Theorem ghmnsgpreima 13792

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 13728	. . 3 ⊢ (𝑉 ∈ (NrmSGrp‘𝑇) → 𝑉 ∈ (SubGrp‘𝑇))
2		ghmpreima 13789	. . 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 13768	. . . . . 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 2229	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2229	. . . . . . . . . . . 12 ⊢ (Base‘𝑇) = (Base‘𝑇)
11	9, 10	ghmf 13770	. . . . . . . . . . 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 5470	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹 Fn (Base‘𝑆))
14		elpreima 5747	. . . . . . . . 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 2229	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
19	9, 18	grpcl 13527	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
20	5, 6, 17, 19	syl3anc 1271	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
21		eqid 2229	. . . . . 6 ⊢ (-g‘𝑆) = (-g‘𝑆)
22	9, 21	grpsubcl 13599	. . . . 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 1271	. . . 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 2229	. . . . . . . 8 ⊢ (-g‘𝑇) = (-g‘𝑇)
25	9, 21, 24	ghmsub 13774	. . . . . . 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 1271	. . . . . 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 2229	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
28	9, 18, 27	ghmlin 13771	. . . . . . . 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 1271	. . . . . . 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 6009	. . . . . 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 2262	. . . . 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 5764	. . . . . 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 13729	. . . . . 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 1271	. . . . 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 2306	. . . 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 5747	. . . . 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 950	. . 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 2612	. 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 13730	. 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 1395   ∈ wcel 2200  ∀wral 2508  ^◡ccnv 4715   “ cima 4719   Fn wfn 5309  ⟶wf 5310  ‘cfv 5314  (class class class)co 5994  Basecbs 13018  +_gcplusg 13096  Grpcgrp 13519  -_gcsg 13521  SubGrpcsubg 13690  NrmSGrpcnsg 13691   GrpHom cghm 13763

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-pre-ltirr 8099  ax-pre-ltadd 8103

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-pnf 8171  df-mnf 8172  df-ltxr 8174  df-inn 9099  df-2 9157  df-ndx 13021  df-slot 13022  df-base 13024  df-sets 13025  df-iress 13026  df-plusg 13109  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-grp 13522  df-minusg 13523  df-sbg 13524  df-subg 13693  df-nsg 13694  df-ghm 13764

This theorem is referenced by:  ghmker  13793