Theorem efghgrpilem 8719

Description: Lemma for efghgrpi 8720,

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 8126	. 2 |- + e. Abel
2		efghgrpi.5	. 2 |- ( + |` (X X. X)) e. (SubGrp` + )
3		subgabl 8123	. . 3 |- (( + e. Abel /\ ( + |` (X X. X)) e. (SubGrp` + )) -> ( + |` (X X. X)) e. Abel)
4		issubg 8116	. . . . . . . 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 791	. . . . . 6 |- + e. Grp
7		axaddopr 5265	. . . . . . 7 |- + :(CC X. CC)-->CC
8	7	fdmi 3632	. . . . . 6 |- dom + = (CC X. CC)
9	6, 8	grprn 8056	. . . . 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 5273	. . . . . . . 8 |- ((A e. CC /\ x e. CC) -> (A x. x) e. CC)
13	11, 12	mpan 695	. . . . . . 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 3827	. . . . 5 |- F:CC-->CC
17		ssid 2080	. . . . 5 |- CC (_ CC
18		axmulopr 5266	. . . . . 6 |- x. :(CC X. CC)-->CC
19		ffn 3627	. . . . . 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 8718	. . . . . 6 |- ((A e. CC /\ z e. CC /\ w e. CC) -> (F` (z + w)) = ((F` z) x. (F` w)))
22	11, 21	mp3an1 903	. . . . 5 |- ((z e. CC /\ w e. CC) -> (F` (z + w)) = ((F` z) x. (F` w)))
23	5	3simp2i 792	. . . . . 6 |- ( + |` (X X. X)) e. Grp
24		efghgrpi.4	. . . . . 6 |- X (_ CC
25		eqid 1475	. . . . . . 7 |- ( + |` (X X. X)) = ( + |` (X X. X))
26	25	resgrprn 8095	. . . . . 6 |- ((dom + = (CC X. CC) /\ ( + |` (X X. X)) e. Grp /\ X (_ CC) -> X = ran ( + |` (X X. X)))
27	8, 23, 24, 26	mp3an 916	. . . . 5 |- X = ran ( + |` (X X. X))
28		fvex 3732	. . . . . . . 8 |- (exp` (A x. x)) e. V
29	28, 10	fnopab2 3618	. . . . . . 7 |- F Fn CC
30		fnssres 3600	. . . . . . . 8 |- ((F Fn CC /\ X (_ CC) -> (F |` X) Fn X)
31		fnrnfv 3759	. . . . . . . 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 697	. . . . . 6 |- ran ( F |` X) = {w | E.z e. X w = ((F |` X)` z)}
34		df-ima 3191	. . . . . 6 |- (F"X) = ran ( F |` X)
35		eqeq1 1481	. . . . . . . . . 10 |- (y = w -> (y = (exp` (A x. x)) <-> w = (exp` (A x. x))))
36	35	rexbidv 1664	. . . . . . . . 9 |- (y = w -> (E.x e. X y = (exp` (A x. x)) <-> E.x e. X w = (exp` (A x. x))))
37		opreq2 3969	. . . . . . . . . . . 12 |- (x = z -> (A x. x) = (A x. z))
38	37	fveq2d 3728	. . . . . . . . . . 11 |- (x = z -> (exp` (A x. x)) = (exp` (A x. z)))
39	38	eqeq2d 1486	. . . . . . . . . 10 |- (x = z -> (w = (exp` (A x. x)) <-> w = (exp` (A x. z))))
40	39	cbvrexv 1801	. . . . . . . . 9 |- (E.x e. X w = (exp` (A x. x)) <-> E.z e. X w = (exp` (A x. z)))
41	36, 40	syl6bb 536	. . . . . . . 8 |- (y = w -> (E.x e. X y = (exp` (A x. x)) <-> E.z e. X w = (exp` (A x. z))))
42	41	cbvabv 1909	. . . . . . 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 3734	. . . . . . . . . . 11 |- (z e. X -> ((F |` X)` z) = (F` z))
45	24	sseli 2065	. . . . . . . . . . . 12 |- (z e. X -> z e. CC)
46		fvex 3732	. . . . . . . . . . . . 13 |- (exp` (A x. z)) e. V
47	38, 10, 46	fvopab4 3780	. . . . . . . . . . . 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 1507	. . . . . . . . . 10 |- (z e. X -> ((F |` X)` z) = (exp` (A x. z)))
50	49	eqeq2d 1486	. . . . . . . . 9 |- (z e. X -> (w = ((F |` X)` z) <-> w = (exp` (A x. z))))
51	50	rexbiia 1674	. . . . . . . 8 |- (E.z e. X w = ((F |` X)` z) <-> E.z e. X w = (exp` (A x. z)))
52	51	abbii 1575	. . . . . . 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 1505	. . . . . 6 |- S = {w | E.z e. X w = ((F |` X)` z)}
54	33, 34, 53	3eqtr4r 1506	. . . . 5 |- S = (F"X)
55		efghgrpi.2	. . . . 5 |- G = ( x. |` (S X. S))
56	2, 9, 16, 17, 20, 22, 27, 54, 55	ghsubgi 8138	. . . 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 697	1 |- G e. Abel

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  {cab 1463  E.wrex 1646   (_ wss 2047  {copab 2666   X. cxp 3168  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173   Fn wfn 3177  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232   + caddc 5237   x. cmul 5239  expce 7293  Grpcgr 8033  Abelcabl 8099  SubGrpcsubg 8114

This theorem is referenced by:  efghgrpi 8720

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-sup 4574  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-2 5970  df-3 5971  df-4 5972  df-n0 6100  df-z 6136  df-fl 6224  df-seq1 6308  df-shft 6341  df-uz 6418  df-fz 6468  df-seqz 6533  df-seq0 6534  df-exp 6569  df-sqr 6670  df-re 6751  df-im 6752  df-cj 6753  df-abs 6754  df-fac 6932  df-bc 6957  df-clim 6975  df-sum 6980  df-ef 7298  df-grp 8037  df-gid 8038  df-ginv 8039  df-gdiv 8040  df-abl 8100  df-subg 8115

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