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| Description: A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.) | 
| Ref | Expression | 
|---|---|
| mhmf.b | ⊢ 𝐵 = (Base‘𝑆) | 
| mhmf.c | ⊢ 𝐶 = (Base‘𝑇) | 
| Ref | Expression | 
|---|---|
| mhmf | ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:𝐵⟶𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mhmf.b | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | |
| 2 | mhmf.c | . . . 4 ⊢ 𝐶 = (Base‘𝑇) | |
| 3 | eqid 2737 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
| 4 | eqid 2737 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
| 5 | eqid 2737 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
| 6 | eqid 2737 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
| 7 | 1, 2, 3, 4, 5, 6 | ismhm 18798 | . . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇)))) | 
| 8 | 7 | simprbi 496 | . 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) | 
| 9 | 8 | simp1d 1143 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:𝐵⟶𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 MndHom cmhm 18794 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-mhm 18796 | 
| This theorem is referenced by: mhmf1o 18809 mhmvlin 18814 resmhm 18833 resmhm2 18834 resmhm2b 18835 mhmco 18836 mhmimalem 18837 mhmima 18838 mhmeql 18839 pwsco2mhm 18846 gsumwmhm 18858 frmdup3lem 18879 frmdup3 18880 mhmmulg 19133 ghmmhmb 19245 cntzmhm 19359 cntzmhm2 19360 frgpup3lem 19795 gsumzmhm 19955 gsummhm2 19957 gsummptmhm 19958 rhmimasubrnglem 20565 mhmcompl 22384 mdetleib2 22594 mdetf 22601 mdetdiaglem 22604 mdetrlin 22608 mdetrsca 22609 mdetralt 22614 mdetunilem7 22624 mdetunilem8 22625 dchrelbas2 27281 dchrn0 27294 mhmhmeotmd 33926 mhmcopsr 42559 | 
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