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Mirrors > Home > MPE Home > Th. List > c0ghm | Structured version Visualization version GIF version |
Description: The constant mapping to zero is a group homomorphism. (Contributed by AV, 16-Apr-2020.) |
Ref | Expression |
---|---|
c0mhm.b | ⊢ 𝐵 = (Base‘𝑆) |
c0mhm.0 | ⊢ 0 = (0g‘𝑇) |
c0mhm.h | ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) |
Ref | Expression |
---|---|
c0ghm | ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 18971 | . . . 4 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd) | |
2 | grpmnd 18971 | . . . 4 ⊢ (𝑇 ∈ Grp → 𝑇 ∈ Mnd) | |
3 | 1, 2 | anim12i 613 | . . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd)) |
4 | c0mhm.b | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | |
5 | c0mhm.0 | . . . 4 ⊢ 0 = (0g‘𝑇) | |
6 | c0mhm.h | . . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) | |
7 | 4, 5, 6 | c0mhm 20477 | . . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇)) |
8 | 3, 7 | syl 17 | . 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 MndHom 𝑇)) |
9 | ghmmhmb 19258 | . . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇)) | |
10 | 9 | eleq2d 2825 | . 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐻 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐻 ∈ (𝑆 MndHom 𝑇))) |
11 | 8, 10 | mpbird 257 | 1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 0gc0g 17486 Mndcmnd 18760 MndHom cmhm 18807 Grpcgrp 18964 GrpHom cghm 19243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-ghm 19244 |
This theorem is referenced by: c0rhm 20551 c0rnghm 20552 |
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