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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 17783 | . . 3 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) | |
2 | grpmnd 17783 | . . 3 ⊢ (𝑁 ∈ Grp → 𝑁 ∈ Mnd) | |
3 | 0ghm.z | . . . 4 ⊢ 0 = (0g‘𝑁) | |
4 | 0ghm.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
5 | 3, 4 | 0mhm 17711 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁)) |
6 | 1, 2, 5 | syl2an 591 | . 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁)) |
7 | ghmmhmb 18022 | . 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝑀 GrpHom 𝑁) = (𝑀 MndHom 𝑁)) | |
8 | 6, 7 | eleqtrrd 2909 | 1 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {csn 4397 × cxp 5340 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 0gc0g 16453 Mndcmnd 17647 MndHom cmhm 17686 Grpcgrp 17776 GrpHom cghm 18008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-map 8124 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-grp 17779 df-ghm 18009 |
This theorem is referenced by: 0frgp 18545 0lmhm 19399 nmo0 22909 0nghm 22915 |
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