of Theorem pinq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1pi 9705	. . . 4 |- 1o e. N.
2		opelxpi 5148	. . . 4 |- ( ( A e. N. /\ 1o e. N. ) -> <. A , 1o >. e. ( N. X. N. ) )
3	1, 2	mpan2 707	. . 3 |- ( A e. N. -> <. A , 1o >. e. ( N. X. N. ) )
4		nlt1pi 9728	. . . . . 6 |- -. ( 2nd ` y ) <N 1o
5	1	elexi 3213	. . . . . . . 8 |- 1o e. _V
6		op2ndg 7181	. . . . . . . 8 |- ( ( A e. N. /\ 1o e. _V ) -> ( 2nd ` <. A , 1o >. ) = 1o )
7	5, 6	mpan2 707	. . . . . . 7 |- ( A e. N. -> ( 2nd ` <. A , 1o >. ) = 1o )
8	7	breq2d 4665	. . . . . 6 |- ( A e. N. -> ( ( 2nd ` y ) <N ( 2nd ` <. A , 1o >. ) <-> ( 2nd ` y ) <N 1o ) )
9	4, 8	mtbiri 317	. . . . 5 |- ( A e. N. -> -. ( 2nd ` y ) <N ( 2nd ` <. A , 1o >. ) )
10	9	a1d 25	. . . 4 |- ( A e. N. -> ( <. A , 1o >. ~Q y -> -. ( 2nd ` y ) <N ( 2nd ` <. A , 1o >. ) ) )
11	10	ralrimivw 2967	. . 3 |- ( A e. N. -> A. y e. ( N. X. N. ) ( <. A , 1o >. ~Q y -> -. ( 2nd ` y ) <N ( 2nd ` <. A , 1o >. ) ) )
12		breq1 4656	. . . . . 6 |- ( x = <. A , 1o >. -> ( x ~Q y <-> <. A , 1o >. ~Q y ) )
13		fveq2 6191	. . . . . . . 8 |- ( x = <. A , 1o >. -> ( 2nd ` x ) = ( 2nd ` <. A , 1o >. ) )
14	13	breq2d 4665	. . . . . . 7 |- ( x = <. A , 1o >. -> ( ( 2nd ` y ) <N ( 2nd ` x ) <-> ( 2nd ` y ) <N ( 2nd ` <. A , 1o >. ) ) )
15	14	notbid 308	. . . . . 6 |- ( x = <. A , 1o >. -> ( -. ( 2nd ` y ) <N ( 2nd ` x ) <-> -. ( 2nd ` y ) <N ( 2nd ` <. A , 1o >. ) ) )
16	12, 15	imbi12d 334	. . . . 5 |- ( x = <. A , 1o >. -> ( ( x ~Q y -> -. ( 2nd ` y ) <N ( 2nd ` x ) ) <-> ( <. A , 1o >. ~Q y -> -. ( 2nd ` y ) <N ( 2nd ` <. A , 1o >. ) ) ) )
17	16	ralbidv 2986	. . . 4 |- ( x = <. A , 1o >. -> ( A. y e. ( N. X. N. ) ( x ~Q y -> -. ( 2nd ` y ) <N ( 2nd ` x ) ) <-> A. y e. ( N. X. N. ) ( <. A , 1o >. ~Q y -> -. ( 2nd ` y ) <N ( 2nd ` <. A , 1o >. ) ) ) )
18	17	elrab 3363	. . 3 |- ( <. A , 1o >. e. { x e. ( N. X. N. ) | A. y e. ( N. X. N. ) ( x ~Q y -> -. ( 2nd ` y ) <N ( 2nd ` x ) ) } <-> ( <. A , 1o >. e. ( N. X. N. ) /\ A. y e. ( N. X. N. ) ( <. A , 1o >. ~Q y -> -. ( 2nd ` y ) <N ( 2nd ` <. A , 1o >. ) ) ) )
19	3, 11, 18	sylanbrc 698	. 2 |- ( A e. N. -> <. A , 1o >. e. { x e. ( N. X. N. ) | A. y e. ( N. X. N. ) ( x ~Q y -> -. ( 2nd ` y ) <N ( 2nd ` x ) ) } )
20		df-nq 9734	. 2 |- Q. = { x e. ( N. X. N. ) | A. y e. ( N. X. N. ) ( x ~Q y -> -. ( 2nd ` y ) <N ( 2nd ` x ) ) }
21	19, 20	syl6eleqr 2712	1 |- ( A e. N. -> <. A , 1o >. e. Q. )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   <.cop 4183   class class class wbr 4653   X. cxp 5112   ` cfv 5888   2ndc2nd 7167   1oc1o 7553   N.cnpi 9666   <N clti 9669   ~Q ceq 9673   Q.cnq 9674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-om 7066  df-2nd 7169  df-1o 7560  df-ni 9694  df-lti 9697  df-nq 9734

This theorem is referenced by:  1nq  9750  archnq  9802  prlem934  9855