Theorem ghomgrplem 10323

Description: Lemma for ghomgrp 10324.

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 3360	. . 3 |- (H = if(ph, H, S) -> (H |` (ran F X. ran F)) = (if(ph, H, S) |` (ran F X. ran F)))
2		fveq2 3715	. . 3 |- (H = if(ph, H, S) -> (SubGrp` H) = (SubGrp` if(ph, H, S)))
3	1, 2	eleq12d 1539	. 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 3334	. . . 4 |- (F = if(ph, F, J) -> ran F = ran if(ph, F, J))
5		xpeq1 3195	. . . . . 6 |- (ran F = ran if(ph, F, J) -> (ran F X. ran F) = (ran if(ph, F, J) X. ran F))
6		xpeq2 3196	. . . . . 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 1504	. . . . 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 3361	. . . . 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 1537	. 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 793	. . . 4 |- (ph -> G e. Grp)
14		eleq1 1531	. . . 4 |- (G = if(ph, G, S) -> (G e. Grp <-> if(ph, G, S) e. Grp))
15		eleq1 1531	. . . 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 1809	. . . . . 6 |- z e. V
18	17	grpsn 8076	. . . . 5 |- {<.<.z, z>., z>.} e. Grp
19	16, 18	eqeltr 1541	. . . 4 |- S e. Grp
20	13, 14, 15, 19	elimdhyp 2391	. . 3 |- if(ph, G, S) e. Grp
21	12	3simp2d 794	. . . 4 |- (ph -> H e. Grp)
22		eleq1 1531	. . . 4 |- (H = if(ph, H, S) -> (H e. Grp <-> if(ph, H, S) e. Grp))
23		eleq1 1531	. . . 4 |- (S = if(ph, H, S) -> (S e. Grp <-> if(ph, H, S) e. Grp))
24	21, 22, 23, 19	elimdhyp 2391	. . 3 |- if(ph, H, S) e. Grp
25	12	3simp3d 795	. . . 4 |- (ph -> F e. (G GrpHom H))
26		ghomgrplem.3	. . . . 5 |- J = (I |` {z})
27	17, 16	ghomsn 10322	. . . . 5 |- (I |` {z}) e. (S GrpHom S)
28	26, 27	eqeltr 1541	. . . 4 |- J e. (S GrpHom S)
29	25, 28	elimdeloprv 3992	. . 3 |- if(ph, F, J) e. (if(ph, G, S) GrpHom if(ph, H, S))
30		eqid 1473	. . 3 |- ran if(ph, F, J) = ran if(ph, F, J)
31		eqid 1473	. . 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 10321	. 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 2380	1 |- (ph -> (H |` (ran F X. ran F)) e. (SubGrp` H))

Syntax hints:   -> wi 3   /\ w3a 774   = wceq 954   e. wcel 956  ifcif 2357  {csn 2405  <.cop 2407  Icid 2826   X. cxp 3163  ran crn 3166   |` cres 3167  ` cfv 3177  (class class class)co 3954  Grpcgr 7983  SubGrpcsubg 8066   GrpHom cghom 10312

This theorem is referenced by:  ghomgrp 10324

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-opr 3956  df-oprab 3957  df-grp 7987  df-gid 7988  df-ginv 7989  df-subg 8067  df-ghom 10314

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