Theorem rerecclap 8689

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 7959	. . . . . 6 |- 0 e. RR
2		apreap 8546	. . . . . 6 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> A #ℝ 0 ) )
3	1, 2	mpan2 425	. . . . 5 |- ( A e. RR -> ( A # 0 <-> A #ℝ 0 ) )
4	3	pm5.32i 454	. . . 4 |- ( ( A e. RR /\ A # 0 ) <-> ( A e. RR /\ A #ℝ 0 ) )
5		recexre 8537	. . . 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 2179	. . . . 5 |- ( x = ( 1 / A ) <-> ( 1 / A ) = x )
8		1cnd 7975	. . . . . 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 7988	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> x e. CC )
11		simpll 527	. . . . . . 7 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A e. RR )
12	11	recnd 7988	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A e. CC )
13		simplr 528	. . . . . 6 |- ( ( ( A e. RR /\ A # 0 ) /\ x e. RR ) -> A # 0 )
14		divmulap 8634	. . . . . 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 1242	. . . . 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 2474	. . 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 2505	. 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 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4005  (class class class)co 5877   CCcc 7811   RRcr 7812   0cc0 7813   1c1 7814   x. cmul 7818   #_ℝ creap 8533   # cap 8540   / cdiv 8631

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632

This theorem is referenced by:  redivclap  8690  rerecclapzi  8735  rerecclapd  8793  rerecapb  8802  ltdiv2  8846  recnz  9348  reexpclzap  10542  redivap  10885  imdivap  10892  caucvgrelemrec  10990  trirec0  14877