Theorem dmaddpqlem 7525

Description: Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7527. (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 6697	. . 3 |- ( x e. ( ( N. X. N. ) /. ~Q ) -> E. a e. ( N. X. N. ) x = [ a ] ~Q )
2		elxpi 4709	. . . . . . . 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 1625	. . . . . . . 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 1928	. . . . . 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 6678	. . . . . . . 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 2240	. . . . . 6 |- ( ( a = <. w , v >. /\ x = [ a ] ~Q ) -> x = [ <. w , v >. ] ~Q )
13	12	2eximi 1625	. . . . 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 2620	. . 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 7496	. 2 |- Q. = ( ( N. X. N. ) /. ~Q )
18	16, 17	eleq2s 2302	1 |- ( x e. Q. -> E. w E. v x = [ <. w , v >. ] ~Q )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   E.wex 1516   e. wcel 2178   E.wrex 2487   <.cop 3646   X. cxp 4691   [cec 6641   /.cqs 6642   N.cnpi 7420   ~Q ceq 7427   Q.cnq 7428

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-ec 6645  df-qs 6649  df-nqqs 7496

This theorem is referenced by:  dmaddpq  7527  dmmulpq  7528