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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 18326 | . . . 4 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd) | |
2 | grpmnd 18326 | . . . 4 ⊢ (𝑇 ∈ Grp → 𝑇 ∈ Mnd) | |
3 | 1, 2 | anim12i 616 | . . 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 45084 | . . 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 18587 | . . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇)) | |
10 | 9 | eleq2d 2816 | . 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐻 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐻 ∈ (𝑆 MndHom 𝑇))) |
11 | 8, 10 | mpbird 260 | 1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ↦ cmpt 5120 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 0gc0g 16898 Mndcmnd 18127 MndHom cmhm 18170 Grpcgrp 18319 GrpHom cghm 18573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-map 8488 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-grp 18322 df-ghm 18574 |
This theorem is referenced by: c0rhm 45086 c0rnghm 45087 |
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