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Theorem idghm 17992
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 . . 3 (𝐺 ∈ Grp → 𝐺 ∈ Grp)
21ancli 545 . 2 (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ 𝐺 ∈ Grp))
3 idghm.b . . . . . . . 8 𝐵 = (Base‘𝐺)
4 eqid 2803 . . . . . . . 8 (+g𝐺) = (+g𝐺)
53, 4grpcl 17750 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝐺)𝑏) ∈ 𝐵)
653expb 1150 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝐺)𝑏) ∈ 𝐵)
7 fvresi 6672 . . . . . 6 ((𝑎(+g𝐺)𝑏) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = (𝑎(+g𝐺)𝑏))
86, 7syl 17 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = (𝑎(+g𝐺)𝑏))
9 fvresi 6672 . . . . . . 7 (𝑎𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎)
10 fvresi 6672 . . . . . . 7 (𝑏𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏)
119, 10oveqan12d 6901 . . . . . 6 ((𝑎𝐵𝑏𝐵) → ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g𝐺)𝑏))
1211adantl 474 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g𝐺)𝑏))
138, 12eqtr4d 2840 . . . 4 ((𝐺 ∈ Grp ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)))
1413ralrimivva 3156 . . 3 (𝐺 ∈ Grp → ∀𝑎𝐵𝑏𝐵 (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏)))
15 f1oi 6397 . . . 4 ( I ↾ 𝐵):𝐵1-1-onto𝐵
16 f1of 6360 . . . 4 (( I ↾ 𝐵):𝐵1-1-onto𝐵 → ( I ↾ 𝐵):𝐵𝐵)
1715, 16ax-mp 5 . . 3 ( I ↾ 𝐵):𝐵𝐵
1814, 17jctil 516 . 2 (𝐺 ∈ Grp → (( I ↾ 𝐵):𝐵𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (( I ↾ 𝐵)‘(𝑎(+g𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g𝐺)(( I ↾ 𝐵)‘𝑏))))
193, 3, 4, 4isghm 17977 . 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)))))
202, 18, 19sylanbrc 579 1 (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3093   I cid 5223  cres 5318  wf 6101  1-1-ontowf1o 6104  cfv 6105  (class class class)co 6882  Basecbs 16188  +gcplusg 16271  Grpcgrp 17742   GrpHom cghm 17974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-rep 4968  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-reu 3100  df-rab 3102  df-v 3391  df-sbc 3638  df-csb 3733  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-pw 4355  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-iun 4716  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-res 5328  df-ima 5329  df-iota 6068  df-fun 6107  df-fn 6108  df-f 6109  df-f1 6110  df-fo 6111  df-f1o 6112  df-fv 6113  df-ov 6885  df-oprab 6886  df-mpt2 6887  df-mgm 17561  df-sgrp 17603  df-mnd 17614  df-grp 17745  df-ghm 17975
This theorem is referenced by:  gicref  18030  symgga  18142  0frgp  18511  idrhm  19053  idlmhm  19366  frgpcyg  20247  nmoid  22878  idnghm  22879  idrnghm  42711
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