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Mirrors > Home > MPE Home > Th. List > 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 2735 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2735 | . . 3 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | ismgmhm 18722 | . 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 771 | . 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 217 | 1 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Mgmcmgm 18664 MgmHom cmgmhm 18716 |
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-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-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-mgmhm 18718 |
This theorem is referenced by: mgmhmf1o 18726 resmgmhm 18737 resmgmhm2 18738 resmgmhm2b 18739 mgmhmco 18740 mgmhmima 18741 mgmhmeql 18742 |
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