Theorem recexpr 7918

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 4098	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( z <Q w <-> u <Q v ) )
2		simpr 110	. . . . . . . . 9 |- ( ( z = u /\ w = v ) -> w = v )
3	2	fveq2d 5652	. . . . . . . 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 2357	. . . 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 4106	. . . . . . 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 2357	. . . 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 3872	. . 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 7912	. 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 7917	. 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 6036	. . . 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 2911	. 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 2202   {cab 2217   E.wrex 2512   <.cop 3676   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   1stc1st 6310   2ndc2nd 6311   *Qcrq 7564   <Q cltq 7565   P.cnp 7571   1Pc1p 7572   .P. cmp 7574

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-enq0 7704  df-nq0 7705  df-0nq0 7706  df-plq0 7707  df-mq0 7708  df-inp 7746  df-i1p 7747  df-imp 7749

This theorem is referenced by:  ltmprr  7922  recexgt0sr  8053