![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > c0snghm | Structured version Visualization version GIF version |
Description: The constant mapping to zero is a group homomorphism from the trivial group (consisting of the zero only) to any group. (Contributed by AV, 17-Apr-2020.) |
Ref | Expression |
---|---|
zrrhm.b | ⊢ 𝐵 = (Base‘𝑇) |
zrrhm.0 | ⊢ 0 = (0g‘𝑆) |
zrrhm.h | ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) |
c0snmhm.z | ⊢ 𝑍 = (0g‘𝑇) |
Ref | Expression |
---|---|
c0snghm | ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 GrpHom 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 18755 | . . 3 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd) | |
2 | grpmnd 18755 | . . 3 ⊢ (𝑇 ∈ Grp → 𝑇 ∈ Mnd) | |
3 | id 22 | . . 3 ⊢ (𝐵 = {𝑍} → 𝐵 = {𝑍}) | |
4 | zrrhm.b | . . . 4 ⊢ 𝐵 = (Base‘𝑇) | |
5 | zrrhm.0 | . . . 4 ⊢ 0 = (0g‘𝑆) | |
6 | zrrhm.h | . . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) | |
7 | c0snmhm.z | . . . 4 ⊢ 𝑍 = (0g‘𝑇) | |
8 | 4, 5, 6, 7 | c0snmhm 46203 | . . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MndHom 𝑆)) |
9 | 1, 2, 3, 8 | syl3an 1160 | . 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MndHom 𝑆)) |
10 | ghmmhmb 19019 | . . . . 5 ⊢ ((𝑇 ∈ Grp ∧ 𝑆 ∈ Grp) → (𝑇 GrpHom 𝑆) = (𝑇 MndHom 𝑆)) | |
11 | 10 | eleq2d 2823 | . . . 4 ⊢ ((𝑇 ∈ Grp ∧ 𝑆 ∈ Grp) → (𝐻 ∈ (𝑇 GrpHom 𝑆) ↔ 𝐻 ∈ (𝑇 MndHom 𝑆))) |
12 | 11 | ancoms 459 | . . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐻 ∈ (𝑇 GrpHom 𝑆) ↔ 𝐻 ∈ (𝑇 MndHom 𝑆))) |
13 | 12 | 3adant3 1132 | . 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {𝑍}) → (𝐻 ∈ (𝑇 GrpHom 𝑆) ↔ 𝐻 ∈ (𝑇 MndHom 𝑆))) |
14 | 9, 13 | mpbird 256 | 1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 GrpHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {csn 4586 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 0gc0g 17321 Mndcmnd 18556 MndHom cmhm 18599 Grpcgrp 18748 GrpHom cghm 19005 |
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 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 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-nel 3050 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-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 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-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 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 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-ghm 19006 df-mgmhm 46063 |
This theorem is referenced by: zrrnghm 46205 |
Copyright terms: Public domain | W3C validator |