Theorem recexre 8537

Description: Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)

Assertion

Ref	Expression
recexre	|- ( ( A e. RR /\ A #ℝ 0 ) -> E. x e. RR ( A x. x ) = 1 )

Distinct variable group:   x, A

Proof of Theorem recexre

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0re 7959	. . . 4 |- 0 e. RR
2		reapval 8535	. . . 4 |- ( ( A e. RR /\ 0 e. RR ) -> ( A #ℝ 0 <-> ( A < 0 \/ 0 < A ) ) )
3	1, 2	mpan2 425	. . 3 |- ( A e. RR -> ( A #ℝ 0 <-> ( A < 0 \/ 0 < A ) ) )
4		lt0neg1 8427	. . . . . . . . . 10 |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
5		renegcl 8220	. . . . . . . . . . 11 |- ( A e. RR -> -u A e. RR )
6		ltxrlt 8025	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ -u A e. RR ) -> ( 0 < -u A <-> 0 <RR -u A ) )
7	1, 5, 6	sylancr 414	. . . . . . . . . 10 |- ( A e. RR -> ( 0 < -u A <-> 0 <RR -u A ) )
8	4, 7	bitrd 188	. . . . . . . . 9 |- ( A e. RR -> ( A < 0 <-> 0 <RR -u A ) )
9	8	pm5.32i 454	. . . . . . . 8 |- ( ( A e. RR /\ A < 0 ) <-> ( A e. RR /\ 0 <RR -u A ) )
10		ax-precex 7923	. . . . . . . . . 10 |- ( ( -u A e. RR /\ 0 <RR -u A ) -> E. y e. RR ( 0 <RR y /\ ( -u A x. y ) = 1 ) )
11		simpr 110	. . . . . . . . . . 11 |- ( ( 0 <RR y /\ ( -u A x. y ) = 1 ) -> ( -u A x. y ) = 1 )
12	11	reximi 2574	. . . . . . . . . 10 |- ( E. y e. RR ( 0 <RR y /\ ( -u A x. y ) = 1 ) -> E. y e. RR ( -u A x. y ) = 1 )
13	10, 12	syl 14	. . . . . . . . 9 |- ( ( -u A e. RR /\ 0 <RR -u A ) -> E. y e. RR ( -u A x. y ) = 1 )
14	5, 13	sylan 283	. . . . . . . 8 |- ( ( A e. RR /\ 0 <RR -u A ) -> E. y e. RR ( -u A x. y ) = 1 )
15	9, 14	sylbi 121	. . . . . . 7 |- ( ( A e. RR /\ A < 0 ) -> E. y e. RR ( -u A x. y ) = 1 )
16		recn 7946	. . . . . . . . . . . . 13 |- ( y e. RR -> y e. CC )
17	16	negnegd 8261	. . . . . . . . . . . 12 |- ( y e. RR -> -u -u y = y )
18	17	oveq2d 5893	. . . . . . . . . . 11 |- ( y e. RR -> ( -u A x. -u -u y ) = ( -u A x. y ) )
19	18	eqeq1d 2186	. . . . . . . . . 10 |- ( y e. RR -> ( ( -u A x. -u -u y ) = 1 <-> ( -u A x. y ) = 1 ) )
20	19	pm5.32i 454	. . . . . . . . 9 |- ( ( y e. RR /\ ( -u A x. -u -u y ) = 1 ) <-> ( y e. RR /\ ( -u A x. y ) = 1 ) )
21		renegcl 8220	. . . . . . . . . 10 |- ( y e. RR -> -u y e. RR )
22		negeq 8152	. . . . . . . . . . . . 13 |- ( x = -u y -> -u x = -u -u y )
23	22	oveq2d 5893	. . . . . . . . . . . 12 |- ( x = -u y -> ( -u A x. -u x ) = ( -u A x. -u -u y ) )
24	23	eqeq1d 2186	. . . . . . . . . . 11 |- ( x = -u y -> ( ( -u A x. -u x ) = 1 <-> ( -u A x. -u -u y ) = 1 ) )
25	24	rspcev 2843	. . . . . . . . . 10 |- ( ( -u y e. RR /\ ( -u A x. -u -u y ) = 1 ) -> E. x e. RR ( -u A x. -u x ) = 1 )
26	21, 25	sylan 283	. . . . . . . . 9 |- ( ( y e. RR /\ ( -u A x. -u -u y ) = 1 ) -> E. x e. RR ( -u A x. -u x ) = 1 )
27	20, 26	sylbir 135	. . . . . . . 8 |- ( ( y e. RR /\ ( -u A x. y ) = 1 ) -> E. x e. RR ( -u A x. -u x ) = 1 )
28	27	adantl 277	. . . . . . 7 |- ( ( ( A e. RR /\ A < 0 ) /\ ( y e. RR /\ ( -u A x. y ) = 1 ) ) -> E. x e. RR ( -u A x. -u x ) = 1 )
29	15, 28	rexlimddv 2599	. . . . . 6 |- ( ( A e. RR /\ A < 0 ) -> E. x e. RR ( -u A x. -u x ) = 1 )
30		recn 7946	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
31		recn 7946	. . . . . . . . . 10 |- ( x e. RR -> x e. CC )
32		mul2neg 8357	. . . . . . . . . 10 |- ( ( A e. CC /\ x e. CC ) -> ( -u A x. -u x ) = ( A x. x ) )
33	30, 31, 32	syl2an 289	. . . . . . . . 9 |- ( ( A e. RR /\ x e. RR ) -> ( -u A x. -u x ) = ( A x. x ) )
34	33	eqeq1d 2186	. . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( ( -u A x. -u x ) = 1 <-> ( A x. x ) = 1 ) )
35	34	rexbidva 2474	. . . . . . 7 |- ( A e. RR -> ( E. x e. RR ( -u A x. -u x ) = 1 <-> E. x e. RR ( A x. x ) = 1 ) )
36	35	adantr 276	. . . . . 6 |- ( ( A e. RR /\ A < 0 ) -> ( E. x e. RR ( -u A x. -u x ) = 1 <-> E. x e. RR ( A x. x ) = 1 ) )
37	29, 36	mpbid 147	. . . . 5 |- ( ( A e. RR /\ A < 0 ) -> E. x e. RR ( A x. x ) = 1 )
38	37	ex 115	. . . 4 |- ( A e. RR -> ( A < 0 -> E. x e. RR ( A x. x ) = 1 ) )
39		ltxrlt 8025	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> 0 <RR A ) )
40	1, 39	mpan 424	. . . . . . 7 |- ( A e. RR -> ( 0 < A <-> 0 <RR A ) )
41	40	pm5.32i 454	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) <-> ( A e. RR /\ 0 <RR A ) )
42		ax-precex 7923	. . . . . . 7 |- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
43		simpr 110	. . . . . . . 8 |- ( ( 0 <RR x /\ ( A x. x ) = 1 ) -> ( A x. x ) = 1 )
44	43	reximi 2574	. . . . . . 7 |- ( E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) -> E. x e. RR ( A x. x ) = 1 )
45	42, 44	syl 14	. . . . . 6 |- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( A x. x ) = 1 )
46	41, 45	sylbi 121	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> E. x e. RR ( A x. x ) = 1 )
47	46	ex 115	. . . 4 |- ( A e. RR -> ( 0 < A -> E. x e. RR ( A x. x ) = 1 ) )
48	38, 47	jaod 717	. . 3 |- ( A e. RR -> ( ( A < 0 \/ 0 < A ) -> E. x e. RR ( A x. x ) = 1 ) )
49	3, 48	sylbid 150	. 2 |- ( A e. RR -> ( A #ℝ 0 -> E. x e. RR ( A x. x ) = 1 ) )
50	49	imp 124	1 |- ( ( A e. RR /\ A #ℝ 0 ) -> E. x e. RR ( A x. x ) = 1 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   = 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   <RR cltrr 7817   x. cmul 7818   < clt 7994   -ucneg 8131   #_ℝ creap 8533

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-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltadd 7929

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-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-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-ltxr 7999  df-sub 8132  df-neg 8133  df-reap 8534

This theorem is referenced by:  rimul  8544  recexap  8612  rerecclap  8689