Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mhmf | Structured version Visualization version GIF version |
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 2821 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2821 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | eqid 2821 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
6 | eqid 2821 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | ismhm 17957 | . . 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 499 | . 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 1138 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 0gc0g 16712 Mndcmnd 17910 MndHom cmhm 17953 |
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 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8407 df-mhm 17955 |
This theorem is referenced by: mhmf1o 17965 resmhm 17984 resmhm2 17985 resmhm2b 17986 mhmco 17987 mhmima 17988 mhmeql 17989 pwsco2mhm 17996 gsumwmhm 18009 frmdup3lem 18030 frmdup3 18031 mhmmulg 18267 ghmmhmb 18368 cntzmhm 18468 cntzmhm2 18469 frgpup3lem 18902 gsumzmhm 19056 gsummhm2 19058 gsummptmhm 19059 mhmvlin 21007 mdetleib2 21196 mdetf 21203 mdetdiaglem 21206 mdetrlin 21210 mdetrsca 21211 mdetralt 21216 mdetunilem7 21226 mdetunilem8 21227 dchrelbas2 25812 dchrn0 25825 mhmhmeotmd 31170 |
Copyright terms: Public domain | W3C validator |