Theorem ghmrn 19260

Description: The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Assertion

Ref	Expression
ghmrn	⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))

Proof of Theorem ghmrn

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2761	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	ghmf 19251	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4	3	frnd 6695	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ⊆ (Base‘𝑇))
5	3	fdmd 6697	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 = (Base‘𝑆))
6		ghmgrp1 19249	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
7	1	grpbn0 18999	. . . . 5 ⊢ (𝑆 ∈ Grp → (Base‘𝑆) ≠ ∅)
8	6, 7	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (Base‘𝑆) ≠ ∅)
9	5, 8	eqnetrd 3023	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 ≠ ∅)
10		dm0rn0 5896	. . . 4 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
11	10	necon3bii 3008	. . 3 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
12	9, 11	sylib 220	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ≠ ∅)
13		eqid 2761	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
14		eqid 2761	. . . . . . . . . 10 ⊢ (+g‘𝑇) = (+g‘𝑇)
15	1, 13, 14	ghmlin 19252	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) = ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)))
16	3	ffnd 6687	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn (Base‘𝑆))
17	16	3ad2ant1 1145	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
18	1, 13	grpcl 18974	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆))
19	6, 18	syl3an1 1175	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆))
20		fnfvelrn 7056	. . . . . . . . . 10 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ (𝑐(+g‘𝑆)𝑎) ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) ∈ ran 𝐹)
21	17, 19, 20	syl2anc 593	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g‘𝑆)𝑎)) ∈ ran 𝐹)
22	15, 21	eqeltrrd 2862	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹)
23	22	3expia 1133	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝑎 ∈ (Base‘𝑆) → ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
24	23	ralrimiv 3152	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (Base‘𝑆)((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹)
25		oveq2 7399	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑎) → ((𝐹‘𝑐)(+g‘𝑇)𝑏) = ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)))
26	25	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑎) → (((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
27	26	ralrn 7064	. . . . . . . 8 ⊢ (𝐹 Fn (Base‘𝑆) → (∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
28	16, 27	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
29	28	adantr 484	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹‘𝑐)(+g‘𝑇)(𝐹‘𝑎)) ∈ ran 𝐹))
30	24, 29	mpbird 259	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹)
31		eqid 2761	. . . . . . 7 ⊢ (invg‘𝑆) = (invg‘𝑆)
32		eqid 2761	. . . . . . 7 ⊢ (invg‘𝑇) = (invg‘𝑇)
33	1, 31, 32	ghminv 19254	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) = ((invg‘𝑇)‘(𝐹‘𝑐)))
34	16	adantr 484	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
35	1, 31	grpinvcl 19020	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆))
36	6, 35	sylan 589	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆))
37		fnfvelrn 7056	. . . . . . 7 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ ((invg‘𝑆)‘𝑐) ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) ∈ ran 𝐹)
38	34, 36, 37	syl2anc 593	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg‘𝑆)‘𝑐)) ∈ ran 𝐹)
39	33, 38	eqeltrrd 2862	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹)
40	30, 39	jca 519	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹))
41	40	ralrimiva 3153	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹))
42		oveq1 7398	. . . . . . . 8 ⊢ (𝑎 = (𝐹‘𝑐) → (𝑎(+g‘𝑇)𝑏) = ((𝐹‘𝑐)(+g‘𝑇)𝑏))
43	42	eleq1d 2846	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑐) → ((𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹))
44	43	ralbidv 3184	. . . . . 6 ⊢ (𝑎 = (𝐹‘𝑐) → (∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹))
45		fveq2 6862	. . . . . . 7 ⊢ (𝑎 = (𝐹‘𝑐) → ((invg‘𝑇)‘𝑎) = ((invg‘𝑇)‘(𝐹‘𝑐)))
46	45	eleq1d 2846	. . . . . 6 ⊢ (𝑎 = (𝐹‘𝑐) → (((invg‘𝑇)‘𝑎) ∈ ran 𝐹 ↔ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹))
47	44, 46	anbi12d 641	. . . . 5 ⊢ (𝑎 = (𝐹‘𝑐) → ((∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹) ↔ (∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹)))
48	47	ralrn 7064	. . . 4 ⊢ (𝐹 Fn (Base‘𝑆) → (∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹) ↔ ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹)))
49	16, 48	syl 17	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹) ↔ ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹‘𝑐)(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘(𝐹‘𝑐)) ∈ ran 𝐹)))
50	41, 49	mpbird 259	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹))
51		ghmgrp2 19250	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
52	2, 14, 32	issubg2 19174	. . 3 ⊢ (𝑇 ∈ Grp → (ran 𝐹 ∈ (SubGrp‘𝑇) ↔ (ran 𝐹 ⊆ (Base‘𝑇) ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹))))
53	51, 52	syl 17	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (ran 𝐹 ∈ (SubGrp‘𝑇) ↔ (ran 𝐹 ⊆ (Base‘𝑇) ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g‘𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg‘𝑇)‘𝑎) ∈ ran 𝐹))))
54	4, 12, 50, 53	mpbir3and 1355	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075   ⊆ wss 3902  ∅c0 4283  dom cdm 5643  ran crn 5644   Fn wfn 6511  ‘cfv 6516  (class class class)co 7391  Basecbs 17236  +_gcplusg 17277  Grpcgrp 18966  inv_gcminusg 18967  SubGrpcsubg 19153   GrpHom cghm 19244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-er 8672  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-sets 17191  df-slot 17209  df-ndx 17221  df-base 17237  df-ress 17258  df-plusg 17290  df-0g 17461  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18969  df-minusg 18970  df-subg 19156  df-ghm 19245

This theorem is referenced by:  ghmghmrn  19266  ghmima  19268  ghmqusnsg  19313  ghmquskerlem3  19317  cayley  19445