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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 2728 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2728 | . . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | eqid 2728 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2728 | . . . 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 18735 | . . 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 1142 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘ 0 ) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 +gcplusg 17226 0gc0g 17414 Mndcmnd 18687 MndHom cmhm 18731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8840 df-mhm 18733 |
This theorem is referenced by: mhmf1o 18746 resmhm 18765 resmhm2 18766 resmhm2b 18767 mhmco 18768 mhmima 18770 mhmeql 18771 pwsco2mhm 18778 gsumwmhm 18790 mhmmulg 19063 gsumzmhm 19885 rhm1 20421 madetsumid 22356 mdetunilem7 22513 pm2mp 22720 dchrzrh1 27170 dchrmulcl 27175 dchrn0 27176 dchrinvcl 27179 dchrfi 27181 dchrabs 27186 sumdchr2 27196 rpvmasum2 27438 rhmpreimaidl 33127 mhmcompl 41775 selvvvval 41812 |
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