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Theorem idghm 19250
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 2736 . . . . . . . 8 (+g𝐺) = (+g𝐺)
42, 3grpcl 18960 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝐺)𝑏) ∈ 𝐵)
543expb 1120 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝐺)𝑏) ∈ 𝐵)
6 fvresi 7194 . . . . . 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 7194 . . . . . . 7 (𝑎𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎)
9 fvresi 7194 . . . . . . 7 (𝑏𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏)
108, 9oveqan12d 7451 . . . . . 6 ((𝑎𝐵𝑏𝐵) → ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g𝐺)𝑏))
1110adantl 481 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g𝐺)𝑏))
127, 11eqtr4d 2779 . . . 4 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)))
1312ralrimivva 3201 . . 3 (𝐺 ∈ Grp → ∀𝑎𝐵𝑏𝐵 (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)))
14 f1oi 6885 . . . 4 ( I ↾ 𝐵):𝐵1-1-onto𝐵
15 f1of 6847 . . . 4 (( I ↾ 𝐵):𝐵1-1-onto𝐵 → ( I ↾ 𝐵):𝐵𝐵)
1614, 15ax-mp 5 . . 3 ( I ↾ 𝐵):𝐵𝐵
1713, 16jctil 519 . 2 (𝐺 ∈ Grp → (( I ↾ 𝐵):𝐵𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏))))
182, 2, 3, 3isghm 19234 . 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 1344 1 (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060   I cid 5576  cres 5686  wf 6556  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  Grpcgrp 18952   GrpHom cghm 19231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-ghm 19232
This theorem is referenced by:  gicref  19291  symgga  19426  0frgp  19798  idrnghm  20459  idrhm  20491  idlmhm  21041  frgpcyg  21593  nmoid  24764  idnghm  24765
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