Theorem dmaddpqlem 7444

Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7446. (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 6646	. . 3 |- ( x e. ( ( N. X. N. ) /. ~Q ) -> E. a e. ( N. X. N. ) x = [ a ] ~Q )
2		elxpi 4679	. . . . . . . 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 1615	. . . . . . . 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 1918	. . . . . 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 6627	. . . . . . . 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 2229	. . . . . 6 |- ( ( a = <. w , v >. /\ x = [ a ] ~Q ) -> x = [ <. w , v >. ] ~Q )
13	12	2eximi 1615	. . . . 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 2609	. . 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 7415	. 2 |- Q. = ( ( N. X. N. ) /. ~Q )
18	16, 17	eleq2s 2291	1 |- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1506   e. wcel 2167   E.wrex 2476   <.cop 3625   X. cxp 4661   [cec 6590   /.cqs 6591   N.cnpi 7339   ~Q ceq 7346   Q.cnq 7347

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-ec 6594  df-qs 6598  df-nqqs 7415

This theorem is referenced by:  dmaddpq  7446  dmmulpq  7447