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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 2727 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2727 | . . 3 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | ismgmhm 18647 | . 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 770 | . 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 216 | 1 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 Basecbs 17171 +gcplusg 17224 Mgmcmgm 18589 MgmHom cmgmhm 18641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8838 df-mgmhm 18643 |
This theorem is referenced by: mgmhmf1o 18651 resmgmhm 18662 resmgmhm2 18663 resmgmhm2b 18664 mgmhmco 18665 mgmhmima 18666 mgmhmeql 18667 |
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