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Theorem idghm 19152
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 2724 . . . . . . . 8 (+g𝐺) = (+g𝐺)
42, 3grpcl 18867 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝐺)𝑏) ∈ 𝐵)
543expb 1117 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝐺)𝑏) ∈ 𝐵)
6 fvresi 7164 . . . . . 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 7164 . . . . . . 7 (𝑎𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎)
9 fvresi 7164 . . . . . . 7 (𝑏𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏)
108, 9oveqan12d 7421 . . . . . 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 2767 . . . 4 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)))
1312ralrimivva 3192 . . 3 (𝐺 ∈ Grp → ∀𝑎𝐵𝑏𝐵 (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)))
14 f1oi 6862 . . . 4 ( I ↾ 𝐵):𝐵1-1-onto𝐵
15 f1of 6824 . . . 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 19137 . 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 1341 1 (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3053   I cid 5564  cres 5669  wf 6530  1-1-ontowf1o 6533  cfv 6534  (class class class)co 7402  Basecbs 17149  +gcplusg 17202  Grpcgrp 18859   GrpHom cghm 19134
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-grp 18862  df-ghm 19135
This theorem is referenced by:  gicref  19193  symgga  19323  0frgp  19695  idrnghm  20356  idrhm  20388  idlmhm  20885  frgpcyg  21457  nmoid  24603  idnghm  24604
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