Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  c0ghm Structured version   Visualization version   GIF version

Theorem c0ghm 44528
 Description: The constant mapping to zero is a group homomorphism. (Contributed by AV, 16-Apr-2020.)
Hypotheses
Ref Expression
c0mhm.b 𝐵 = (Base‘𝑆)
c0mhm.0 0 = (0g𝑇)
c0mhm.h 𝐻 = (𝑥𝐵0 )
Assertion
Ref Expression
c0ghm ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑆   𝑥,𝑇   𝑥, 0
Allowed substitution hint:   𝐻(𝑥)

Proof of Theorem c0ghm
StepHypRef Expression
1 grpmnd 18106 . . . 4 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
2 grpmnd 18106 . . . 4 (𝑇 ∈ Grp → 𝑇 ∈ Mnd)
31, 2anim12i 615 . . 3 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
4 c0mhm.b . . . 4 𝐵 = (Base‘𝑆)
5 c0mhm.0 . . . 4 0 = (0g𝑇)
6 c0mhm.h . . . 4 𝐻 = (𝑥𝐵0 )
74, 5, 6c0mhm 44527 . . 3 ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))
83, 7syl 17 . 2 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 MndHom 𝑇))
9 ghmmhmb 18365 . . 3 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
109eleq2d 2878 . 2 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐻 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐻 ∈ (𝑆 MndHom 𝑇)))
118, 10mpbird 260 1 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112   ↦ cmpt 5113  ‘cfv 6328  (class class class)co 7139  Basecbs 16479  0gc0g 16709  Mndcmnd 17907   MndHom cmhm 17950  Grpcgrp 18099   GrpHom cghm 18351 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-0g 16711  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-mhm 17952  df-grp 18102  df-ghm 18352 This theorem is referenced by:  c0rhm  44529  c0rnghm  44530
 Copyright terms: Public domain W3C validator