Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mhm0 | Structured version Visualization version GIF version |
Description: A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
mhm0.z | ⊢ 0 = (0g‘𝑆) |
mhm0.y | ⊢ 𝑌 = (0g‘𝑇) |
Ref | Expression |
---|---|
mhm0 | ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘ 0 ) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2821 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | eqid 2821 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2821 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | mhm0.z | . . . 4 ⊢ 0 = (0g‘𝑆) | |
6 | mhm0.y | . . . 4 ⊢ 𝑌 = (0g‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | ismhm 17952 | . . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌))) |
8 | 7 | simprbi 499 | . 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)) |
9 | 8 | simp3d 1140 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘ 0 ) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Mndcmnd 17905 MndHom cmhm 17948 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
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 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-mhm 17950 |
This theorem is referenced by: mhmf1o 17960 resmhm 17979 resmhm2 17980 resmhm2b 17981 mhmco 17982 mhmima 17983 mhmeql 17984 pwsco2mhm 17991 gsumwmhm 18004 mhmmulg 18262 gsumzmhm 19051 rhm1 19476 madetsumid 21064 mdetunilem7 21221 pm2mp 21427 dchrzrh1 25814 dchrmulcl 25819 dchrn0 25820 dchrinvcl 25823 dchrfi 25825 dchrabs 25830 sumdchr2 25840 rpvmasum2 26082 |
Copyright terms: Public domain | W3C validator |