Theorem rerecclap 8294

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 7585	. . . . . 6 |- 0 e. RR
2		apreap 8161	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> A #ℝ 0 ) )
3	1, 2	mpan2 417	. . . . 5 |- ( A e. RR -> ( A # 0 <-> A #ℝ 0 ) )
4	3	pm5.32i 443	. . . 4 |- ( ( A e. RR /\ A # 0 ) <-> ( A e. RR /\ A #ℝ 0 ) )
5		recexre 8152	. . . 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 2097	. . . . 5 |- ( x = ( 1 / A ) <-> ( 1 / A ) = x )
8		1cnd 7601	. . . . . 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 7613	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> x e. CC )
11		simpll 497	. . . . . . 7 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A e. RR )
12	11	recnd 7613	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A e. CC )
13		simplr 498	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A # 0 )
14		divmulap 8239	. . . . . 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 1185	. . . . 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 2388	. . 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 2417	. 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 1296   e. wcel 1445   E.wrex 2371   class class class wbr 3867  (class class class)co 5690   CCcc 7445   RRcr 7446   0cc0 7447   1c1 7448   x. cmul 7452   #_ℝ creap 8148   # cap 8155   / cdiv 8236

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-po 4147  df-iso 4148  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237

This theorem is referenced by:  redivclap  8295  rerecclapzi  8340  rerecclapd  8397  ltdiv2  8445  recnz  8938  reexpclzap  10090  redivap  10423  imdivap  10430  caucvgrelemrec  10527