Theorem recexpr 7579

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 3987	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( z <Q w <-> u <Q v ) )
2		simpr 109	. . . . . . . . 9 |- ( ( z = u /\ w = v ) -> w = v )
3	2	fveq2d 5490	. . . . . . . 8 |- ( ( z = u /\ w = v ) -> ( *Q ` w ) = ( *Q ` v ) )
4	3	eleq1d 2235	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( ( *Q ` w ) e. ( 2nd ` A ) <-> ( *Q ` v ) e. ( 2nd ` A ) ) )
5	1, 4	anbi12d 465	. . . . . 6 |- ( ( z = u /\ w = v ) -> ( ( z <Q w /\ ( *Q ` w ) e. ( 2nd ` A ) ) <-> ( u <Q v /\ ( *Q ` v ) e. ( 2nd ` A ) ) ) )
6	5	cbvexdva 1917	. . . . 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 2291	. . . 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 108	. . . . . . . 8 |- ( ( z = u /\ w = v ) -> z = u )
9	2, 8	breq12d 3995	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( w <Q z <-> v <Q u ) )
10	3	eleq1d 2235	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( ( *Q ` w ) e. ( 1st ` A ) <-> ( *Q ` v ) e. ( 1st ` A ) ) )
11	9, 10	anbi12d 465	. . . . . 6 |- ( ( z = u /\ w = v ) -> ( ( w <Q z /\ ( *Q ` w ) e. ( 1st ` A ) ) <-> ( v <Q u /\ ( *Q ` v ) e. ( 1st ` A ) ) ) )
12	11	cbvexdva 1917	. . . . 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 2291	. . . 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 3763	. . 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 7573	. 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 7578	. 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 5850	. . . 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 2174	. . 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 2830	. 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 409	1 |- ( A e. P. -> E. x e. P. ( A .P. x ) = 1P )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   E.wrex 2445   <.cop 3579   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   *Qcrq 7225   <Q cltq 7226   P.cnp 7232   1Pc1p 7233   .P. cmp 7235

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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-i1p 7408  df-imp 7410

This theorem is referenced by:  ltmprr  7583  recexgt0sr  7714