Theorem ghmnsgpreima 19216

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 19133	. . 3 ⊢ (𝑉 ∈ (NrmSGrp‘𝑇) → 𝑉 ∈ (SubGrp‘𝑇))
2		ghmpreima 19213	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))
3	1, 2	sylan2 594	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))
4		ghmgrp1 19193	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
5	4	ad2antrr 727	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑆 ∈ Grp)
6		simprl 771	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑥 ∈ (Base‘𝑆))
7		simprr 773	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑦 ∈ (◡𝐹 “ 𝑉))
8		simpll 767	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝑇) = (Base‘𝑇)
11	9, 10	ghmf 19195	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12	8, 11	syl 17	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
13	12	ffnd 6669	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹 Fn (Base‘𝑆))
14		elpreima 7010	. . . . . . . . 9 ⊢ (𝐹 Fn (Base‘𝑆) → (𝑦 ∈ (◡𝐹 “ 𝑉) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) ∈ 𝑉)))
15	13, 14	syl 17	. . . . . . . 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 232	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) ∈ 𝑉))
17	16	simpld 494	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑦 ∈ (Base‘𝑆))
18		eqid 2736	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
19	9, 18	grpcl 18917	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
20	5, 6, 17, 19	syl3anc 1374	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
21		eqid 2736	. . . . . 6 ⊢ (-g‘𝑆) = (-g‘𝑆)
22	9, 21	grpsubcl 18996	. . . . 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 1374	. . . 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 2736	. . . . . . . 8 ⊢ (-g‘𝑇) = (-g‘𝑇)
25	9, 21, 24	ghmsub 19199	. . . . . . 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 1374	. . . . . 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 2736	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
28	9, 18, 27	ghmlin 19196	. . . . . . . 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 1374	. . . . . . 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 7382	. . . . . 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 2771	. . . . 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 769	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑉 ∈ (NrmSGrp‘𝑇))
33	12, 6	ffvelcdmd 7037	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝐹‘𝑥) ∈ (Base‘𝑇))
34	16	simprd 495	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝐹‘𝑦) ∈ 𝑉)
35	10, 27, 24	nsgconj 19134	. . . . . 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 1374	. . . . 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 2836	. . . 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 7010	. . . . 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 17	. . . 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 714	. . 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 3180	. 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 19135	. 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 584	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (NrmSGrp‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ^◡ccnv 5630   “ cima 5634   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  Grpcgrp 18909  -_gcsg 18911  SubGrpcsubg 19096  NrmSGrpcnsg 19097   GrpHom cghm 19187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-sbg 18914  df-subg 19099  df-nsg 19100  df-ghm 19188

This theorem is referenced by:  ghmker  19217