![]() |
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 2731 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2731 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | eqid 2731 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2731 | . . . 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 18710 | . . 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 496 | . 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 1143 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘ 0 ) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 0gc0g 17392 Mndcmnd 18662 MndHom cmhm 18706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8828 df-mhm 18708 |
This theorem is referenced by: mhmf1o 18721 resmhm 18740 resmhm2 18741 resmhm2b 18742 mhmco 18743 mhmima 18745 mhmeql 18746 pwsco2mhm 18753 gsumwmhm 18765 mhmmulg 19035 gsumzmhm 19850 rhm1 20384 madetsumid 22196 mdetunilem7 22353 pm2mp 22560 dchrzrh1 26998 dchrmulcl 27003 dchrn0 27004 dchrinvcl 27007 dchrfi 27009 dchrabs 27014 sumdchr2 27024 rpvmasum2 27266 rhmpreimaidl 32826 mhmcompl 41435 selvvvval 41472 |
Copyright terms: Public domain | W3C validator |