Theorem recexpr 7700

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 4035	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( z <Q w <-> u <Q v ) )
2		simpr 110	. . . . . . . . 9 |- ( ( z = u /\ w = v ) -> w = v )
3	2	fveq2d 5559	. . . . . . . 8 |- ( ( z = u /\ w = v ) -> ( *Q ` w ) = ( *Q ` v ) )
4	3	eleq1d 2262	. . . . . . 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 1941	. . . . 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 2318	. . . 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 4043	. . . . . . 7 |- ( ( z = u /\ w = v ) -> ( w <Q z <-> v <Q u ) )
10	3	eleq1d 2262	. . . . . . 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 1941	. . . . 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 2318	. . . 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 3810	. . 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 7694	. 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 7699	. 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 5927	. . . 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 2202	. . 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 2865	. 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 1364   E.wex 1503   e. wcel 2164   {cab 2179   E.wrex 2473   <.cop 3622   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   1stc1st 6193   2ndc2nd 6194   *Qcrq 7346   <Q cltq 7347   P.cnp 7353   1Pc1p 7354   .P. cmp 7356

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-i1p 7529  df-imp 7531

This theorem is referenced by:  ltmprr  7704  recexgt0sr  7835