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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 17252 | . . . 4 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd) | |
2 | grpmnd 17252 | . . . 4 ⊢ (𝑇 ∈ Grp → 𝑇 ∈ Mnd) | |
3 | 1, 2 | anim12i 588 | . . 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 41700 | . . 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 17494 | . . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇)) | |
10 | 9 | eleq2d 2673 | . 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐻 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐻 ∈ (𝑆 MndHom 𝑇))) |
11 | 8, 10 | mpbird 246 | 1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 0gc0g 15923 Mndcmnd 17117 MndHom cmhm 17156 Grpcgrp 17245 GrpHom cghm 17480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-ghm 17481 |
This theorem is referenced by: c0rhm 41702 c0rnghm 41703 |
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