Theorem recexpr 7953

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 4114	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( z <Q w <-> u <Q v ) )
2		simpr 110	. . . . . . . . 9 |- ( ( z = u /\ w = v ) -> w = v )
3	2	fveq2d 5674	. . . . . . . 8 |- ( ( z = u /\ w = v ) -> ( *Q ` w ) = ( *Q ` v ) )
4	3	eleq1d 2301	. . . . . . 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 1979	. . . . 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 2359	. . . 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 4122	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( w <Q z <-> v <Q u ) )
10	3	eleq1d 2301	. . . . . . 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 1979	. . . . 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 2359	. . . 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 3888	. . 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 7947	. 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 7952	. 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 6058	. . . 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 2241	. . 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 2921	. 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 1398   E.wex 1541   e. wcel 2203   {cab 2218   E.wrex 2521   <.cop 3692   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   1stc1st 6332   2ndc2nd 6333   *Qcrq 7599   <Q cltq 7600   P.cnp 7606   1Pc1p 7607   .P. cmp 7609

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 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-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-i1p 7782  df-imp 7784

This theorem is referenced by:  ltmprr  7957  recexgt0sr  8088