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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 2733 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2733 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | eqid 2733 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2733 | . . . 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 18608 | . . 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 498 | . 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 1145 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘ 0 ) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 0gc0g 17326 Mndcmnd 18561 MndHom cmhm 18604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8770 df-mhm 18606 |
This theorem is referenced by: mhmf1o 18617 resmhm 18636 resmhm2 18637 resmhm2b 18638 mhmco 18639 mhmima 18640 mhmeql 18641 pwsco2mhm 18648 gsumwmhm 18660 mhmmulg 18922 gsumzmhm 19719 rhm1 20169 madetsumid 21826 mdetunilem7 21983 pm2mp 22190 dchrzrh1 26608 dchrmulcl 26613 dchrn0 26614 dchrinvcl 26617 dchrfi 26619 dchrabs 26624 sumdchr2 26634 rpvmasum2 26876 rhmpreimaidl 32248 mhmcompl 40779 |
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