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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 2724 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2724 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | eqid 2724 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2724 | . . . 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 18707 | . . 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 1141 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘ 0 ) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 +gcplusg 17198 0gc0g 17386 Mndcmnd 18659 MndHom cmhm 18703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-map 8819 df-mhm 18705 |
This theorem is referenced by: mhmf1o 18718 resmhm 18737 resmhm2 18738 resmhm2b 18739 mhmco 18740 mhmima 18742 mhmeql 18743 pwsco2mhm 18750 gsumwmhm 18762 mhmmulg 19034 gsumzmhm 19849 rhm1 20383 madetsumid 22287 mdetunilem7 22444 pm2mp 22651 dchrzrh1 27096 dchrmulcl 27101 dchrn0 27102 dchrinvcl 27105 dchrfi 27107 dchrabs 27112 sumdchr2 27122 rpvmasum2 27364 rhmpreimaidl 33009 mhmcompl 41613 selvvvval 41650 |
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