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Mirrors > Home > MPE Home > Th. List > idghm | Structured version Visualization version GIF version |
Description: The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
idghm.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
idghm | ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp) | |
2 | idghm.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2735 | . . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | 2, 3 | grpcl 18972 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) |
5 | 4 | 3expb 1119 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝐺)𝑏) ∈ 𝐵) |
6 | fvresi 7193 | . . . . . 6 ⊢ ((𝑎(+g‘𝐺)𝑏) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = (𝑎(+g‘𝐺)𝑏)) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = (𝑎(+g‘𝐺)𝑏)) |
8 | fvresi 7193 | . . . . . . 7 ⊢ (𝑎 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎) | |
9 | fvresi 7193 | . . . . . . 7 ⊢ (𝑏 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏) | |
10 | 8, 9 | oveqan12d 7450 | . . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝐺)𝑏)) |
11 | 10 | adantl 481 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝐺)𝑏)) |
12 | 7, 11 | eqtr4d 2778 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏))) |
13 | 12 | ralrimivva 3200 | . . 3 ⊢ (𝐺 ∈ Grp → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏))) |
14 | f1oi 6887 | . . . 4 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
15 | f1of 6849 | . . . 4 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ ( I ↾ 𝐵):𝐵⟶𝐵 |
17 | 13, 16 | jctil 519 | . 2 ⊢ (𝐺 ∈ Grp → (( I ↾ 𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏)))) |
18 | 2, 2, 3, 3 | isghm 19246 | . 2 ⊢ (( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ Grp) ∧ (( I ↾ 𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝐺)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝐺)(( I ↾ 𝐵)‘𝑏))))) |
19 | 1, 1, 17, 18 | syl21anbrc 1343 | 1 ⊢ (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpHom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 I cid 5582 ↾ cres 5691 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Grpcgrp 18964 GrpHom cghm 19243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-ghm 19244 |
This theorem is referenced by: gicref 19303 symgga 19440 0frgp 19812 idrnghm 20475 idrhm 20507 idlmhm 21058 frgpcyg 21610 nmoid 24779 idnghm 24780 |
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