Theorem receui 6686

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
receui	|- E!x e. CC (A x. x) = B

Distinct variable groups:   x,A   x,B

Proof of Theorem receui

Step	Hyp	Ref	Expression
1		opreq2 4701	. . . 4 |- (x = y -> (A x. x) = (A x. y))
2	1	eqeq1d 1729	. . 3 |- (x = y -> ((A x. x) = B <-> (A x. y) = B))
3	2	reu4 2279	. 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	recexi 6673	. . 3 |- E.y e. CC (A x. y) = 1
7		receu.2	. . . . . . 7 |- B e. CC
8		mulcl 6252	. . . . . . 7 |- ((y e. CC /\ B e. CC) -> (y x. B) e. CC)
9	7, 8	mpan2 757	. . . . . 6 |- (y e. CC -> (y x. B) e. CC)
10		risset 1979	. . . . . 6 |- ((y x. B) e. CC <-> E.x e. CC x = (y x. B))
11	9, 10	sylib 214	. . . . 5 |- (y e. CC -> E.x e. CC x = (y x. B))
12		opreq2 4701	. . . . . . . . . . . . 13 |- (x = (y x. B) -> (A x. x) = (A x. (y x. B)))
13		mulass 6257	. . . . . . . . . . . . . . 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 1029	. . . . . . . . . . . . . 14 |- (y e. CC -> ((A x. y) x. B) = (A x. (y x. B)))
15	14	eqcomd 1726	. . . . . . . . . . . . 13 |- (y e. CC -> (A x. (y x. B)) = ((A x. y) x. B))
16	12, 15	sylan9eqr 1788	. . . . . . . . . . . 12 |- ((y e. CC /\ x = (y x. B)) -> (A x. x) = ((A x. y) x. B))
17		opreq1 4700	. . . . . . . . . . . . 13 |- ((A x. y) = 1 -> ((A x. y) x. B) = (1 x. B))
18	7	mulid2i 6282	. . . . . . . . . . . . 13 |- (1 x. B) = B
19	17, 18	syl6eq 1781	. . . . . . . . . . . 12 |- ((A x. y) = 1 -> ((A x. y) x. B) = B)
20	16, 19	sylan9eqr 1788	. . . . . . . . . . 11 |- (((A x. y) = 1 /\ (y e. CC /\ x = (y x. B))) -> (A x. x) = B)
21	20	exp32 406	. . . . . . . . . 10 |- ((A x. y) = 1 -> (y e. CC -> (x = (y x. B) -> (A x. x) = B)))
22	21	impcom 376	. . . . . . . . 9 |- ((y e. CC /\ (A x. y) = 1) -> (x = (y x. B) -> (A x. x) = B))
23	22	a1d 15	. . . . . . . 8 |- ((y e. CC /\ (A x. y) = 1) -> (x e. CC -> (x = (y x. B) -> (A x. x) = B)))
24	23	r19.21aiv 2009	. . . . . . 7 |- ((y e. CC /\ (A x. y) = 1) -> A.x e. CC (x = (y x. B) -> (A x. x) = B))
25	24	ex 400	. . . . . 6 |- (y e. CC -> ((A x. y) = 1 -> A.x e. CC (x = (y x. B) -> (A x. x) = B)))
26		rexim 2028	. . . . . 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 25	. . . . 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 58	. . . 4 |- (y e. CC -> ((A x. y) = 1 -> E.x e. CC (A x. x) = B))
29	28	r19.23aiv 2045	. . 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	mulcant2i 6675	. . . . 5 |- ((x e. CC /\ y e. CC /\ A e. CC) -> ((A x. x) = (A x. y) <-> x = y))
32		eqtr3 1744	. . . . 5 |- (((A x. x) = B /\ (A x. y) = B) -> (A x. x) = (A x. y))
33	31, 32	syl5bi 224	. . . 4 |- ((x e. CC /\ y e. CC /\ A e. CC) -> (((A x. x) = B /\ (A x. y) = B) -> x = y))
34	4, 33	mp3an3 1027	. . 3 |- ((x e. CC /\ y e. CC) -> (((A x. x) = B /\ (A x. y) = B) -> x = y))
35	34	rgen2a 1994	. 2 |- A.x e. CC A.y e. CC (((A x. x) = B /\ (A x. y) = B) -> x = y)
36	3, 30, 35	mpbir2an 797	1 |- E!x e. CC (A x. x) = B

Syntax hints:   -> wi 3   /\ wa 239   /\ w3a 855   = wceq 1136   e. wcel 1138   =/= wne 1854  A.wral 1939  E.wrex 1940  E!wreu 1941  (class class class)co 4695  CCcc 6180  0cc0 6182  1c1 6183   x. cmul 6187

This theorem is referenced by:  divmuli 6690  divcli 6695

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-13 1149  ax-14 1150  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426  ax-ext 1702  ax-rep 3243  ax-sep 3253  ax-nul 3260  ax-pow 3296  ax-pr 3339  ax-un 3601  ax-inf2 5540

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-3or 856  df-3an 857  df-ex 1165  df-sb 1374  df-eu 1613  df-mo 1614  df-clab 1709  df-cleq 1714  df-clel 1717  df-ne 1856  df-nel 1857  df-ral 1943  df-rex 1944  df-reu 1945  df-rab 1946  df-v 2127  df-sbc 2287  df-csb 2374  df-dif 2430  df-un 2433  df-in 2436  df-ss 2438  df-pss 2440  df-nul 2702  df-if 2807  df-pw 2859  df-sn 2873  df-pr 2874  df-tp 2876  df-op 2877  df-uni 3000  df-int 3037  df-iun 3079  df-br 3159  df-opab 3214  df-tr 3230  df-eprel 3398  df-id 3401  df-po 3406  df-so 3419  df-fr 3440  df-we 3459  df-ord 3475  df-on 3476  df-lim 3477  df-suc 3478  df-om 3761  df-xp 3811  df-rel 3812  df-cnv 3813  df-co 3814  df-dm 3815  df-rn 3816  df-res 3817  df-ima 3818  df-fun 3819  df-fn 3820  df-f 3821  df-f1 3822  df-fo 3823  df-f1o 3824  df-fv 3825  df-opr 4697  df-oprab 4698  df-mpt 4817  df-1st 4831  df-2nd 4832  df-iota 4900  df-rdg 4951  df-1o 4988  df-oadd 4990  df-omul 4991  df-er 5129  df-ec 5131  df-qs 5134  df-en 5238  df-dom 5239  df-sdom 5240  df-undef 5367  df-riota 5371  df-ni 5948  df-pli 5949  df-mi 5950  df-lti 5951  df-plpq 5983  df-mpq 5984  df-enq 5985  df-nq 5986  df-plq 5987  df-mq 5988  df-rq 5989  df-ltq 5990  df-1q 5991  df-np 6034  df-1p 6035  df-plp 6036  df-mp 6037  df-ltp 6038  df-plpr 6112  df-mpr 6113  df-enr 6114  df-nr 6115  df-plr 6116  df-mr 6117  df-ltr 6118  df-0r 6119  df-1r 6120  df-m1r 6121  df-c 6188  df-0 6189  df-1 6190  df-i 6191  df-r 6192  df-plus 6193  df-mul 6194  df-lt 6195  df-sub 6307  df-neg 6309  df-pnf 6450  df-mnf 6451  df-xr 6452  df-ltxr 6453  df-le 6454

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