Theorem receu 9623

Description: Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)

Assertion

Ref	Expression
receu	|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> E! x e. CC ( B x. x ) = A )

Distinct variable groups:   x, A   x, B

Proof of Theorem receu

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		recex 9610	. . . 4 |- ( ( B e. CC /\ B =/= 0 ) -> E. y e. CC ( B x. y ) = 1 )
2	1	3adant1 975	. . 3 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> E. y e. CC ( B x. y ) = 1 )
3		simprl 733	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> y e. CC )
4		simpll 731	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> A e. CC )
5	3, 4	mulcld 9064	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( y x. A ) e. CC )
6		oveq1 6047	. . . . . . . 8 |- ( ( B x. y ) = 1 -> ( ( B x. y ) x. A ) = ( 1 x. A ) )
7	6	ad2antll 710	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( ( B x. y ) x. A ) = ( 1 x. A ) )
8		simplr 732	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> B e. CC )
9	8, 3, 4	mulassd 9067	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( ( B x. y ) x. A ) = ( B x. ( y x. A ) ) )
10	4	mulid2d 9062	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( 1 x. A ) = A )
11	7, 9, 10	3eqtr3d 2444	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( B x. ( y x. A ) ) = A )
12		oveq2 6048	. . . . . . . 8 |- ( x = ( y x. A ) -> ( B x. x ) = ( B x. ( y x. A ) ) )
13	12	eqeq1d 2412	. . . . . . 7 |- ( x = ( y x. A ) -> ( ( B x. x ) = A <-> ( B x. ( y x. A ) ) = A ) )
14	13	rspcev 3012	. . . . . 6 |- ( ( ( y x. A ) e. CC /\ ( B x. ( y x. A ) ) = A ) -> E. x e. CC ( B x. x ) = A )
15	5, 11, 14	syl2anc 643	. . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> E. x e. CC ( B x. x ) = A )
16	15	rexlimdvaa 2791	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( E. y e. CC ( B x. y ) = 1 -> E. x e. CC ( B x. x ) = A ) )
17	16	3adant3 977	. . 3 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( E. y e. CC ( B x. y ) = 1 -> E. x e. CC ( B x. x ) = A ) )
18	2, 17	mpd 15	. 2 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> E. x e. CC ( B x. x ) = A )
19		eqtr3 2423	. . . . . . 7 |- ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> ( B x. x ) = ( B x. y ) )
20		mulcan 9615	. . . . . . 7 |- ( ( x e. CC /\ y e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( B x. x ) = ( B x. y ) <-> x = y ) )
21	19, 20	syl5ib 211	. . . . . 6 |- ( ( x e. CC /\ y e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) )
22	21	3expa 1153	. . . . 5 |- ( ( ( x e. CC /\ y e. CC ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) )
23	22	expcom 425	. . . 4 |- ( ( B e. CC /\ B =/= 0 ) -> ( ( x e. CC /\ y e. CC ) -> ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) ) )
24	23	3adant1 975	. . 3 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( x e. CC /\ y e. CC ) -> ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) ) )
25	24	ralrimivv 2757	. 2 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> A. x e. CC A. y e. CC ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) )
26		oveq2 6048	. . . 4 |- ( x = y -> ( B x. x ) = ( B x. y ) )
27	26	eqeq1d 2412	. . 3 |- ( x = y -> ( ( B x. x ) = A <-> ( B x. y ) = A ) )
28	27	reu4 3088	. 2 |- ( E! x e. CC ( B x. x ) = A <-> ( E. x e. CC ( B x. x ) = A /\ A. x e. CC A. y e. CC ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) ) )
29	18, 25, 28	sylanbrc 646	1 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> E! x e. CC ( B x. x ) = A )

Syntax hints:   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E.wrex 2667   E!wreu 2668  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947   x. cmul 8951

This theorem is referenced by:  divmul  9637  divcl  9640  rexdiv  24125

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250