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Theorem ghmrn 18944
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.
StepHypRef Expression
1 eqid 2737 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2737 . . . 4 (Base‘𝑇) = (Base‘𝑇)
31, 2ghmf 18935 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
43frnd 6664 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ⊆ (Base‘𝑇))
53fdmd 6667 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 = (Base‘𝑆))
6 ghmgrp1 18933 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
71grpbn0 18705 . . . . 5 (𝑆 ∈ Grp → (Base‘𝑆) ≠ ∅)
86, 7syl 17 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (Base‘𝑆) ≠ ∅)
95, 8eqnetrd 3009 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 ≠ ∅)
10 dm0rn0 5871 . . . 4 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
1110necon3bii 2994 . . 3 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
129, 11sylib 217 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ≠ ∅)
13 eqid 2737 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
14 eqid 2737 . . . . . . . . . 10 (+g𝑇) = (+g𝑇)
151, 13, 14ghmlin 18936 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) = ((𝐹𝑐)(+g𝑇)(𝐹𝑎)))
163ffnd 6657 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn (Base‘𝑆))
17163ad2ant1 1133 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
181, 13grpcl 18682 . . . . . . . . . . 11 ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆))
196, 18syl3an1 1163 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆))
20 fnfvelrn 7019 . . . . . . . . . 10 ((𝐹 Fn (Base‘𝑆) ∧ (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) ∈ ran 𝐹)
2117, 19, 20syl2anc 585 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) ∈ ran 𝐹)
2215, 21eqeltrrd 2839 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹)
23223expia 1121 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝑎 ∈ (Base‘𝑆) → ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
2423ralrimiv 3139 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹)
25 oveq2 7350 . . . . . . . . . 10 (𝑏 = (𝐹𝑎) → ((𝐹𝑐)(+g𝑇)𝑏) = ((𝐹𝑐)(+g𝑇)(𝐹𝑎)))
2625eleq1d 2822 . . . . . . . . 9 (𝑏 = (𝐹𝑎) → (((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
2726ralrn 7025 . . . . . . . 8 (𝐹 Fn (Base‘𝑆) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
2816, 27syl 17 . . . . . . 7 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
2928adantr 482 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
3024, 29mpbird 257 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹)
31 eqid 2737 . . . . . . 7 (invg𝑆) = (invg𝑆)
32 eqid 2737 . . . . . . 7 (invg𝑇) = (invg𝑇)
331, 31, 32ghminv 18938 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) = ((invg𝑇)‘(𝐹𝑐)))
3416adantr 482 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
351, 31grpinvcl 18724 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑐) ∈ (Base‘𝑆))
366, 35sylan 581 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑐) ∈ (Base‘𝑆))
37 fnfvelrn 7019 . . . . . . 7 ((𝐹 Fn (Base‘𝑆) ∧ ((invg𝑆)‘𝑐) ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) ∈ ran 𝐹)
3834, 36, 37syl2anc 585 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) ∈ ran 𝐹)
3933, 38eqeltrrd 2839 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹)
4030, 39jca 513 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹))
4140ralrimiva 3140 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹))
42 oveq1 7349 . . . . . . . 8 (𝑎 = (𝐹𝑐) → (𝑎(+g𝑇)𝑏) = ((𝐹𝑐)(+g𝑇)𝑏))
4342eleq1d 2822 . . . . . . 7 (𝑎 = (𝐹𝑐) → ((𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹))
4443ralbidv 3171 . . . . . 6 (𝑎 = (𝐹𝑐) → (∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹))
45 fveq2 6830 . . . . . . 7 (𝑎 = (𝐹𝑐) → ((invg𝑇)‘𝑎) = ((invg𝑇)‘(𝐹𝑐)))
4645eleq1d 2822 . . . . . 6 (𝑎 = (𝐹𝑐) → (((invg𝑇)‘𝑎) ∈ ran 𝐹 ↔ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹))
4744, 46anbi12d 632 . . . . 5 (𝑎 = (𝐹𝑐) → ((∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹) ↔ (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹)))
4847ralrn 7025 . . . 4 (𝐹 Fn (Base‘𝑆) → (∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹) ↔ ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹)))
4916, 48syl 17 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹) ↔ ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹)))
5041, 49mpbird 257 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹))
51 ghmgrp2 18934 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
522, 14, 32issubg2 18867 . . 3 (𝑇 ∈ Grp → (ran 𝐹 ∈ (SubGrp‘𝑇) ↔ (ran 𝐹 ⊆ (Base‘𝑇) ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹))))
5351, 52syl 17 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (ran 𝐹 ∈ (SubGrp‘𝑇) ↔ (ran 𝐹 ⊆ (Base‘𝑇) ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑎 ∈ ran 𝐹(∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘𝑎) ∈ ran 𝐹))))
544, 12, 50, 53mpbir3and 1342 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1087   = wceq 1541  wcel 2106  wne 2941  wral 3062  wss 3902  c0 4274  dom cdm 5625  ran crn 5626   Fn wfn 6479  cfv 6484  (class class class)co 7342  Basecbs 17010  +gcplusg 17060  Grpcgrp 18674  invgcminusg 18675  SubGrpcsubg 18846   GrpHom cghm 18928
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-om 7786  df-2nd 7905  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-er 8574  df-en 8810  df-dom 8811  df-sdom 8812  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-nn 12080  df-2 12142  df-sets 16963  df-slot 16981  df-ndx 16993  df-base 17011  df-ress 17040  df-plusg 17073  df-0g 17250  df-mgm 18424  df-sgrp 18473  df-mnd 18484  df-grp 18677  df-minusg 18678  df-subg 18849  df-ghm 18929
This theorem is referenced by:  ghmghmrn  18950  ghmima  18952  cayley  19119
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