Theorem recexre 8497

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 7920	. . . 4 |- 0 e. RR
2		reapval 8495	. . . 4 |- ( ( A e. RR /\ 0 e. RR ) -> ( A #ℝ 0 <-> ( A < 0 \/ 0 < A ) ) )
3	1, 2	mpan2 423	. . 3 |- ( A e. RR -> ( A #ℝ 0 <-> ( A < 0 \/ 0 < A ) ) )
4		lt0neg1 8387	. . . . . . . . . 10 |- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
5		renegcl 8180	. . . . . . . . . . 11 |- ( A e. RR -> -u A e. RR )
6		ltxrlt 7985	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ -u A e. RR ) -> ( 0 < -u A <-> 0 <RR -u A ) )
7	1, 5, 6	sylancr 412	. . . . . . . . . 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 451	. . . . . . . 8 |- ( ( A e. RR /\ A < 0 ) <-> ( A e. RR /\ 0 <RR -u A ) )
10		ax-precex 7884	. . . . . . . . . 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 2567	. . . . . . . . . 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 7907	. . . . . . . . . . . . 13 |- ( y e. RR -> y e. CC )
17	16	negnegd 8221	. . . . . . . . . . . 12 |- ( y e. RR -> -u -u y = y )
18	17	oveq2d 5869	. . . . . . . . . . 11 |- ( y e. RR -> ( -u A x. -u -u y ) = ( -u A x. y ) )
19	18	eqeq1d 2179	. . . . . . . . . 10 |- ( y e. RR -> ( ( -u A x. -u -u y ) = 1 <-> ( -u A x. y ) = 1 ) )
20	19	pm5.32i 451	. . . . . . . . 9 |- ( ( y e. RR /\ ( -u A x. -u -u y ) = 1 ) <-> ( y e. RR /\ ( -u A x. y ) = 1 ) )
21		renegcl 8180	. . . . . . . . . 10 |- ( y e. RR -> -u y e. RR )
22		negeq 8112	. . . . . . . . . . . . 13 |- ( x = -u y -> -u x = -u -u y )
23	22	oveq2d 5869	. . . . . . . . . . . 12 |- ( x = -u y -> ( -u A x. -u x ) = ( -u A x. -u -u y ) )
24	23	eqeq1d 2179	. . . . . . . . . . 11 |- ( x = -u y -> ( ( -u A x. -u x ) = 1 <-> ( -u A x. -u -u y ) = 1 ) )
25	24	rspcev 2834	. . . . . . . . . 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 2592	. . . . . 6 |- ( ( A e. RR /\ A < 0 ) -> E. x e. RR ( -u A x. -u x ) = 1 )
30		recn 7907	. . . . . . . . . 10 |- ( A e. RR -> A e. CC )
31		recn 7907	. . . . . . . . . 10 |- ( x e. RR -> x e. CC )
32		mul2neg 8317	. . . . . . . . . 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 2179	. . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( ( -u A x. -u x ) = 1 <-> ( A x. x ) = 1 ) )
35	34	rexbidva 2467	. . . . . . 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 7985	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> 0 <RR A ) )
40	1, 39	mpan 422	. . . . . . 7 |- ( A e. RR -> ( 0 < A <-> 0 <RR A ) )
41	40	pm5.32i 451	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) <-> ( A e. RR /\ 0 <RR A ) )
42		ax-precex 7884	. . . . . . 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 2567	. . . . . . 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 712	. . 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 703   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775   <RR cltrr 7778   x. cmul 7779   < clt 7954   -ucneg 8091   #_ℝ creap 8493

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093  df-reap 8494

This theorem is referenced by:  rimul  8504  recexap  8571  rerecclap  8647