Theorem ghomgrplem 10384

Description: Lemma for ghomgrp 10385.

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 3374	. . 3 |- (H = if(ph, H, S) -> (H |` (ran F X. ran F)) = (if(ph, H, S) |` (ran F X. ran F)))
2		fveq2 3730	. . 3 |- (H = if(ph, H, S) -> (SubGrp` H) = (SubGrp` if(ph, H, S)))
3	1, 2	eleq12d 1545	. 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 3345	. . . 4 |- (F = if(ph, F, J) -> ran F = ran if(ph, F, J))
5		xpeq1 3206	. . . . . 6 |- (ran F = ran if(ph, F, J) -> (ran F X. ran F) = (ran if(ph, F, J) X. ran F))
6		xpeq2 3207	. . . . . 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 1510	. . . . 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 3375	. . . . 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 10	. . . 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 10	. . 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 1543	. 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	3simp1d 796	. . . 4 |- (ph -> G e. Grp)
14		eleq1 1537	. . . 4 |- (G = if(ph, G, S) -> (G e. Grp <-> if(ph, G, S) e. Grp))
15		eleq1 1537	. . . 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 1816	. . . . . 6 |- z e. V
18	17	grpsn 8120	. . . . 5 |- {<.<.z, z>., z>.} e. Grp
19	16, 18	eqeltr 1547	. . . 4 |- S e. Grp
20	13, 14, 15, 19	elimdhyp 2399	. . 3 |- if(ph, G, S) e. Grp
21	12	3simp2d 797	. . . 4 |- (ph -> H e. Grp)
22		eleq1 1537	. . . 4 |- (H = if(ph, H, S) -> (H e. Grp <-> if(ph, H, S) e. Grp))
23		eleq1 1537	. . . 4 |- (S = if(ph, H, S) -> (S e. Grp <-> if(ph, H, S) e. Grp))
24	21, 22, 23, 19	elimdhyp 2399	. . 3 |- if(ph, H, S) e. Grp
25	12	3simp3d 798	. . . 4 |- (ph -> F e. (G GrpHom H))
26		ghomgrplem.3	. . . . 5 |- J = (I |` {z})
27	17, 16	ghomsn 10383	. . . . 5 |- (I |` {z}) e. (S GrpHom S)
28	26, 27	eqeltr 1547	. . . 4 |- J e. (S GrpHom S)
29	25, 28	elimdeloprv 4007	. . 3 |- if(ph, F, J) e. (if(ph, G, S) GrpHom if(ph, H, S))
30		eqid 1478	. . 3 |- ran if(ph, F, J) = ran if(ph, F, J)
31		eqid 1478	. . 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 10382	. 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 2388	1 |- (ph -> (H |` (ran F X. ran F)) e. (SubGrp` H))

Syntax hints:   -> wi 3   /\ w3a 777   = wceq 958   e. wcel 960  ifcif 2365  {csn 2413  <.cop 2415  Icid 2837   X. cxp 3174  ran crn 3177   |` cres 3178  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  SubGrpcsubg 8110   GrpHom cghom 10373

This theorem is referenced by:  ghomgrp 10385

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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-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-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  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-opr 3971  df-oprab 3972  df-grp 8034  df-gid 8035  df-ginv 8036  df-subg 8111  df-ghom 10375

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