Theorem dmaddpqlem 7309

Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7311. (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 6544	. . 3 |- ( x e. ( ( N. X. N. ) /. ~Q ) -> E. a e. ( N. X. N. ) x = [ a ] ~Q )
2		elxpi 4614	. . . . . . . 8 |- ( a e. ( N. X. N. ) -> E. w E. v ( a = <. w , v >. /\ ( w e. N. /\ v e. N. ) ) )
3		simpl 108	. . . . . . . . 9 |- ( ( a = <. w , v >. /\ ( w e. N. /\ v e. N. ) ) -> a = <. w , v >. )
4	3	2eximi 1588	. . . . . . . 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 338	. . . . . 6 |- ( ( a e. ( N. X. N. ) /\ x = [ a ] ~Q ) -> ( E. w E. v a = <. w , v >. /\ x = [ a ] ~Q ) )
7		19.41vv 1890	. . . . . 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 133	. . . . 5 |- ( ( a e. ( N. X. N. ) /\ x = [ a ] ~Q ) -> E. w E. v ( a = <. w , v >. /\ x = [ a ] ~Q ) )
9		simpr 109	. . . . . . 7 |- ( ( a = <. w , v >. /\ x = [ a ] ~Q ) -> x = [ a ] ~Q )
10		eceq1 6527	. . . . . . . 8 |- ( a = <. w , v >. -> [ a ] ~Q = [ <. w , v >. ] ~Q )
11	10	adantr 274	. . . . . . 7 |- ( ( a = <. w , v >. /\ x = [ a ] ~Q ) -> [ a ] ~Q = [ <. w , v >. ] ~Q )
12	9, 11	eqtrd 2197	. . . . . 6 |- ( ( a = <. w , v >. /\ x = [ a ] ~Q ) -> x = [ <. w , v >. ] ~Q )
13	12	2eximi 1588	. . . . 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 2576	. . 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 7280	. 2 |- Q. = ( ( N. X. N. ) /. ~Q )
18	16, 17	eleq2s 2259	1 |- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1342   E.wex 1479   e. wcel 2135   E.wrex 2443   <.cop 3573   X. cxp 4596   [cec 6490   /.cqs 6491   N.cnpi 7204   ~Q ceq 7211   Q.cnq 7212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-xp 4604  df-cnv 4606  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-ec 6494  df-qs 6498  df-nqqs 7280

This theorem is referenced by:  dmaddpq  7311  dmmulpq  7312