Theorem ghomgrplem 13632

Description: Lemma for ghomgrp 13633.

Hypotheses

Ref	Expression
ghomgrplem.1	|- (ph -> (G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)))
ghomgrplem.2	|- S = {<.<.z, z>., z>.}
ghomgrplem.3	|- J = ( _I |` {z})

Assertion

Ref	Expression
ghomgrplem	|- (ph -> (H |` (ran F X. ran F)) e. (SubGrp` H))

Proof of Theorem ghomgrplem

Step	Hyp	Ref	Expression
1		reseq1 4218	. . 3 |- (H = if(ph, H, S) -> (H |` (ran F X. ran F)) = (if(ph, H, S) |` (ran F X. ran F)))
2		fveq2 4681	. . 3 |- (H = if(ph, H, S) -> (SubGrp` H) = (SubGrp` if(ph, H, S)))
3	1, 2	eleq12d 1965	. 2 |- (H = if(ph, H, S) -> ((H |` (ran F X. ran F)) e. (SubGrp` H) <-> (if(ph, H, S) |` (ran F X. ran F)) e. (SubGrp` if(ph, H, S))))
4		rneq 4186	. . . 4 |- (F = if(ph, F, J) -> ran F = ran if(ph, F, J))
5		xpeq1 4016	. . . . . 6 |- (ran F = ran if(ph, F, J) -> (ran F X. ran F) = (ran if(ph, F, J) X. ran F))
6		xpeq2 4017	. . . . . 6 |- (ran F = ran if(ph, F, J) -> (ran if(ph, F, J) X. ran F) = (ran if(ph, F, J) X. ran if(ph, F, J)))
7	5, 6	eqtrd 1925	. . . . 5 |- (ran F = ran if(ph, F, J) -> (ran F X. ran F) = (ran if(ph, F, J) X. ran if(ph, F, J)))
8		reseq2 4219	. . . . 5 |- ((ran F X. ran F) = (ran if(ph, F, J) X. ran if(ph, F, J)) -> (if(ph, H, S) |` (ran F X. ran F)) = (if(ph, H, S) |` (ran if(ph, F, J) X. ran if(ph, F, J))))
9	7, 8	syl 12	. . . 4 |- (ran F = ran if(ph, F, J) -> (if(ph, H, S) |` (ran F X. ran F)) = (if(ph, H, S) |` (ran if(ph, F, J) X. ran if(ph, F, J))))
10	4, 9	syl 12	. . 3 |- (F = if(ph, F, J) -> (if(ph, H, S) |` (ran F X. ran F)) = (if(ph, H, S) |` (ran if(ph, F, J) X. ran if(ph, F, J))))
11	10	eleq1d 1963	. 2 |- (F = if(ph, F, J) -> ((if(ph, H, S) |` (ran F X. ran F)) e. (SubGrp` if(ph, H, S)) <-> (if(ph, H, S) |` (ran if(ph, F, J) X. ran if(ph, F, J))) e. (SubGrp` if(ph, H, S))))
12		ghomgrplem.1	. . . . 5 |- (ph -> (G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)))
13	12	simp1d 888	. . . 4 |- (ph -> G e. Grp)
14		eleq1 1957	. . . 4 |- (G = if(ph, G, S) -> (G e. Grp <-> if(ph, G, S) e. Grp))
15		eleq1 1957	. . . 4 |- (S = if(ph, G, S) -> (S e. Grp <-> if(ph, G, S) e. Grp))
16		ghomgrplem.2	. . . . 5 |- S = {<.<.z, z>., z>.}
17		visset 2295	. . . . . 6 |- z e. _V
18	17	grpsn 9340	. . . . 5 |- {<.<.z, z>., z>.} e. Grp
19	16, 18	eqeltri 1967	. . . 4 |- S e. Grp
20	13, 14, 15, 19	elimdhyp 3026	. . 3 |- if(ph, G, S) e. Grp
21	12	simp2d 889	. . . 4 |- (ph -> H e. Grp)
22		eleq1 1957	. . . 4 |- (H = if(ph, H, S) -> (H e. Grp <-> if(ph, H, S) e. Grp))
23		eleq1 1957	. . . 4 |- (S = if(ph, H, S) -> (S e. Grp <-> if(ph, H, S) e. Grp))
24	21, 22, 23, 19	elimdhyp 3026	. . 3 |- if(ph, H, S) e. Grp
25	12	simp3d 890	. . . 4 |- (ph -> F e. (G GrpHom H))
26		ghomgrplem.3	. . . . 5 |- J = ( _I |` {z})
27	17, 16	ghomsn 13631	. . . . 5 |- ( _I |` {z}) e. (S GrpHom S)
28	26, 27	eqeltri 1967	. . . 4 |- J e. (S GrpHom S)
29	25, 28	elimdeloprv 4930	. . 3 |- if(ph, F, J) e. (if(ph, G, S) GrpHom if(ph, H, S))
30		eqid 1884	. . 3 |- ran if(ph, F, J) = ran if(ph, F, J)
31		eqid 1884	. . 3 |- (if(ph, H, S) |` (ran if(ph, F, J) X. ran if(ph, F, J))) = (if(ph, H, S) |` (ran if(ph, F, J) X. ran if(ph, F, J)))
32	20, 24, 29, 30, 31	ghomgrpi 13630	. 2 |- (if(ph, H, S) |` (ran if(ph, F, J) X. ran if(ph, F, J))) e. (SubGrp` if(ph, H, S))
33	3, 11, 32	dedth2v 3018	1 |- (ph -> (H |` (ran F X. ran F)) e. (SubGrp` H))

Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300  ifcif 2982  {csn 3044  <.cop 3046   _I cid 3582   X. cxp 3984  ran crn 3987   |` cres 3988  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  SubGrpcsubg 9423   GrpHom cghom 10189

This theorem is referenced by:  ghomgrp 13633

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-rep 3428  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-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-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-nul 2876  df-if 2983  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-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-grp 9316  df-gid 9317  df-ginv 9318  df-subg 9424  df-ghom 10190

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