Theorem rerecclap 8626

Description: Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)

Assertion

Ref	Expression
rerecclap	|- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR )

Proof of Theorem rerecclap

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0re 7899	. . . . . 6 |- 0 e. RR
2		apreap 8485	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> A #ℝ 0 ) )
3	1, 2	mpan2 422	. . . . 5 |- ( A e. RR -> ( A # 0 <-> A #ℝ 0 ) )
4	3	pm5.32i 450	. . . 4 |- ( ( A e. RR /\ A # 0 ) <-> ( A e. RR /\ A #ℝ 0 ) )
5		recexre 8476	. . . 4 |- ( ( A e. RR /\ A #ℝ 0 ) -> E. x e. RR ( A x. x ) = 1 )
6	4, 5	sylbi 120	. . 3 |- ( ( A e. RR /\ A # 0 ) -> E. x e. RR ( A x. x ) = 1 )
7		eqcom 2167	. . . . 5 |- ( x = ( 1 / A ) <-> ( 1 / A ) = x )
8		1cnd 7915	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> 1 e. CC )
9		simpr 109	. . . . . . 7 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> x e. RR )
10	9	recnd 7927	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> x e. CC )
11		simpll 519	. . . . . . 7 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A e. RR )
12	11	recnd 7927	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A e. CC )
13		simplr 520	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A # 0 )
14		divmulap 8571	. . . . . 6 |- ( ( 1 e. CC /\ x e. CC /\ ( A e. CC /\ A # 0 ) ) -> ( ( 1 / A ) = x <-> ( A x. x ) = 1 ) )
15	8, 10, 12, 13, 14	syl112anc 1232	. . . . 5 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> ( ( 1 / A ) = x <-> ( A x. x ) = 1 ) )
16	7, 15	syl5bb 191	. . . 4 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> ( x = ( 1 / A ) <-> ( A x. x ) = 1 ) )
17	16	rexbidva 2463	. . 3 |- ( ( A e. RR /\ A # 0 ) -> ( E. x e. RR x = ( 1 / A ) <-> E. x e. RR ( A x. x ) = 1 ) )
18	6, 17	mpbird 166	. 2 |- ( ( A e. RR /\ A # 0 ) -> E. x e. RR x = ( 1 / A ) )
19		risset 2494	. 2 |- ( ( 1 / A ) e. RR <-> E. x e. RR x = ( 1 / A ) )
20	18, 19	sylibr 133	1 |- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   class class class wbr 3982  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   1c1 7754   x. cmul 7758   #_ℝ creap 8472   # cap 8479   / cdiv 8568

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-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  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-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569

This theorem is referenced by:  redivclap  8627  rerecclapzi  8672  rerecclapd  8730  ltdiv2  8782  recnz  9284  reexpclzap  10475  redivap  10816  imdivap  10823  caucvgrelemrec  10921  trirec0  13923