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Mathbox for Alexander van der Vekens |
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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 2825 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2825 | . . 3 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | ismgmhm 42648 | . 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 787 | . 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 209 | 1 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +gcplusg 16312 Mgmcmgm 17600 MgmHom cmgmhm 42642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 df-mgmhm 42644 |
This theorem is referenced by: mgmhmf1o 42652 resmgmhm 42663 resmgmhm2 42664 resmgmhm2b 42665 mgmhmco 42666 mgmhmima 42667 mgmhmeql 42668 |
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