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Mirrors > Home > MPE Home > Th. List > 0ghm | Structured version Visualization version GIF version |
Description: The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
0ghm.z | ⊢ 0 = (0g‘𝑁) |
0ghm.b | ⊢ 𝐵 = (Base‘𝑀) |
Ref | Expression |
---|---|
0ghm | ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 18102 | . . 3 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) | |
2 | grpmnd 18102 | . . 3 ⊢ (𝑁 ∈ Grp → 𝑁 ∈ Mnd) | |
3 | 0ghm.z | . . . 4 ⊢ 0 = (0g‘𝑁) | |
4 | 0ghm.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
5 | 3, 4 | 0mhm 17976 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁)) |
6 | 1, 2, 5 | syl2an 598 | . 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁)) |
7 | ghmmhmb 18361 | . 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝑀 GrpHom 𝑁) = (𝑀 MndHom 𝑁)) | |
8 | 6, 7 | eleqtrrd 2893 | 1 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 × cxp 5517 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 0gc0g 16705 Mndcmnd 17903 MndHom cmhm 17946 Grpcgrp 18095 GrpHom cghm 18347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-ghm 18348 |
This theorem is referenced by: 0frgp 18897 0lmhm 19805 nmo0 23341 0nghm 23347 |
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