Theorem ghmnsgpreima 13475

Description: The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)

Assertion

Ref	Expression
ghmnsgpreima	⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (NrmSGrp‘𝑆))

Proof of Theorem ghmnsgpreima

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nsgsubg 13411	. . 3 ⊢ (𝑉 ∈ (NrmSGrp‘𝑇) → 𝑉 ∈ (SubGrp‘𝑇))
2		ghmpreima 13472	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))
3	1, 2	sylan2 286	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))
4		ghmgrp1 13451	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
5	4	ad2antrr 488	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑆 ∈ Grp)
6		simprl 529	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑥 ∈ (Base‘𝑆))
7		simprr 531	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑦 ∈ (◡𝐹 “ 𝑉))
8		simpll 527	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9		eqid 2196	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2196	. . . . . . . . . . . 12 ⊢ (Base‘𝑇) = (Base‘𝑇)
11	9, 10	ghmf 13453	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12	8, 11	syl 14	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
13	12	ffnd 5411	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹 Fn (Base‘𝑆))
14		elpreima 5684	. . . . . . . . 9 ⊢ (𝐹 Fn (Base‘𝑆) → (𝑦 ∈ (◡𝐹 “ 𝑉) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) ∈ 𝑉)))
15	13, 14	syl 14	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝑦 ∈ (◡𝐹 “ 𝑉) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) ∈ 𝑉)))
16	7, 15	mpbid 147	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) ∈ 𝑉))
17	16	simpld 112	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑦 ∈ (Base‘𝑆))
18		eqid 2196	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
19	9, 18	grpcl 13210	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
20	5, 6, 17, 19	syl3anc 1249	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
21		eqid 2196	. . . . . 6 ⊢ (-g‘𝑆) = (-g‘𝑆)
22	9, 21	grpsubcl 13282	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥) ∈ (Base‘𝑆))
23	5, 20, 6, 22	syl3anc 1249	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → ((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥) ∈ (Base‘𝑆))
24		eqid 2196	. . . . . . . 8 ⊢ (-g‘𝑇) = (-g‘𝑇)
25	9, 21, 24	ghmsub 13457	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥)) = ((𝐹‘(𝑥(+g‘𝑆)𝑦))(-g‘𝑇)(𝐹‘𝑥)))
26	8, 20, 6, 25	syl3anc 1249	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝐹‘((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥)) = ((𝐹‘(𝑥(+g‘𝑆)𝑦))(-g‘𝑇)(𝐹‘𝑥)))
27		eqid 2196	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
28	9, 18, 27	ghmlin 13454	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
29	8, 6, 17, 28	syl3anc 1249	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
30	29	oveq1d 5940	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → ((𝐹‘(𝑥(+g‘𝑆)𝑦))(-g‘𝑇)(𝐹‘𝑥)) = (((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))(-g‘𝑇)(𝐹‘𝑥)))
31	26, 30	eqtrd 2229	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝐹‘((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥)) = (((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))(-g‘𝑇)(𝐹‘𝑥)))
32		simplr 528	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑉 ∈ (NrmSGrp‘𝑇))
33	12, 6	ffvelcdmd 5701	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝐹‘𝑥) ∈ (Base‘𝑇))
34	16	simprd 114	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝐹‘𝑦) ∈ 𝑉)
35	10, 27, 24	nsgconj 13412	. . . . . 6 ⊢ ((𝑉 ∈ (NrmSGrp‘𝑇) ∧ (𝐹‘𝑥) ∈ (Base‘𝑇) ∧ (𝐹‘𝑦) ∈ 𝑉) → (((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))(-g‘𝑇)(𝐹‘𝑥)) ∈ 𝑉)
36	32, 33, 34, 35	syl3anc 1249	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))(-g‘𝑇)(𝐹‘𝑥)) ∈ 𝑉)
37	31, 36	eqeltrd 2273	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝐹‘((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥)) ∈ 𝑉)
38		elpreima 5684	. . . . 5 ⊢ (𝐹 Fn (Base‘𝑆) → (((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥) ∈ (◡𝐹 “ 𝑉) ↔ (((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥) ∈ (Base‘𝑆) ∧ (𝐹‘((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥)) ∈ 𝑉)))
39	13, 38	syl 14	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥) ∈ (◡𝐹 “ 𝑉) ↔ (((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥) ∈ (Base‘𝑆) ∧ (𝐹‘((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥)) ∈ 𝑉)))
40	23, 37, 39	mpbir2and 946	. . 3 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → ((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥) ∈ (◡𝐹 “ 𝑉))
41	40	ralrimivva 2579	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (◡𝐹 “ 𝑉)((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥) ∈ (◡𝐹 “ 𝑉))
42	9, 18, 21	isnsg3 13413	. 2 ⊢ ((◡𝐹 “ 𝑉) ∈ (NrmSGrp‘𝑆) ↔ ((◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (◡𝐹 “ 𝑉)((𝑥(+g‘𝑆)𝑦)(-g‘𝑆)𝑥) ∈ (◡𝐹 “ 𝑉)))
43	3, 41, 42	sylanbrc 417	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (NrmSGrp‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ^◡ccnv 4663   “ cima 4667   Fn wfn 5254  ⟶wf 5255  ‘cfv 5259  (class class class)co 5925  Basecbs 12703  +_gcplusg 12780  Grpcgrp 13202  -_gcsg 13204  SubGrpcsubg 13373  NrmSGrpcnsg 13374   GrpHom cghm 13446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-sbg 13207  df-subg 13376  df-nsg 13377  df-ghm 13447

This theorem is referenced by:  ghmker  13476