Theorem efghgrpilem 10073

Description: Lemma for efghgrpi 10074,

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 C_ 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 9434	. 2 |- + e. Abel
2		efghgrpi.5	. 2 |- ( + |` (X X. X)) e. (SubGrp` + )
3		subgabl 9432	. . 3 |- (( + e. Abel /\ ( + |` (X X. X)) e. (SubGrp` + )) -> ( + |` (X X. X)) e. Abel)
4		issubg 9425	. . . . . . . 8 |- (( + |` (X X. X)) e. (SubGrp` + ) <-> ( + e. Grp /\ ( + |` (X X. X)) e. Grp /\ ( + |` (X X. X)) C_ + ))
5	2, 4	mpbi 206	. . . . . . 7 |- ( + e. Grp /\ ( + |` (X X. X)) e. Grp /\ ( + |` (X X. X)) C_ + )
6	5	simp1i 885	. . . . . 6 |- + e. Grp
7		axaddopr 6417	. . . . . . 7 |- + :(CC X. CC)-->CC
8	7	fdmi 4568	. . . . . 6 |- dom + = (CC X. CC)
9	6, 8	grprn 9336	. . . . 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		mulcl 6456	. . . . . . . 8 |- ((A e. CC /\ x e. CC) -> (A x. x) e. CC)
13	11, 12	mpan 759	. . . . . . 7 |- (x e. CC -> (A x. x) e. CC)
14		efcl 8574	. . . . . . 7 |- ((A x. x) e. CC -> (exp` (A x. x)) e. CC)
15	13, 14	syl 12	. . . . . 6 |- (x e. CC -> (exp` (A x. x)) e. CC)
16	10, 15	fopab 4800	. . . . 5 |- F:CC-->CC
17		ssid 2634	. . . . 5 |- CC C_ CC
18		axmulopr 6418	. . . . . 6 |- x. :(CC X. CC)-->CC
19		ffn 4562	. . . . . 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 10072	. . . . . 6 |- ((A e. CC /\ z e. CC /\ w e. CC) -> (F` (z + w)) = ((F` z) x. (F` w)))
22	11, 21	mp3an1 1178	. . . . 5 |- ((z e. CC /\ w e. CC) -> (F` (z + w)) = ((F` z) x. (F` w)))
23	5	simp2i 886	. . . . . 6 |- ( + |` (X X. X)) e. Grp
24		efghgrpi.4	. . . . . 6 |- X C_ CC
25		eqid 1884	. . . . . . 7 |- ( + |` (X X. X)) = ( + |` (X X. X))
26	25	resgrprn 9403	. . . . . 6 |- ((dom + = (CC X. CC) /\ ( + |` (X X. X)) e. Grp /\ X C_ CC) -> X = ran ( + |` (X X. X)))
27	8, 23, 24, 26	mp3an 1191	. . . . 5 |- X = ran ( + |` (X X. X))
28		fvex 4689	. . . . . . . 8 |- (exp` (A x. x)) e. _V
29	28, 10	fnopab2 4549	. . . . . . 7 |- F Fn CC
30		fnssres 4526	. . . . . . . 8 |- ((F Fn CC /\ X C_ CC) -> (F |` X) Fn X)
31		fnrnfv 4718	. . . . . . . 8 |- ((F |` X) Fn X -> ran ( F |` X) = {w | E.z e. X w = ((F |` X)` z)})
32	30, 31	syl 12	. . . . . . 7 |- ((F Fn CC /\ X C_ CC) -> ran ( F |` X) = {w | E.z e. X w = ((F |` X)` z)})
33	29, 24, 32	mp2an 761	. . . . . 6 |- ran ( F |` X) = {w | E.z e. X w = ((F |` X)` z)}
34		df-ima 4007	. . . . . 6 |- (F"X) = ran ( F |` X)
35		eqeq1 1890	. . . . . . . . . 10 |- (y = w -> (y = (exp` (A x. x)) <-> w = (exp` (A x. x))))
36	35	rexbidv 2124	. . . . . . . . 9 |- (y = w -> (E.x e. X y = (exp` (A x. x)) <-> E.x e. X w = (exp` (A x. x))))
37		opreq2 4890	. . . . . . . . . . . 12 |- (x = z -> (A x. x) = (A x. z))
38	37	fveq2d 4685	. . . . . . . . . . 11 |- (x = z -> (exp` (A x. x)) = (exp` (A x. z)))
39	38	eqeq2d 1895	. . . . . . . . . 10 |- (x = z -> (w = (exp` (A x. x)) <-> w = (exp` (A x. z))))
40	39	cbvrexv 2281	. . . . . . . . 9 |- (E.x e. X w = (exp` (A x. x)) <-> E.z e. X w = (exp` (A x. z)))
41	36, 40	syl6bb 595	. . . . . . . 8 |- (y = w -> (E.x e. X y = (exp` (A x. x)) <-> E.z e. X w = (exp` (A x. z))))
42	41	cbvabv 2420	. . . . . . 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 4691	. . . . . . . . . . 11 |- (z e. X -> ((F |` X)` z) = (F` z))
45	24	sseli 2617	. . . . . . . . . . . 12 |- (z e. X -> z e. CC)
46		fvex 4689	. . . . . . . . . . . . 13 |- (exp` (A x. z)) e. _V
47	38, 10, 46	fvopab4 4743	. . . . . . . . . . . 12 |- (z e. CC -> (F` z) = (exp` (A x. z)))
48	45, 47	syl 12	. . . . . . . . . . 11 |- (z e. X -> (F` z) = (exp` (A x. z)))
49	44, 48	eqtrd 1925	. . . . . . . . . 10 |- (z e. X -> ((F |` X)` z) = (exp` (A x. z)))
50	49	eqeq2d 1895	. . . . . . . . 9 |- (z e. X -> (w = ((F |` X)` z) <-> w = (exp` (A x. z))))
51	50	rexbiia 2134	. . . . . . . 8 |- (E.z e. X w = ((F |` X)` z) <-> E.z e. X w = (exp` (A x. z)))
52	51	abbii 2006	. . . . . . 7 |- {w | E.z e. X w = ((F |` X)` z)} = {w | E.z e. X w = (exp` (A x. z))}
53	42, 43, 52	3eqtr4i 1921	. . . . . 6 |- S = {w | E.z e. X w = ((F |` X)` z)}
54	33, 34, 53	3eqtr4ri 1923	. . . . 5 |- S = (F"X)
55		efghgrpi.2	. . . . 5 |- G = ( x. |` (S X. S))
56	2, 9, 16, 17, 20, 22, 27, 54, 55	ghsubgi 9446	. . . 4 |- (G e. Grp /\ (( + |` (X X. X)) e. Abel -> G e. Abel))
57	56	simpri 351	. . 3 |- (( + |` (X X. X)) e. Abel -> G e. Abel)
58	3, 57	syl 12	. 2 |- (( + e. Abel /\ ( + |` (X X. X)) e. (SubGrp` + )) -> G e. Abel)
59	1, 2, 58	mp2an 761	1 |- G e. Abel

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106   C_ wss 2593  {copab 3395   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988  "cima 3989   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384   + caddc 6389   x. cmul 6391  expce 8555  Grpcgr 9311  Abelcabl 9407  SubGrpcsubg 9423

This theorem is referenced by:  efghgrpi 10074

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  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-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-fac 8184  df-bc 8209  df-clim 8235  df-sum 8240  df-ef 8560  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-subg 9424

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