Theorem dmaddpqlem 7692

Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7694. (Contributed by Jim Kingdon, 15-Sep-2019.)

Assertion

Ref	Expression
dmaddpqlem	|- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )

Distinct variable group:   w, v, x

Proof of Theorem dmaddpqlem

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqsi 6821	. . 3 |- ( x e. ( ( N. X. N. ) /. ~Q ) -> E. a e. ( N. X. N. ) x = [ a ] ~Q )
2		elxpi 4765	. . . . . . . 8 |- ( a e. ( N. X. N. ) -> E. w E. v ( a = <. w , v >. /\ ( w e. N. /\ v e. N. ) ) )
3		simpl 109	. . . . . . . . 9 |- ( ( a = <. w , v >. /\ ( w e. N. /\ v e. N. ) ) -> a = <. w , v >. )
4	3	2eximi 1650	. . . . . . . 8 |- ( E. w E. v ( a = <. w , v >. /\ ( w e. N. /\ v e. N. ) ) -> E. w E. v a = <. w , v >. )
5	2, 4	syl 14	. . . . . . 7 |- ( a e. ( N. X. N. ) -> E. w E. v a = <. w , v >. )
6	5	anim1i 340	. . . . . 6 |- ( ( a e. ( N. X. N. ) /\ x = [ a ] ~Q ) -> ( E. w E. v a = <. w , v >. /\ x = [ a ] ~Q ) )
7		19.41vv 1953	. . . . . 6 |- ( E. w E. v ( a = <. w , v >. /\ x = [ a ] ~Q ) <-> ( E. w E. v a = <. w , v >. /\ x = [ a ] ~Q ) )
8	6, 7	sylibr 134	. . . . 5 |- ( ( a e. ( N. X. N. ) /\ x = [ a ] ~Q ) -> E. w E. v ( a = <. w , v >. /\ x = [ a ] ~Q ) )
9		simpr 110	. . . . . . 7 |- ( ( a = <. w , v >. /\ x = [ a ] ~Q ) -> x = [ a ] ~Q )
10		eceq1 6802	. . . . . . . 8 |- ( a = <. w , v >. -> [ a ] ~Q = [ <. w , v >. ] ~Q )
11	10	adantr 276	. . . . . . 7 |- ( ( a = <. w , v >. /\ x = [ a ] ~Q ) -> [ a ] ~Q = [ <. w , v >. ] ~Q )
12	9, 11	eqtrd 2265	. . . . . 6 |- ( ( a = <. w , v >. /\ x = [ a ] ~Q ) -> x = [ <. w , v >. ] ~Q )
13	12	2eximi 1650	. . . . 5 |- ( E. w E. v ( a = <. w , v >. /\ x = [ a ] ~Q ) -> E. w E. v x = [ <. w , v >. ] ~Q )
14	8, 13	syl 14	. . . 4 |- ( ( a e. ( N. X. N. ) /\ x = [ a ] ~Q ) -> E. w E. v x = [ <. w , v >. ] ~Q )
15	14	rexlimiva 2655	. . 3 |- ( E. a e. ( N. X. N. ) x = [ a ] ~Q -> E. w E. v x = [ <. w , v >. ] ~Q )
16	1, 15	syl 14	. 2 |- ( x e. ( ( N. X. N. ) /. ~Q ) -> E. w E. v x = [ <. w , v >. ] ~Q )
17		df-nqqs 7663	. 2 |- Q. = ( ( N. X. N. ) /. ~Q )
18	16, 17	eleq2s 2327	1 |- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2203   E.wrex 2521   <.cop 3692   X. cxp 4747   [cec 6765   /.cqs 6766   N.cnpi 7587   ~Q ceq 7594   Q.cnq 7595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-ec 6769  df-qs 6773  df-nqqs 7663

This theorem is referenced by:  dmaddpq  7694  dmmulpq  7695