Theorem ismgm 10367

Description: The predicate "is a magma". (Contributed by FL, 2-Nov-2009.)

Hypothesis

Ref	Expression
ismgm.1	|- X = dom dom G

Assertion

Ref	Expression
ismgm	|- (G e. A -> (G e. Magma <-> G:(X X. X)-->X))

Proof of Theorem ismgm

Step	Hyp	Ref	Expression
1		feq1 4551	. . . . 5 |- (g = G -> (g:(t X. t)-->t <-> G:(t X. t)-->t))
2	1	exbidv 1657	. . . 4 |- (g = G -> (E.t g:(t X. t)-->t <-> E.t G:(t X. t)-->t))
3		df-mgm 10366	. . . 4 |- Magma = {g | E.t g:(t X. t)-->t}
4	2, 3	elab2g 2406	. . 3 |- (G e. A -> (G e. Magma <-> E.t G:(t X. t)-->t))
5		f00 4601	. . . . . . . 8 |- (G:((/) X. (/))-->(/) <-> (G = (/) /\ ((/) X. (/)) = (/)))
6		dmeq 4157	. . . . . . . . . 10 |- (G = (/) -> dom G = dom (/))
7		dmeq 4157	. . . . . . . . . . 11 |- (dom G = dom (/) -> dom dom G = dom dom (/))
8		dm0 4170	. . . . . . . . . . . . 13 |- dom (/) = (/)
9	8	dmeqi 4158	. . . . . . . . . . . 12 |- dom dom (/) = dom (/)
10	9, 8	eqtri 1908	. . . . . . . . . . 11 |- dom dom (/) = (/)
11	7, 10	syl6req 1945	. . . . . . . . . 10 |- (dom G = dom (/) -> (/) = dom dom G)
12	6, 11	syl 12	. . . . . . . . 9 |- (G = (/) -> (/) = dom dom G)
13	12	adantr 425	. . . . . . . 8 |- ((G = (/) /\ ((/) X. (/)) = (/)) -> (/) = dom dom G)
14	5, 13	sylbi 216	. . . . . . 7 |- (G:((/) X. (/))-->(/) -> (/) = dom dom G)
15		xpeq12 4020	. . . . . . . . . 10 |- ((t = (/) /\ t = (/)) -> (t X. t) = ((/) X. (/)))
16	15	anidms 480	. . . . . . . . 9 |- (t = (/) -> (t X. t) = ((/) X. (/)))
17		feq23 4554	. . . . . . . . 9 |- (((t X. t) = ((/) X. (/)) /\ t = (/)) -> (G:(t X. t)-->t <-> G:((/) X. (/))-->(/)))
18	16, 17	mpancom 769	. . . . . . . 8 |- (t = (/) -> (G:(t X. t)-->t <-> G:((/) X. (/))-->(/)))
19		eqeq1 1890	. . . . . . . 8 |- (t = (/) -> (t = dom dom G <-> (/) = dom dom G))
20	18, 19	imbi12d 688	. . . . . . 7 |- (t = (/) -> ((G:(t X. t)-->t -> t = dom dom G) <-> (G:((/) X. (/))-->(/) -> (/) = dom dom G)))
21	14, 20	mpbiri 211	. . . . . 6 |- (t = (/) -> (G:(t X. t)-->t -> t = dom dom G))
22		fdm 4567	. . . . . . . 8 |- (G:(t X. t)-->t -> dom G = (t X. t))
23		dmeq 4157	. . . . . . . . 9 |- (dom G = (t X. t) -> dom dom G = dom ( t X. t))
24		df-ne 2019	. . . . . . . . . . . . 13 |- (t =/= (/) <-> -. t = (/))
25		dmxp 4177	. . . . . . . . . . . . 13 |- (t =/= (/) -> dom ( t X. t) = t)
26	24, 25	sylbir 218	. . . . . . . . . . . 12 |- (-. t = (/) -> dom ( t X. t) = t)
27	26	eqeq1d 1892	. . . . . . . . . . 11 |- (-. t = (/) -> (dom ( t X. t) = dom dom G <-> t = dom dom G))
28	27	biimpcd 172	. . . . . . . . . 10 |- (dom ( t X. t) = dom dom G -> (-. t = (/) -> t = dom dom G))
29	28	eqcoms 1887	. . . . . . . . 9 |- (dom dom G = dom ( t X. t) -> (-. t = (/) -> t = dom dom G))
30	23, 29	syl 12	. . . . . . . 8 |- (dom G = (t X. t) -> (-. t = (/) -> t = dom dom G))
31	22, 30	syl 12	. . . . . . 7 |- (G:(t X. t)-->t -> (-. t = (/) -> t = dom dom G))
32	31	com12 14	. . . . . 6 |- (-. t = (/) -> (G:(t X. t)-->t -> t = dom dom G))
33	21, 32	pm2.61i 140	. . . . 5 |- (G:(t X. t)-->t -> t = dom dom G)
34	33	pm4.71ri 700	. . . 4 |- (G:(t X. t)-->t <-> (t = dom dom G /\ G:(t X. t)-->t))
35	34	exbii 1398	. . 3 |- (E.t G:(t X. t)-->t <-> E.t(t = dom dom G /\ G:(t X. t)-->t))
36	4, 35	syl6bb 595	. 2 |- (G e. A -> (G e. Magma <-> E.t(t = dom dom G /\ G:(t X. t)-->t)))
37		dmexg 4206	. . . 4 |- (G e. A -> dom G e. _V)
38		dmexg 4206	. . . 4 |- (dom G e. _V -> dom dom G e. _V)
39	37, 38	syl 12	. . 3 |- (G e. A -> dom dom G e. _V)
40		xpeq12 4020	. . . . . . 7 |- ((t = dom dom G /\ t = dom dom G) -> (t X. t) = (dom dom G X. dom dom G))
41	40	anidms 480	. . . . . 6 |- (t = dom dom G -> (t X. t) = (dom dom G X. dom dom G))
42		feq23 4554	. . . . . 6 |- (((t X. t) = (dom dom G X. dom dom G) /\ t = dom dom G) -> (G:(t X. t)-->t <-> G:(dom dom G X. dom dom G)-->dom dom G))
43	41, 42	mpancom 769	. . . . 5 |- (t = dom dom G -> (G:(t X. t)-->t <-> G:(dom dom G X. dom dom G)-->dom dom G))
44		ismgm.1	. . . . . . . 8 |- X = dom dom G
45	44	eqcomi 1888	. . . . . . 7 |- dom dom G = X
46	45, 45	xpeq12i 4023	. . . . . 6 |- (dom dom G X. dom dom G) = (X X. X)
47		feq23 4554	. . . . . 6 |- (((dom dom G X. dom dom G) = (X X. X) /\ dom dom G = X) -> (G:(dom dom G X. dom dom G)-->dom dom G <-> G:(X X. X)-->X))
48	46, 45, 47	mp2an 761	. . . . 5 |- (G:(dom dom G X. dom dom G)-->dom dom G <-> G:(X X. X)-->X)
49	43, 48	syl6bb 595	. . . 4 |- (t = dom dom G -> (G:(t X. t)-->t <-> G:(X X. X)-->X))
50	49	ceqsexgv 2393	. . 3 |- (dom dom G e. _V -> (E.t(t = dom dom G /\ G:(t X. t)-->t) <-> G:(X X. X)-->X))
51	39, 50	syl 12	. 2 |- (G e. A -> (E.t(t = dom dom G /\ G:(t X. t)-->t) <-> G:(X X. X)-->X))
52	36, 51	bitrd 587	1 |- (G e. A -> (G e. Magma <-> G:(X X. X)-->X))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  _Vcvv 2292  (/)c0 2875   X. cxp 3984  dom cdm 3986  -->wf 3994  Magmacmagm 10365

This theorem is referenced by:  clmgm 10368  opidon 10369  issmgrp 10381  mgmlion 14697  isppm 14715  mgmrddd 14727  symgfo 14730

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-13 1311  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  ax-un 3790

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-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010  df-mgm 10366

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