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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 18754 | . . . 4 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd) | |
2 | grpmnd 18754 | . . . 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 46180 | . . 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 19017 | . . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇)) | |
10 | 9 | eleq2d 2823 | . 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐻 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐻 ∈ (𝑆 MndHom 𝑇))) |
11 | 8, 10 | mpbird 256 | 1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 0gc0g 17320 Mndcmnd 18555 MndHom cmhm 18598 Grpcgrp 18747 GrpHom cghm 19003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-map 8766 df-0g 17322 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-mhm 18600 df-grp 18750 df-ghm 19004 |
This theorem is referenced by: c0rhm 46182 c0rnghm 46183 |
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