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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.
StepHypRef Expression
1 eqid 2735 . . . 4 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2735 . . . 4 (Base‘𝑇) = (Base‘𝑇)
31, 2ghmf 19251 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
43frnd 6745 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ⊆ (Base‘𝑇))
53fdmd 6747 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 = (Base‘𝑆))
6 ghmgrp1 19249 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
71grpbn0 18997 . . . . 5 (𝑆 ∈ Grp → (Base‘𝑆) ≠ ∅)
86, 7syl 17 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (Base‘𝑆) ≠ ∅)
95, 8eqnetrd 3006 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → dom 𝐹 ≠ ∅)
10 dm0rn0 5938 . . . 4 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
1110necon3bii 2991 . . 3 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
129, 11sylib 218 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ≠ ∅)
13 eqid 2735 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
14 eqid 2735 . . . . . . . . . 10 (+g𝑇) = (+g𝑇)
151, 13, 14ghmlin 19252 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) = ((𝐹𝑐)(+g𝑇)(𝐹𝑎)))
163ffnd 6738 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 Fn (Base‘𝑆))
17163ad2ant1 1132 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
181, 13grpcl 18972 . . . . . . . . . . 11 ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆))
196, 18syl3an1 1162 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆))
20 fnfvelrn 7100 . . . . . . . . . 10 ((𝐹 Fn (Base‘𝑆) ∧ (𝑐(+g𝑆)𝑎) ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) ∈ ran 𝐹)
2117, 19, 20syl2anc 584 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → (𝐹‘(𝑐(+g𝑆)𝑎)) ∈ ran 𝐹)
2215, 21eqeltrrd 2840 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆) ∧ 𝑎 ∈ (Base‘𝑆)) → ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹)
23223expia 1120 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝑎 ∈ (Base‘𝑆) → ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
2423ralrimiv 3143 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (Base‘𝑆)((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹)
25 oveq2 7439 . . . . . . . . . 10 (𝑏 = (𝐹𝑎) → ((𝐹𝑐)(+g𝑇)𝑏) = ((𝐹𝑐)(+g𝑇)(𝐹𝑎)))
2625eleq1d 2824 . . . . . . . . 9 (𝑏 = (𝐹𝑎) → (((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹𝑐)(+g𝑇)(𝐹𝑎)) ∈ ran 𝐹))
2726ralrn 7108 . . . . . . . 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 480 . . . . . 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 2735 . . . . . . 7 (invg𝑆) = (invg𝑆)
32 eqid 2735 . . . . . . 7 (invg𝑇) = (invg𝑇)
331, 31, 32ghminv 19254 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) = ((invg𝑇)‘(𝐹𝑐)))
3416adantr 480 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → 𝐹 Fn (Base‘𝑆))
351, 31grpinvcl 19018 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑐) ∈ (Base‘𝑆))
366, 35sylan 580 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑆)‘𝑐) ∈ (Base‘𝑆))
37 fnfvelrn 7100 . . . . . . 7 ((𝐹 Fn (Base‘𝑆) ∧ ((invg𝑆)‘𝑐) ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) ∈ ran 𝐹)
3834, 36, 37syl2anc 584 . . . . . 6 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘((invg𝑆)‘𝑐)) ∈ ran 𝐹)
3933, 38eqeltrrd 2840 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹)
4030, 39jca 511 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑐 ∈ (Base‘𝑆)) → (∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹))
4140ralrimiva 3144 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑐 ∈ (Base‘𝑆)(∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹 ∧ ((invg𝑇)‘(𝐹𝑐)) ∈ ran 𝐹))
42 oveq1 7438 . . . . . . . 8 (𝑎 = (𝐹𝑐) → (𝑎(+g𝑇)𝑏) = ((𝐹𝑐)(+g𝑇)𝑏))
4342eleq1d 2824 . . . . . . 7 (𝑎 = (𝐹𝑐) → ((𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹))
4443ralbidv 3176 . . . . . 6 (𝑎 = (𝐹𝑐) → (∀𝑏 ∈ ran 𝐹(𝑎(+g𝑇)𝑏) ∈ ran 𝐹 ↔ ∀𝑏 ∈ ran 𝐹((𝐹𝑐)(+g𝑇)𝑏) ∈ ran 𝐹))
45 fveq2 6907 . . . . . . 7 (𝑎 = (𝐹𝑐) → ((invg𝑇)‘𝑎) = ((invg𝑇)‘(𝐹𝑐)))
4645eleq1d 2824 . . . . . 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 7108 . . . 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 19250 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
522, 14, 32issubg2 19172 . . 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 1341 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ran 𝐹 ∈ (SubGrp‘𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wss 3963  c0 4339  dom cdm 5689  ran crn 5690   Fn wfn 6558  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  invgcminusg 18965  SubGrpcsubg 19151   GrpHom cghm 19243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-ghm 19244
This theorem is referenced by:  ghmghmrn  19266  ghmima  19268  ghmqusnsg  19313  ghmquskerlem3  19317  cayley  19447
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