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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgmhmf | Structured version Visualization version GIF version |
Description: A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.) |
Ref | Expression |
---|---|
mgmhmf.b | ⊢ 𝐵 = (Base‘𝑆) |
mgmhmf.c | ⊢ 𝐶 = (Base‘𝑇) |
Ref | Expression |
---|---|
mgmhmf | ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgmhmf.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
2 | mgmhmf.c | . . 3 ⊢ 𝐶 = (Base‘𝑇) | |
3 | eqid 2821 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2821 | . . 3 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | ismgmhm 44044 | . 2 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))))) |
6 | simprl 769 | . 2 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))) → 𝐹:𝐵⟶𝐶) | |
7 | 5, 6 | sylbi 219 | 1 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 Mgmcmgm 17844 MgmHom cmgmhm 44038 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-mgmhm 44040 |
This theorem is referenced by: mgmhmf1o 44048 resmgmhm 44059 resmgmhm2 44060 resmgmhm2b 44061 mgmhmco 44062 mgmhmima 44063 mgmhmeql 44064 |
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