Theorem recexpr 7857

Description: The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)

Assertion

Ref	Expression
recexpr	|- ( A e. P. -> E. x e. P. ( A .P. x ) = 1P )

Distinct variable group:   x, A

Proof of Theorem recexpr

Dummy variables u v w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq12 4093	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( z <Q w <-> u <Q v ) )
2		simpr 110	. . . . . . . . 9 |- ( ( z = u /\ w = v ) -> w = v )
3	2	fveq2d 5643	. . . . . . . 8 |- ( ( z = u /\ w = v ) -> ( *Q ` w ) = ( *Q ` v ) )
4	3	eleq1d 2300	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( ( *Q ` w ) e. ( 2nd ` A ) <-> ( *Q ` v ) e. ( 2nd ` A ) ) )
5	1, 4	anbi12d 473	. . . . . 6 |- ( ( z = u /\ w = v ) -> ( ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) <-> ( u <Q v /\ ( *Q ` v ) e. ( 2nd ` A ) ) ) )
6	5	cbvexdva 1978	. . . . 5 |- ( z = u -> ( E. w ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) <-> E. v ( u <Q v /\ ( *Q ` v ) e. ( 2nd ` A ) ) ) )
7	6	cbvabv 2356	. . . 4 |- { z | E. w ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) } = { u | E. v ( u <Q v /\ ( *Q ` v ) e. ( 2nd ` A ) ) }
8		simpl 109	. . . . . . . 8 |- ( ( z = u /\ w = v ) -> z = u )
9	2, 8	breq12d 4101	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( w <Q z <-> v <Q u ) )
10	3	eleq1d 2300	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( ( *Q ` w ) e. ( 1st ` A ) <-> ( *Q ` v ) e. ( 1st ` A ) ) )
11	9, 10	anbi12d 473	. . . . . 6 |- ( ( z = u /\ w = v ) -> ( ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) <-> ( v <Q u /\ ( *Q ` v ) e. ( 1st ` A ) ) ) )
12	11	cbvexdva 1978	. . . . 5 |- ( z = u -> ( E. w ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) <-> E. v ( v <Q u /\ ( *Q ` v ) e. ( 1st ` A ) ) ) )
13	12	cbvabv 2356	. . . 4 |- { z | E. w ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) } = { u | E. v ( v <Q u /\ ( *Q ` v ) e. ( 1st ` A ) ) }
14	7, 13	opeq12i 3867	. . 3 |- <. { z | E. w ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) } , { z | E. w ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) } >. = <. { u | E. v ( u <Q v /\ ( *Q ` v ) e. ( 2nd ` A ) ) } , { u | E. v ( v <Q u /\ ( *Q ` v ) e. ( 1st ` A ) ) } >.
15	14	recexprlempr 7851	. 2 |- ( A e. P. -> <. { z | E. w ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) } , { z | E. w ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) } >. e. P. )
16	14	recexprlemex 7856	. 2 |- ( A e. P. -> ( A .P. <. { z | E. w ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) } , { z | E. w ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) } >. ) = 1P )
17		oveq2 6025	. . . 4 |- ( x = <. { z | E. w ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) } , { z | E. w ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) } >. -> ( A .P. x ) = ( A .P. <. { z | E. w ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) } , { z | E. w ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) } >. ) )
18	17	eqeq1d 2240	. . 3 |- ( x = <. { z | E. w ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) } , { z | E. w ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) } >. -> ( ( A .P. x ) = 1P <-> ( A .P. <. { z | E. w ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) } , { z | E. w ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) } >. ) = 1P ) )
19	18	rspcev 2910	. 2 |- ( ( <. { z | E. w ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) } , { z | E. w ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) } >. e. P. /\ ( A .P. <. { z | E. w ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) } , { z | E. w ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) } >. ) = 1P ) -> E. x e. P. ( A .P. x ) = 1P )
20	15, 16, 19	syl2anc 411	1 |- ( A e. P. -> E. x e. P. ( A .P. x ) = 1P )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   E.wrex 2511   <.cop 3672   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   1stc1st 6300   2ndc2nd 6301   *Qcrq 7503   <Q cltq 7504   P.cnp 7510   1Pc1p 7511   .P. cmp 7513

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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-i1p 7686  df-imp 7688

This theorem is referenced by:  ltmprr  7861  recexgt0sr  7992