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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 18756 | . . 3 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) | |
2 | grpmnd 18756 | . . 3 ⊢ (𝑁 ∈ Grp → 𝑁 ∈ Mnd) | |
3 | 0ghm.z | . . . 4 ⊢ 0 = (0g‘𝑁) | |
4 | 0ghm.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
5 | 3, 4 | 0mhm 18631 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁)) |
6 | 1, 2, 5 | syl2an 597 | . 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁)) |
7 | ghmmhmb 19020 | . 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝑀 GrpHom 𝑁) = (𝑀 MndHom 𝑁)) | |
8 | 6, 7 | eleqtrrd 2841 | 1 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4587 × cxp 5632 ‘cfv 6497 (class class class)co 7358 Basecbs 17084 0gc0g 17322 Mndcmnd 18557 MndHom cmhm 18600 Grpcgrp 18749 GrpHom cghm 19006 |
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 2708 ax-rep 5243 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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 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-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8768 df-0g 17324 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-mhm 18602 df-grp 18752 df-ghm 19007 |
This theorem is referenced by: 0frgp 19562 0lmhm 20504 nmo0 24102 0nghm 24108 |
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