Theorem receu 5670

Description: Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18.

Hypotheses

Ref	Expression
receu.1	|- A e. CC
receu.2	|- B e. CC
receu.3	|- A =/= 0

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem receu

Step	Hyp	Ref	Expression
1		opreq2 3954	. . . 4 |- (x = y -> (A x. x) = (A x. y))
2	1	eqeq1d 1475	. . 3 |- (x = y -> ((A x. x) = B <-> (A x. y) = B))
3	2	reu4 1924	. 2 |- (E!x e. CC (A x. x) = B <-> (E.x e. CC (A x. x) = B /\ A.x e. CC A.y e. CC (((A x. x) = B /\ (A x. y) = B) -> x = y)))
4		receu.1	. . . 4 |- A e. CC
5		receu.3	. . . 4 |- A =/= 0
6	4, 5	recex 5658	. . 3 |- E.y e. CC (A x. y) = 1
7		receu.2	. . . . . . 7 |- B e. CC
8		axmulcl 5245	. . . . . . 7 |- ((y e. CC /\ B e. CC) -> (y x. B) e. CC)
9	7, 8	mpan2 694	. . . . . 6 |- (y e. CC -> (y x. B) e. CC)
10		risset 1677	. . . . . 6 |- ((y x. B) e. CC <-> E.x e. CC x = (y x. B))
11	9, 10	sylib 198	. . . . 5 |- (y e. CC -> E.x e. CC x = (y x. B))
12		opreq2 3954	. . . . . . . . . . . . 13 |- (x = (y x. B) -> (A x. x) = (A x. (y x. B)))
13		axmulass 5250	. . . . . . . . . . . . . . 15 |- ((A e. CC /\ y e. CC /\ B e. CC) -> ((A x. y) x. B) = (A x. (y x. B)))
14	4, 7, 13	mp3an13 904	. . . . . . . . . . . . . 14 |- (y e. CC -> ((A x. y) x. B) = (A x. (y x. B)))
15	14	eqcomd 1472	. . . . . . . . . . . . 13 |- (y e. CC -> (A x. (y x. B)) = ((A x. y) x. B))
16	12, 15	sylan9eqr 1521	. . . . . . . . . . . 12 |- ((y e. CC /\ x = (y x. B)) -> (A x. x) = ((A x. y) x. B))
17		opreq1 3953	. . . . . . . . . . . . 13 |- ((A x. y) = 1 -> ((A x. y) x. B) = (1 x. B))
18	7	mulid2 5305	. . . . . . . . . . . . 13 |- (1 x. B) = B
19	17, 18	syl6eq 1515	. . . . . . . . . . . 12 |- ((A x. y) = 1 -> ((A x. y) x. B) = B)
20	16, 19	sylan9eqr 1521	. . . . . . . . . . 11 |- (((A x. y) = 1 /\ (y e. CC /\ x = (y x. B))) -> (A x. x) = B)
21	20	exp32 377	. . . . . . . . . 10 |- ((A x. y) = 1 -> (y e. CC -> (x = (y x. B) -> (A x. x) = B)))
22	21	impcom 351	. . . . . . . . 9 |- ((y e. CC /\ (A x. y) = 1) -> (x = (y x. B) -> (A x. x) = B))
23	22	a1d 12	. . . . . . . 8 |- ((y e. CC /\ (A x. y) = 1) -> (x e. CC -> (x = (y x. B) -> (A x. x) = B)))
24	23	r19.21aiv 1705	. . . . . . 7 |- ((y e. CC /\ (A x. y) = 1) -> A.x e. CC (x = (y x. B) -> (A x. x) = B))
25	24	ex 373	. . . . . 6 |- (y e. CC -> ((A x. y) = 1 -> A.x e. CC (x = (y x. B) -> (A x. x) = B)))
26		r19.22 1723	. . . . . 6 |- (A.x e. CC (x = (y x. B) -> (A x. x) = B) -> (E.x e. CC x = (y x. B) -> E.x e. CC (A x. x) = B))
27	25, 26	syl6 22	. . . . 5 |- (y e. CC -> ((A x. y) = 1 -> (E.x e. CC x = (y x. B) -> E.x e. CC (A x. x) = B)))
28	11, 27	mpid 47	. . . 4 |- (y e. CC -> ((A x. y) = 1 -> E.x e. CC (A x. x) = B))
29	28	r19.23aiv 1735	. . 3 |- (E.y e. CC (A x. y) = 1 -> E.x e. CC (A x. x) = B)
30	6, 29	ax-mp 7	. 2 |- E.x e. CC (A x. x) = B
31	5	mulcant2 5660	. . . . 5 |- ((A e. CC /\ x e. CC /\ y e. CC) -> ((A x. x) = (A x. y) <-> x = y))
32		eqtr3t 1486	. . . . 5 |- (((A x. x) = B /\ (A x. y) = B) -> (A x. x) = (A x. y))
33	31, 32	syl5bi 208	. . . 4 |- ((A e. CC /\ x e. CC /\ y e. CC) -> (((A x. x) = B /\ (A x. y) = B) -> x = y))
34	4, 33	mp3an1 900	. . 3 |- ((x e. CC /\ y e. CC) -> (((A x. x) = B /\ (A x. y) = B) -> x = y))
35	34	rgen2a 1691	. 2 |- A.x e. CC A.y e. CC (((A x. x) = B /\ (A x. y) = B) -> x = y)
36	3, 30, 35	mpbir2an 728	1 |- E!x e. CC (A x. x) = B

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   =/= wne 1577  A.wral 1637  E.wrex 1638  E!wreu 1639  (class class class)co 3948  CCcc 5204  0cc0 5206  1c1 5207   x. cmul 5211

This theorem is referenced by:  divmul 5674  divcl 5679

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463

Copyright terms: Public domain