Theorem efghgrpilem 8714

Description: Lemma for efghgrpi 8715,

Hypotheses

Ref	Expression
efghgrpi.1	|- S = {y | E.x e. X y = (exp` (A x. x))}
efghgrpi.2	|- G = ( x. |` (S X. S))
efghgrpi.3	|- A e. CC
efghgrpi.4	|- X (_ CC
efghgrpi.5	|- ( + |` (X X. X)) e. (SubGrp` + )
efghgrpi.6	|- F = {<.x, y>. | (x e. CC /\ y = (exp` (A x. x)))}

Assertion

Ref	Expression
efghgrpilem	|- G e. Abel

Distinct variable groups:   x,A,y   x,X,y

Proof of Theorem efghgrpilem

Step	Hyp	Ref	Expression
1		cnaddabl 8122	. 2 |- + e. Abel
2		efghgrpi.5	. 2 |- ( + |` (X X. X)) e. (SubGrp` + )
3		subgabl 8119	. . 3 |- (( + e. Abel /\ ( + |` (X X. X)) e. (SubGrp` + )) -> ( + |` (X X. X)) e. Abel)
4		issubg 8112	. . . . . . . 8 |- (( + |` (X X. X)) e. (SubGrp` + ) <-> ( + e. Grp /\ ( + |` (X X. X)) e. Grp /\ ( + |` (X X. X)) (_ + ))
5	2, 4	mpbi 189	. . . . . . 7 |- ( + e. Grp /\ ( + |` (X X. X)) e. Grp /\ ( + |` (X X. X)) (_ + )
6	5	3simp1i 793	. . . . . 6 |- + e. Grp
7		axaddopr 5277	. . . . . . 7 |- + :(CC X. CC)-->CC
8	7	fdmi 3638	. . . . . 6 |- dom + = (CC X. CC)
9	6, 8	grprn 8053	. . . . 5 |- CC = ran +
10		efghgrpi.6	. . . . . 6 |- F = {<.x, y>. | (x e. CC /\ y = (exp` (A x. x)))}
11		efghgrpi.3	. . . . . . . 8 |- A e. CC
12		axmulcl 5285	. . . . . . . 8 |- ((A e. CC /\ x e. CC) -> (A x. x) e. CC)
13	11, 12	mpan 697	. . . . . . 7 |- (x e. CC -> (A x. x) e. CC)
14		efclt 7312	. . . . . . 7 |- ((A x. x) e. CC -> (exp` (A x. x)) e. CC)
15	13, 14	syl 10	. . . . . 6 |- (x e. CC -> (exp` (A x. x)) e. CC)
16	10, 15	fopab 3833	. . . . 5 |- F:CC-->CC
17		ssid 2083	. . . . 5 |- CC (_ CC
18		axmulopr 5278	. . . . . 6 |- x. :(CC X. CC)-->CC
19		ffn 3633	. . . . . 6 |- ( x. :(CC X. CC)-->CC -> x. Fn (CC X. CC))
20	18, 19	ax-mp 7	. . . . 5 |- x. Fn (CC X. CC)
21	10	efgh 8713	. . . . . 6 |- ((A e. CC /\ z e. CC /\ w e. CC) -> (F` (z + w)) = ((F` z) x. (F` w)))
22	11, 21	mp3an1 905	. . . . 5 |- ((z e. CC /\ w e. CC) -> (F` (z + w)) = ((F` z) x. (F` w)))
23	5	3simp2i 794	. . . . . 6 |- ( + |` (X X. X)) e. Grp
24		efghgrpi.4	. . . . . 6 |- X (_ CC
25		eqid 1478	. . . . . . 7 |- ( + |` (X X. X)) = ( + |` (X X. X))
26	25	resgrprn 8091	. . . . . 6 |- ((dom + = (CC X. CC) /\ ( + |` (X X. X)) e. Grp /\ X (_ CC) -> X = ran ( + |` (X X. X)))
27	8, 23, 24, 26	mp3an 918	. . . . 5 |- X = ran ( + |` (X X. X))
28		fvex 3738	. . . . . . . 8 |- (exp` (A x. x)) e. V
29	28, 10	fnopab2 3624	. . . . . . 7 |- F Fn CC
30		fnssres 3606	. . . . . . . 8 |- ((F Fn CC /\ X (_ CC) -> (F |` X) Fn X)
31		fnrnfv 3765	. . . . . . . 8 |- ((F |` X) Fn X -> ran ( F |` X) = {w | E.z e. X w = ((F |` X)` z)})
32	30, 31	syl 10	. . . . . . 7 |- ((F Fn CC /\ X (_ CC) -> ran ( F |` X) = {w | E.z e. X w = ((F |` X)` z)})
33	29, 24, 32	mp2an 699	. . . . . 6 |- ran ( F |` X) = {w | E.z e. X w = ((F |` X)` z)}
34		df-ima 3197	. . . . . 6 |- (F"X) = ran ( F |` X)
35		eqeq1 1484	. . . . . . . . . 10 |- (y = w -> (y = (exp` (A x. x)) <-> w = (exp` (A x. x))))
36	35	rexbidv 1667	. . . . . . . . 9 |- (y = w -> (E.x e. X y = (exp` (A x. x)) <-> E.x e. X w = (exp` (A x. x))))
37		opreq2 3975	. . . . . . . . . . . 12 |- (x = z -> (A x. x) = (A x. z))
38	37	fveq2d 3734	. . . . . . . . . . 11 |- (x = z -> (exp` (A x. x)) = (exp` (A x. z)))
39	38	eqeq2d 1489	. . . . . . . . . 10 |- (x = z -> (w = (exp` (A x. x)) <-> w = (exp` (A x. z))))
40	39	cbvrexv 1804	. . . . . . . . 9 |- (E.x e. X w = (exp` (A x. x)) <-> E.z e. X w = (exp` (A x. z)))
41	36, 40	syl6bb 538	. . . . . . . 8 |- (y = w -> (E.x e. X y = (exp` (A x. x)) <-> E.z e. X w = (exp` (A x. z))))
42	41	cbvabv 1912	. . . . . . 7 |- {y | E.x e. X y = (exp` (A x. x))} = {w | E.z e. X w = (exp` (A x. z))}
43		efghgrpi.1	. . . . . . 7 |- S = {y | E.x e. X y = (exp` (A x. x))}
44		fvres 3740	. . . . . . . . . . 11 |- (z e. X -> ((F |` X)` z) = (F` z))
45	24	sseli 2068	. . . . . . . . . . . 12 |- (z e. X -> z e. CC)
46		fvex 3738	. . . . . . . . . . . . 13 |- (exp` (A x. z)) e. V
47	38, 10, 46	fvopab4 3786	. . . . . . . . . . . 12 |- (z e. CC -> (F` z) = (exp` (A x. z)))
48	45, 47	syl 10	. . . . . . . . . . 11 |- (z e. X -> (F` z) = (exp` (A x. z)))
49	44, 48	eqtrd 1510	. . . . . . . . . 10 |- (z e. X -> ((F |` X)` z) = (exp` (A x. z)))
50	49	eqeq2d 1489	. . . . . . . . 9 |- (z e. X -> (w = ((F |` X)` z) <-> w = (exp` (A x. z))))
51	50	rexbiia 1677	. . . . . . . 8 |- (E.z e. X w = ((F |` X)` z) <-> E.z e. X w = (exp` (A x. z)))
52	51	abbii 1578	. . . . . . 7 |- {w | E.z e. X w = ((F |` X)` z)} = {w | E.z e. X w = (exp` (A x. z))}
53	42, 43, 52	3eqtr4 1508	. . . . . 6 |- S = {w | E.z e. X w = ((F |` X)` z)}
54	33, 34, 53	3eqtr4r 1509	. . . . 5 |- S = (F"X)
55		efghgrpi.2	. . . . 5 |- G = ( x. |` (S X. S))
56	2, 9, 16, 17, 20, 22, 27, 54, 55	ghsubgi 8134	. . . 4 |- (G e. Grp /\ (( + |` (X X. X)) e. Abel -> G e. Abel))
57	56	pm3.27i 324	. . 3 |- (( + |` (X X. X)) e. Abel -> G e. Abel)
58	3, 57	syl 10	. 2 |- (( + e. Abel /\ ( + |` (X X. X)) e. (SubGrp` + )) -> G e. Abel)
59	1, 2, 58	mp2an 699	1 |- G e. Abel

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  {cab 1466  E.wrex 1649   (_ wss 2050  {copab 2671   X. cxp 3174  dom cdm 3176  ran crn 3177   |` cres 3178  "cima 3179   Fn wfn 3183  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244   + caddc 5249   x. cmul 5251  expce 7293  Grpcgr 8030  Abelcabl 8095  SubGrpcsubg 8110

This theorem is referenced by:  efghgrpi 8715

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-3 5973  df-4 5974  df-n0 6102  df-z 6138  df-fl 6226  df-seq1 6309  df-shft 6342  df-uz 6419  df-fz 6469  df-seqz 6534  df-seq0 6535  df-exp 6570  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-fac 6932  df-bc 6957  df-clim 6975  df-sum 6980  df-ef 7298  df-grp 8034  df-gid 8035  df-ginv 8036  df-gdiv 8037  df-abl 8096  df-subg 8111

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