Theorem rerecclap 8171

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 7467	. . . . . 6 |- 0 e. RR
2		apreap 8040	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> A #ℝ 0 ) )
3	1, 2	mpan2 416	. . . . 5 |- ( A e. RR -> ( A # 0 <-> A #ℝ 0 ) )
4	3	pm5.32i 442	. . . 4 |- ( ( A e. RR /\ A # 0 ) <-> ( A e. RR /\ A #ℝ 0 ) )
5		recexre 8031	. . . 4 |- ( ( A e. RR /\ A #ℝ 0 ) -> E. x e. RR ( A x. x ) = 1 )
6	4, 5	sylbi 119	. . 3 |- ( ( A e. RR /\ A # 0 ) -> E. x e. RR ( A x. x ) = 1 )
7		eqcom 2090	. . . . 5 |- ( x = ( 1 / A ) <-> ( 1 / A ) = x )
8		1cnd 7483	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> 1 e. CC )
9		simpr 108	. . . . . . 7 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> x e. RR )
10	9	recnd 7495	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> x e. CC )
11		simpll 496	. . . . . . 7 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A e. RR )
12	11	recnd 7495	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A e. CC )
13		simplr 497	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A # 0 )
14		divmulap 8116	. . . . . 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 1178	. . . . 5 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> ( ( 1 / A ) = x <-> ( A x. x ) = 1 ) )
16	7, 15	syl5bb 190	. . . 4 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> ( x = ( 1 / A ) <-> ( A x. x ) = 1 ) )
17	16	rexbidva 2377	. . 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 165	. 2 |- ( ( A e. RR /\ A # 0 ) -> E. x e. RR x = ( 1 / A ) )
19		risset 2406	. 2 |- ( ( 1 / A ) e. RR <-> E. x e. RR x = ( 1 / A ) )
20	18, 19	sylibr 132	1 |- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   E.wrex 2360   class class class wbr 3837  (class class class)co 5634   CCcc 7327   RRcr 7328   0cc0 7329   1c1 7330   x. cmul 7334   #_ℝ creap 8027   # cap 8034   / cdiv 8113

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114

This theorem is referenced by:  redivclap  8172  rerecclapzi  8217  rerecclapd  8272  ltdiv2  8320  recnz  8809  reexpclzap  9940  redivap  10273  imdivap  10280  caucvgrelemrec  10377