Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 18572 | . . 3 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) | |
2 | grpmnd 18572 | . . 3 ⊢ (𝑁 ∈ Grp → 𝑁 ∈ Mnd) | |
3 | 0ghm.z | . . . 4 ⊢ 0 = (0g‘𝑁) | |
4 | 0ghm.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
5 | 3, 4 | 0mhm 18446 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁)) |
6 | 1, 2, 5 | syl2an 596 | . 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁)) |
7 | ghmmhmb 18833 | . 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝑀 GrpHom 𝑁) = (𝑀 MndHom 𝑁)) | |
8 | 6, 7 | eleqtrrd 2842 | 1 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {csn 4562 × cxp 5583 ‘cfv 6427 (class class class)co 7268 Basecbs 16900 0gc0g 17138 Mndcmnd 18373 MndHom cmhm 18416 Grpcgrp 18565 GrpHom cghm 18819 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-map 8605 df-0g 17140 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-mhm 18418 df-grp 18568 df-ghm 18820 |
This theorem is referenced by: 0frgp 19373 0lmhm 20290 nmo0 23887 0nghm 23893 |
Copyright terms: Public domain | W3C validator |