Theorem rerecclap 9050

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 8316	. . . . . 6 |- 0 e. RR
2		apreap 8905	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> A #ℝ 0 ) )
3	1, 2	mpan2 429	. . . . 5 |- ( A e. RR -> ( A # 0 <-> A #ℝ 0 ) )
4	3	pm5.32i 458	. . . 4 |- ( ( A e. RR /\ A # 0 ) <-> ( A e. RR /\ A #ℝ 0 ) )
5		recexre 8896	. . . 4 |- ( ( A e. RR /\ A #ℝ 0 ) -> E. x e. RR ( A x. x ) = 1 )
6	4, 5	sylbi 121	. . 3 |- ( ( A e. RR /\ A # 0 ) -> E. x e. RR ( A x. x ) = 1 )
7		eqcom 2240	. . . . 5 |- ( x = ( 1 / A ) <-> ( 1 / A ) = x )
8		1cnd 8332	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> 1 e. CC )
9		simpr 110	. . . . . . 7 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> x e. RR )
10	9	recnd 8344	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> x e. CC )
11		simpll 531	. . . . . . 7 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A e. RR )
12	11	recnd 8344	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A e. CC )
13		simplr 533	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A # 0 )
14		divmulap 8995	. . . . . 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 1282	. . . . 5 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> ( ( 1 / A ) = x <-> ( A x. x ) = 1 ) )
16	7, 15	bitrid 192	. . . 4 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> ( x = ( 1 / A ) <-> ( A x. x ) = 1 ) )
17	16	rexbidva 2547	. . 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 167	. 2 |- ( ( A e. RR /\ A # 0 ) -> E. x e. RR x = ( 1 / A ) )
19		risset 2578	. 2 |- ( ( 1 / A ) e. RR <-> E. x e. RR x = ( 1 / A ) )
20	18, 19	sylibr 134	1 |- ( ( A e. RR /\ A # 0 ) -> ( 1 / A ) e. RR )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   x. cmul 8174   #_ℝ creap 8892   # cap 8899   / cdiv 8992

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993

This theorem is referenced by:  redivclap  9051  rerecclapzi  9096  rerecclapd  9154  rerecapb  9163  ltdiv2  9207  recnz  9718  reexpclzap  10974  redivap  11617  imdivap  11624  caucvgrelemrec  11723  trirec0  16998