Theorem recexre 8364

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 7790	. . . 4 |- 0 e. RR
2		reapval 8362	. . . 4 |- ( ( A e. RR /\ 0 e. RR ) -> ( A #ℝ 0 <-> ( A < 0 \/ 0 < A ) ) )
3	1, 2	mpan2 422	. . 3 |- ( A e. RR -> ( A #ℝ 0 <-> ( A < 0 \/ 0 < A ) ) )
4		lt0neg1 8254	. . . . . . . . . 10 |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
5		renegcl 8047	. . . . . . . . . . 11 |- ( A e. RR -> -u A e. RR )
6		ltxrlt 7854	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ -u A e. RR ) -> ( 0 < -u A <-> 0 <RR -u A ) )
7	1, 5, 6	sylancr 411	. . . . . . . . . 10 |- ( A e. RR -> ( 0 < -u A <-> 0 <RR -u A ) )
8	4, 7	bitrd 187	. . . . . . . . 9 |- ( A e. RR -> ( A < 0 <-> 0 <RR -u A ) )
9	8	pm5.32i 450	. . . . . . . 8 |- ( ( A e. RR /\ A < 0 ) <-> ( A e. RR /\ 0 <RR -u A ) )
10		ax-precex 7754	. . . . . . . . . 10 |- ( ( -u A e. RR /\ 0 <RR -u A ) -> E. y e. RR ( 0 <RR y /\ ( -u A x. y ) = 1 ) )
11		simpr 109	. . . . . . . . . . 11 |- ( ( 0 <RR y /\ ( -u A x. y ) = 1 ) -> ( -u A x. y ) = 1 )
12	11	reximi 2532	. . . . . . . . . 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 281	. . . . . . . 8 |- ( ( A e. RR /\ 0 <RR -u A ) -> E. y e. RR ( -u A x. y ) = 1 )
15	9, 14	sylbi 120	. . . . . . 7 |- ( ( A e. RR /\ A < 0 ) -> E. y e. RR ( -u A x. y ) = 1 )
16		recn 7777	. . . . . . . . . . . . 13 |- ( y e. RR -> y e. CC )
17	16	negnegd 8088	. . . . . . . . . . . 12 |- ( y e. RR -> -u -u y = y )
18	17	oveq2d 5798	. . . . . . . . . . 11 |- ( y e. RR -> ( -u A x. -u -u y ) = ( -u A x. y ) )
19	18	eqeq1d 2149	. . . . . . . . . 10 |- ( y e. RR -> ( ( -u A x. -u -u y ) = 1 <-> ( -u A x. y ) = 1 ) )
20	19	pm5.32i 450	. . . . . . . . 9 |- ( ( y e. RR /\ ( -u A x. -u -u y ) = 1 ) <-> ( y e. RR /\ ( -u A x. y ) = 1 ) )
21		renegcl 8047	. . . . . . . . . 10 |- ( y e. RR -> -u y e. RR )
22		negeq 7979	. . . . . . . . . . . . 13 |- ( x = -u y -> -u x = -u -u y )
23	22	oveq2d 5798	. . . . . . . . . . . 12 |- ( x = -u y -> ( -u A x. -u x ) = ( -u A x. -u -u y ) )
24	23	eqeq1d 2149	. . . . . . . . . . 11 |- ( x = -u y -> ( ( -u A x. -u x ) = 1 <-> ( -u A x. -u -u y ) = 1 ) )
25	24	rspcev 2793	. . . . . . . . . 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 281	. . . . . . . . 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 134	. . . . . . . 8 |- ( ( y e. RR /\ ( -u A x. y ) = 1 ) -> E. x e. RR ( -u A x. -u x ) = 1 )
28	27	adantl 275	. . . . . . 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 2557	. . . . . 6 |- ( ( A e. RR /\ A < 0 ) -> E. x e. RR ( -u A x. -u x ) = 1 )
30		recn 7777	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
31		recn 7777	. . . . . . . . . 10 |- ( x e. RR -> x e. CC )
32		mul2neg 8184	. . . . . . . . . 10 |- ( ( A e. CC /\ x e. CC ) -> ( -u A x. -u x ) = ( A x. x ) )
33	30, 31, 32	syl2an 287	. . . . . . . . 9 |- ( ( A e. RR /\ x e. RR ) -> ( -u A x. -u x ) = ( A x. x ) )
34	33	eqeq1d 2149	. . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( ( -u A x. -u x ) = 1 <-> ( A x. x ) = 1 ) )
35	34	rexbidva 2435	. . . . . . 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 274	. . . . . 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 146	. . . . 5 |- ( ( A e. RR /\ A < 0 ) -> E. x e. RR ( A x. x ) = 1 )
38	37	ex 114	. . . 4 |- ( A e. RR -> ( A < 0 -> E. x e. RR ( A x. x ) = 1 ) )
39		ltxrlt 7854	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> 0 <RR A ) )
40	1, 39	mpan 421	. . . . . . 7 |- ( A e. RR -> ( 0 < A <-> 0 <RR A ) )
41	40	pm5.32i 450	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) <-> ( A e. RR /\ 0 <RR A ) )
42		ax-precex 7754	. . . . . . 7 |- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
43		simpr 109	. . . . . . . 8 |- ( ( 0 <RR x /\ ( A x. x ) = 1 ) -> ( A x. x ) = 1 )
44	43	reximi 2532	. . . . . . 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 120	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> E. x e. RR ( A x. x ) = 1 )
47	46	ex 114	. . . 4 |- ( A e. RR -> ( 0 < A -> E. x e. RR ( A x. x ) = 1 ) )
48	38, 47	jaod 707	. . 3 |- ( A e. RR -> ( ( A < 0 \/ 0 < A ) -> E. x e. RR ( A x. x ) = 1 ) )
49	3, 48	sylbid 149	. 2 |- ( A e. RR -> ( A #ℝ 0 -> E. x e. RR ( A x. x ) = 1 ) )
50	49	imp 123	1 |- ( ( A e. RR /\ A #ℝ 0 ) -> E. x e. RR ( A x. x ) = 1 )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1332   e. wcel 1481   E.wrex 2418   class class class wbr 3937  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645   <RR cltrr 7648   x. cmul 7649   < clt 7824   -ucneg 7958   #_ℝ creap 8360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-ltxr 7829  df-sub 7959  df-neg 7960  df-reap 8361

This theorem is referenced by:  rimul  8371  recexap  8438  rerecclap  8514