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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 18890 | . . 3 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd) | |
2 | grpmnd 18890 | . . 3 ⊢ (𝑁 ∈ Grp → 𝑁 ∈ Mnd) | |
3 | 0ghm.z | . . . 4 ⊢ 0 = (0g‘𝑁) | |
4 | 0ghm.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
5 | 3, 4 | 0mhm 18764 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁)) |
6 | 1, 2, 5 | syl2an 595 | . 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁)) |
7 | ghmmhmb 19174 | . 2 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝑀 GrpHom 𝑁) = (𝑀 MndHom 𝑁)) | |
8 | 6, 7 | eleqtrrd 2832 | 1 ⊢ ((𝑀 ∈ Grp ∧ 𝑁 ∈ Grp) → (𝐵 × { 0 }) ∈ (𝑀 GrpHom 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {csn 4624 × cxp 5670 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 0gc0g 17414 Mndcmnd 18687 MndHom cmhm 18731 Grpcgrp 18883 GrpHom cghm 19160 |
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-rep 5279 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-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 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-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8840 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-grp 18886 df-ghm 19161 |
This theorem is referenced by: 0frgp 19727 0lmhm 20918 nmo0 24645 0nghm 24651 |
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