Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idmgmhm | Structured version Visualization version GIF version |
Description: The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.) |
Ref | Expression |
---|---|
idmgmhm.b | ⊢ 𝐵 = (Base‘𝑀) |
Ref | Expression |
---|---|
idmgmhm | ⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑀 ∈ Mgm → 𝑀 ∈ Mgm) | |
2 | 1 | ancri 573 | . 2 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ Mgm ∧ 𝑀 ∈ Mgm)) |
3 | f1oi 6086 | . . . 4 ⊢ ( I ↾ 𝐵):𝐵–1-1-onto→𝐵 | |
4 | f1of 6050 | . . . 4 ⊢ (( I ↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵) | |
5 | 3, 4 | mp1i 13 | . . 3 ⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵):𝐵⟶𝐵) |
6 | idmgmhm.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀) | |
7 | eqid 2610 | . . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
8 | 6, 7 | mgmcl 17068 | . . . . . . 7 ⊢ ((𝑀 ∈ Mgm ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵) |
9 | 8 | 3expb 1258 | . . . . . 6 ⊢ ((𝑀 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵) |
10 | fvresi 6344 | . . . . . 6 ⊢ ((𝑎(+g‘𝑀)𝑏) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = (𝑎(+g‘𝑀)𝑏)) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = (𝑎(+g‘𝑀)𝑏)) |
12 | fvresi 6344 | . . . . . . 7 ⊢ (𝑎 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎) | |
13 | fvresi 6344 | . . . . . . 7 ⊢ (𝑏 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏) | |
14 | 12, 13 | oveqan12d 6568 | . . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏)) |
15 | 14 | adantl 481 | . . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏)) |
16 | 11, 15 | eqtr4d 2647 | . . . 4 ⊢ ((𝑀 ∈ Mgm ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏))) |
17 | 16 | ralrimivva 2954 | . . 3 ⊢ (𝑀 ∈ Mgm → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏))) |
18 | 5, 17 | jca 553 | . 2 ⊢ (𝑀 ∈ Mgm → (( I ↾ 𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)))) |
19 | 6, 6, 7, 7 | ismgmhm 41573 | . 2 ⊢ (( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀) ↔ ((𝑀 ∈ Mgm ∧ 𝑀 ∈ Mgm) ∧ (( I ↾ 𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏))))) |
20 | 2, 18, 19 | sylanbrc 695 | 1 ⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 I cid 4948 ↾ cres 5040 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Mgmcmgm 17063 MgmHom cmgmhm 41567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-mgm 17065 df-mgmhm 41569 |
This theorem is referenced by: idrnghm 41698 |
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