Theorem ridlideq 14695

Description: If a magma has a left identity element and a right identity element, they are equal.

Assertion

Ref	Expression
ridlideq	|- ((U e. X /\ V e. X) -> (A.x e. X ((UGx) = x /\ (xGV) = x) -> U = V))

Distinct variable groups:   x,G   x,U   x,V   x,X

Proof of Theorem ridlideq

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . . . . 9 |- (x = U -> (UGx) = (UGU))
2		id 73	. . . . . . . . 9 |- (x = U -> x = U)
3	1, 2	eqeq12d 1899	. . . . . . . 8 |- (x = U -> ((UGx) = x <-> (UGU) = U))
4		opreq1 4889	. . . . . . . . 9 |- (x = U -> (xGV) = (UGV))
5	4, 2	eqeq12d 1899	. . . . . . . 8 |- (x = U -> ((xGV) = x <-> (UGV) = U))
6	3, 5	anbi12d 690	. . . . . . 7 |- (x = U -> (((UGx) = x /\ (xGV) = x) <-> ((UGU) = U /\ (UGV) = U)))
7	6	rcla4va 2378	. . . . . 6 |- ((U e. X /\ A.x e. X ((UGx) = x /\ (xGV) = x)) -> ((UGU) = U /\ (UGV) = U))
8	7	ex 402	. . . . 5 |- (U e. X -> (A.x e. X ((UGx) = x /\ (xGV) = x) -> ((UGU) = U /\ (UGV) = U)))
9		opreq2 4890	. . . . . . . . 9 |- (x = V -> (UGx) = (UGV))
10		id 73	. . . . . . . . 9 |- (x = V -> x = V)
11	9, 10	eqeq12d 1899	. . . . . . . 8 |- (x = V -> ((UGx) = x <-> (UGV) = V))
12		opreq1 4889	. . . . . . . . 9 |- (x = V -> (xGV) = (VGV))
13	12, 10	eqeq12d 1899	. . . . . . . 8 |- (x = V -> ((xGV) = x <-> (VGV) = V))
14	11, 13	anbi12d 690	. . . . . . 7 |- (x = V -> (((UGx) = x /\ (xGV) = x) <-> ((UGV) = V /\ (VGV) = V)))
15	14	rcla4va 2378	. . . . . 6 |- ((V e. X /\ A.x e. X ((UGx) = x /\ (xGV) = x)) -> ((UGV) = V /\ (VGV) = V))
16	15	ex 402	. . . . 5 |- (V e. X -> (A.x e. X ((UGx) = x /\ (xGV) = x) -> ((UGV) = V /\ (VGV) = V)))
17	8, 16	im2anan9 622	. . . 4 |- ((U e. X /\ V e. X) -> ((A.x e. X ((UGx) = x /\ (xGV) = x) /\ A.x e. X ((UGx) = x /\ (xGV) = x)) -> (((UGU) = U /\ (UGV) = U) /\ ((UGV) = V /\ (VGV) = V))))
18		eqtr 1904	. . . . . . . . 9 |- ((U = (UGV) /\ (UGV) = V) -> U = V)
19	18	ex 402	. . . . . . . 8 |- (U = (UGV) -> ((UGV) = V -> U = V))
20	19	eqcoms 1887	. . . . . . 7 |- ((UGV) = U -> ((UGV) = V -> U = V))
21	20	adantrd 427	. . . . . 6 |- ((UGV) = U -> (((UGV) = V /\ (VGV) = V) -> U = V))
22	21	adantl 424	. . . . 5 |- (((UGU) = U /\ (UGV) = U) -> (((UGV) = V /\ (VGV) = V) -> U = V))
23	22	imp 377	. . . 4 |- ((((UGU) = U /\ (UGV) = U) /\ ((UGV) = V /\ (VGV) = V)) -> U = V)
24	17, 23	syl6com 64	. . 3 |- ((A.x e. X ((UGx) = x /\ (xGV) = x) /\ A.x e. X ((UGx) = x /\ (xGV) = x)) -> ((U e. X /\ V e. X) -> U = V))
25	24	anidms 480	. 2 |- (A.x e. X ((UGx) = x /\ (xGV) = x) -> ((U e. X /\ V e. X) -> U = V))
26	25	com12 14	1 |- ((U e. X /\ V e. X) -> (A.x e. X ((UGx) = x /\ (xGV) = x) -> U = V))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  (class class class)co 4884

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886

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