Theorem recexre 8686

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 8107	. . . 4 |- 0 e. RR
2		reapval 8684	. . . 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 8576	. . . . . . . . . 10 |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
5		renegcl 8368	. . . . . . . . . . 11 |- ( A e. RR -> -u A e. RR )
6		ltxrlt 8173	. . . . . . . . . . 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 8070	. . . . . . . . . 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 2605	. . . . . . . . . 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 8093	. . . . . . . . . . . . 13 |- ( y e. RR -> y e. CC )
17	16	negnegd 8409	. . . . . . . . . . . 12 |- ( y e. RR -> -u -u y = y )
18	17	oveq2d 5983	. . . . . . . . . . 11 |- ( y e. RR -> ( -u A x. -u -u y ) = ( -u A x. y ) )
19	18	eqeq1d 2216	. . . . . . . . . 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 8368	. . . . . . . . . 10 |- ( y e. RR -> -u y e. RR )
22		negeq 8300	. . . . . . . . . . . . 13 |- ( x = -u y -> -u x = -u -u y )
23	22	oveq2d 5983	. . . . . . . . . . . 12 |- ( x = -u y -> ( -u A x. -u x ) = ( -u A x. -u -u y ) )
24	23	eqeq1d 2216	. . . . . . . . . . 11 |- ( x = -u y -> ( ( -u A x. -u x ) = 1 <-> ( -u A x. -u -u y ) = 1 ) )
25	24	rspcev 2884	. . . . . . . . . 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 2630	. . . . . 6 |- ( ( A e. RR /\ A < 0 ) -> E. x e. RR ( -u A x. -u x ) = 1 )
30		recn 8093	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
31		recn 8093	. . . . . . . . . 10 |- ( x e. RR -> x e. CC )
32		mul2neg 8505	. . . . . . . . . 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 2216	. . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( ( -u A x. -u x ) = 1 <-> ( A x. x ) = 1 ) )
35	34	rexbidva 2505	. . . . . . 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 8173	. . . . . . . 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 8070	. . . . . . 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 2605	. . . . . . 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 719	. . 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 710   = wceq 1373   e. wcel 2178   E.wrex 2487   class class class wbr 4059  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   1c1 7961   <RR cltrr 7964   x. cmul 7965   < clt 8142   -ucneg 8279   #_ℝ creap 8682

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-sub 8280  df-neg 8281  df-reap 8683

This theorem is referenced by:  rimul  8693  recexap  8761  rerecclap  8838