Theorem ghmnsgpreima 14055

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 13991	. . 3 ⊢ (𝑉 ∈ (NrmSGrp‘𝑇) → 𝑉 ∈ (SubGrp‘𝑇))
2		ghmpreima 14052	. . 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 14031	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
5	4	ad2antrr 492	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑆 ∈ Grp)
6		simprl 535	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑥 ∈ (Base‘𝑆))
7		simprr 537	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑦 ∈ (◡𝐹 “ 𝑉))
8		simpll 531	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9		eqid 2238	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2238	. . . . . . . . . . . 12 ⊢ (Base‘𝑇) = (Base‘𝑇)
11	9, 10	ghmf 14033	. . . . . . . . . . 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 5532	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹 Fn (Base‘𝑆))
14		elpreima 5822	. . . . . . . . 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 2238	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
19	9, 18	grpcl 13796	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
20	5, 6, 17, 19	syl3anc 1278	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
21		eqid 2238	. . . . . 6 ⊢ (-g‘𝑆) = (-g‘𝑆)
22	9, 21	grpsubcl 13868	. . . . 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 1278	. . . 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 2238	. . . . . . . 8 ⊢ (-g‘𝑇) = (-g‘𝑇)
25	9, 21, 24	ghmsub 14037	. . . . . . 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 1278	. . . . . 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 2238	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
28	9, 18, 27	ghmlin 14034	. . . . . . . 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 1278	. . . . . . 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 6094	. . . . . 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 2271	. . . . 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 533	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑉 ∈ (NrmSGrp‘𝑇))
33	12, 6	ffvelcdmd 5838	. . . . . 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 13992	. . . . . 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 1278	. . . . 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 2315	. . . 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 5822	. . . . 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 957	. . 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 2632	. 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 13993	. 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 421	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (NrmSGrp‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ^◡ccnv 4771   “ cima 4775   Fn wfn 5370  ⟶wf 5371  ‘cfv 5375  (class class class)co 6079  Basecbs 13335  +_gcplusg 13414  Grpcgrp 13788  -_gcsg 13790  SubGrpcsubg 13953  NrmSGrpcnsg 13954   GrpHom cghm 14026

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-sbg 13793  df-subg 13956  df-nsg 13957  df-ghm 14027

This theorem is referenced by:  ghmker  14056