Theorem receu 10111

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 10098	. . . 4 |- ( ( B e. CC /\ B =/= 0 ) -> E. y e. CC ( B x. y ) = 1 )
2	1	3adant1 1012	. . 3 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> E. y e. CC ( B x. y ) = 1 )
3		simprl 754	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> y e. CC )
4		simpll 751	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> A e. CC )
5	3, 4	mulcld 9527	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( y x. A ) e. CC )
6		oveq1 6203	. . . . . . . 8 |- ( ( B x. y ) = 1 -> ( ( B x. y ) x. A ) = ( 1 x. A ) )
7	6	ad2antll 726	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( ( B x. y ) x. A ) = ( 1 x. A ) )
8		simplr 753	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> B e. CC )
9	8, 3, 4	mulassd 9530	. . . . . . 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 9525	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( 1 x. A ) = A )
11	7, 9, 10	3eqtr3d 2431	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( B x. ( y x. A ) ) = A )
12		oveq2 6204	. . . . . . . 8 |- ( x = ( y x. A ) -> ( B x. x ) = ( B x. ( y x. A ) ) )
13	12	eqeq1d 2384	. . . . . . 7 |- ( x = ( y x. A ) -> ( ( B x. x ) = A <-> ( B x. ( y x. A ) ) = A ) )
14	13	rspcev 3135	. . . . . 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 659	. . . . 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 2875	. . . 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 1014	. . 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 2410	. . . . . . 7 |- ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> ( B x. x ) = ( B x. y ) )
20		mulcan 10103	. . . . . . 7 |- ( ( x e. CC /\ y e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( B x. x ) = ( B x. y ) <-> x = y ) )
21	19, 20	syl5ib 219	. . . . . 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 1194	. . . . 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 433	. . . 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 1012	. . 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 2802	. 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 6204	. . . 4 |- ( x = y -> ( B x. x ) = ( B x. y ) )
27	26	eqeq1d 2384	. . 3 |- ( x = y -> ( ( B x. x ) = A <-> ( B x. y ) = A ) )
28	27	reu4 3218	. 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 662	1 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> E! x e. CC ( B x. x ) = A )

Syntax hints:   -> wi 4   /\ wa 367   /\ w3a 971   = wceq 1399   e. wcel 1826   =/= wne 2577   A.wral 2732   E.wrex 2733   E!wreu 2734  (class class class)co 6196   CCcc 9401   0cc0 9403   1c1 9404   x. cmul 9408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721

This theorem is referenced by:  divmul  10127  divcl  10130  rexdiv  27775