Theorem ghmnsgpreima 19464

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 19377	. . 3 ⊢ (𝑉 ∈ (NrmSGrp‘𝑇) → 𝑉 ∈ (SubGrp‘𝑇))
2		ghmpreima 19461	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))
3	1, 2	sylan2 605	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (SubGrp‘𝑆))
4		ghmgrp1 19441	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
5	4	ad2antrr 739	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑆 ∈ Grp)
6		simprl 783	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑥 ∈ (Base‘𝑆))
7		simprr 785	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑦 ∈ (◡𝐹 “ 𝑉))
8		simpll 779	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝑇) = (Base‘𝑇)
11	9, 10	ghmf 19443	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12	8, 11	syl 18	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
13	12	ffnd 6715	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝐹 Fn (Base‘𝑆))
14		elpreima 7062	. . . . . . . . 9 ⊢ (𝐹 Fn (Base‘𝑆) → (𝑦 ∈ (◡𝐹 “ 𝑉) ↔ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) ∈ 𝑉)))
15	13, 14	syl 18	. . . . . . . 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 235	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) ∈ 𝑉))
17	16	simpld 500	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑦 ∈ (Base‘𝑆))
18		eqid 2760	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
19	9, 18	grpcl 19161	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
20	5, 6, 17, 19	syl3anc 1398	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
21		eqid 2760	. . . . . 6 ⊢ (-g‘𝑆) = (-g‘𝑆)
22	9, 21	grpsubcl 19239	. . . . 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 1398	. . . 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 2760	. . . . . . . 8 ⊢ (-g‘𝑇) = (-g‘𝑇)
25	9, 21, 24	ghmsub 19447	. . . . . . 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 1398	. . . . . 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 2760	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
28	9, 18, 27	ghmlin 19444	. . . . . . . 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 1398	. . . . . . 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 7439	. . . . . 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 2795	. . . . 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 781	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → 𝑉 ∈ (NrmSGrp‘𝑇))
33	12, 6	ffvelcdmd 7090	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝐹‘𝑥) ∈ (Base‘𝑇))
34	16	simprd 501	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (◡𝐹 “ 𝑉))) → (𝐹‘𝑦) ∈ 𝑉)
35	10, 27, 24	nsgconj 19378	. . . . . 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 1398	. . . . 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 2860	. . . 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 7062	. . . . 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 18	. . . 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 726	. . 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 3205	. 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 19379	. 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 595	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉 ∈ (NrmSGrp‘𝑇)) → (◡𝐹 “ 𝑉) ∈ (NrmSGrp‘𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ^◡ccnv 5650   “ cima 5654   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  +_gcplusg 17437  Grpcgrp 19153  -_gcsg 19155  SubGrpcsubg 19339  NrmSGrpcnsg 19340   GrpHom cghm 19436

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-er 8724  df-map 8856  df-en 8981  df-dom 8982  df-sdom 8983  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-sets 17351  df-slot 17369  df-ndx 17381  df-base 17397  df-ress 17418  df-plusg 17450  df-0g 17621  df-mgm 18825  df-sgrp 18917  df-mnd 18933  df-grp 19156  df-minusg 19157  df-sbg 19158  df-subg 19342  df-nsg 19343  df-ghm 19437

This theorem is used by:  ghmker  19465