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Theorem idghm 19185
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 2728 . . . . . . . 8 (+g𝐺) = (+g𝐺)
42, 3grpcl 18898 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝐺)𝑏) ∈ 𝐵)
543expb 1118 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝐺)𝑏) ∈ 𝐵)
6 fvresi 7182 . . . . . 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 7182 . . . . . . 7 (𝑎𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎)
9 fvresi 7182 . . . . . . 7 (𝑏𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏)
108, 9oveqan12d 7439 . . . . . 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 2771 . . . 4 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)))
1312ralrimivva 3197 . . 3 (𝐺 ∈ Grp → ∀𝑎𝐵𝑏𝐵 (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)))
14 f1oi 6877 . . . 4 ( I ↾ 𝐵):𝐵1-1-onto𝐵
15 f1of 6839 . . . 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 19170 . 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 1342 1 (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3058   I cid 5575  cres 5680  wf 6544  1-1-ontowf1o 6547  cfv 6548  (class class class)co 7420  Basecbs 17180  +gcplusg 17233  Grpcgrp 18890   GrpHom cghm 19167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-grp 18893  df-ghm 19168
This theorem is referenced by:  gicref  19226  symgga  19362  0frgp  19734  idrnghm  20397  idrhm  20429  idlmhm  20926  frgpcyg  21507  nmoid  24672  idnghm  24673
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