Theorem recexre 8899

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 8319	. . . 4 |- 0 e. RR
2		reapval 8897	. . . 4 |- ( ( A e. RR /\ 0 e. RR ) -> ( A #ℝ 0 <-> ( A < 0 \/ 0 < A ) ) )
3	1, 2	mpan2 429	. . 3 |- ( A e. RR -> ( A #ℝ 0 <-> ( A < 0 \/ 0 < A ) ) )
4		lt0neg1 8789	. . . . . . . . . 10 |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
5		renegcl 8580	. . . . . . . . . . 11 |- ( A e. RR -> -u A e. RR )
6		ltxrlt 8384	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ -u A e. RR ) -> ( 0 < -u A <-> 0 <RR -u A ) )
7	1, 5, 6	sylancr 418	. . . . . . . . . 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 458	. . . . . . . 8 |- ( ( A e. RR /\ A < 0 ) <-> ( A e. RR /\ 0 <RR -u A ) )
10		ax-precex 8282	. . . . . . . . . 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 2647	. . . . . . . . . 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 8305	. . . . . . . . . . . . 13 |- ( y e. RR -> y e. CC )
17	16	negnegd 8621	. . . . . . . . . . . 12 |- ( y e. RR -> -u -u y = y )
18	17	oveq2d 6094	. . . . . . . . . . 11 |- ( y e. RR -> ( -u A x. -u -u y ) = ( -u A x. y ) )
19	18	eqeq1d 2247	. . . . . . . . . 10 |- ( y e. RR -> ( ( -u A x. -u -u y ) = 1 <-> ( -u A x. y ) = 1 ) )
20	19	pm5.32i 458	. . . . . . . . 9 |- ( ( y e. RR /\ ( -u A x. -u -u y ) = 1 ) <-> ( y e. RR /\ ( -u A x. y ) = 1 ) )
21		renegcl 8580	. . . . . . . . . 10 |- ( y e. RR -> -u y e. RR )
22		negeq 8512	. . . . . . . . . . . . 13 |- ( x = -u y -> -u x = -u -u y )
23	22	oveq2d 6094	. . . . . . . . . . . 12 |- ( x = -u y -> ( -u A x. -u x ) = ( -u A x. -u -u y ) )
24	23	eqeq1d 2247	. . . . . . . . . . 11 |- ( x = -u y -> ( ( -u A x. -u x ) = 1 <-> ( -u A x. -u -u y ) = 1 ) )
25	24	rspcev 2929	. . . . . . . . . 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 2673	. . . . . 6 |- ( ( A e. RR /\ A < 0 ) -> E. x e. RR ( -u A x. -u x ) = 1 )
30		recn 8305	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
31		recn 8305	. . . . . . . . . 10 |- ( x e. RR -> x e. CC )
32		mul2neg 8718	. . . . . . . . . 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 2247	. . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( ( -u A x. -u x ) = 1 <-> ( A x. x ) = 1 ) )
35	34	rexbidva 2547	. . . . . . 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 8384	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> 0 <RR A ) )
40	1, 39	mpan 428	. . . . . . 7 |- ( A e. RR -> ( 0 < A <-> 0 <RR A ) )
41	40	pm5.32i 458	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) <-> ( A e. RR /\ 0 <RR A ) )
42		ax-precex 8282	. . . . . . 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 2647	. . . . . . 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 729	. . 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 720   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4128  (class class class)co 6078   CCcc 8170   RRcr 8171   0cc0 8172   1c1 8173   <RR cltrr 8176   x. cmul 8177   < clt 8353   -ucneg 8491   #_ℝ creap 8895

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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-distr 8276  ax-i2m1 8277  ax-0id 8280  ax-rnegex 8281  ax-precex 8282  ax-cnre 8283  ax-pre-ltadd 8288

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-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-pnf 8355  df-mnf 8356  df-ltxr 8358  df-sub 8492  df-neg 8493  df-reap 8896

This theorem is referenced by:  rimul  8906  recexap  8974  rerecclap  9053