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Theorem idghm 18305
Description: The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypothesis
Ref Expression
idghm.b 𝐵 = (Base‘𝐺)
Assertion
Ref Expression
idghm (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))

Proof of Theorem idghm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . 2 (𝐺 ∈ Grp → 𝐺 ∈ Grp)
2 idghm.b . . . . . . . 8 𝐵 = (Base‘𝐺)
3 eqid 2825 . . . . . . . 8 (+g𝐺) = (+g𝐺)
42, 3grpcl 18043 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝐺)𝑏) ∈ 𝐵)
543expb 1114 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝐺)𝑏) ∈ 𝐵)
6 fvresi 6930 . . . . . 6 ((𝑎(+g𝐺)𝑏) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = (𝑎(+g𝐺)𝑏))
75, 6syl 17 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = (𝑎(+g𝐺)𝑏))
8 fvresi 6930 . . . . . . 7 (𝑎𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎)
9 fvresi 6930 . . . . . . 7 (𝑏𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏)
108, 9oveqan12d 7170 . . . . . 6 ((𝑎𝐵𝑏𝐵) → ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g𝐺)𝑏))
1110adantl 482 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g𝐺)𝑏))
127, 11eqtr4d 2863 . . . 4 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)))
1312ralrimivva 3195 . . 3 (𝐺 ∈ Grp → ∀𝑎𝐵𝑏𝐵 (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)))
14 f1oi 6648 . . . 4 ( I ↾ 𝐵):𝐵1-1-onto𝐵
15 f1of 6611 . . . 4 (( I ↾ 𝐵):𝐵1-1-onto𝐵 → ( I ↾ 𝐵):𝐵𝐵)
1614, 15ax-mp 5 . . 3 ( I ↾ 𝐵):𝐵𝐵
1713, 16jctil 520 . 2 (𝐺 ∈ Grp → (( I ↾ 𝐵):𝐵𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏))))
182, 2, 3, 3isghm 18290 . 2 (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ Grp) ∧ (( I ↾ 𝐵):𝐵𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)))))
191, 1, 17, 18syl21anbrc 1338 1 (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3142   I cid 5457  cres 5555  wf 6347  1-1-ontowf1o 6350  cfv 6351  (class class class)co 7151  Basecbs 16475  +gcplusg 16557  Grpcgrp 18035   GrpHom cghm 18287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-grp 18038  df-ghm 18288
This theorem is referenced by:  gicref  18343  symgga  18456  0frgp  18827  idrhm  19405  idlmhm  19735  frgpcyg  20638  nmoid  23268  idnghm  23269  idrnghm  44013
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