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Theorem ghmco 18981
Description: The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
Assertion
Ref Expression
ghmco ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpHom 𝑈))

Proof of Theorem ghmco
StepHypRef Expression
1 ghmmhm 18971 . . 3 (𝐹 ∈ (𝑇 GrpHom 𝑈) → 𝐹 ∈ (𝑇 MndHom 𝑈))
2 ghmmhm 18971 . . 3 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺 ∈ (𝑆 MndHom 𝑇))
3 mhmco 18588 . . 3 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MndHom 𝑈))
41, 2, 3syl2an 596 . 2 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MndHom 𝑈))
5 ghmgrp1 18963 . . 3 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
6 ghmgrp2 18964 . . 3 (𝐹 ∈ (𝑇 GrpHom 𝑈) → 𝑈 ∈ Grp)
7 ghmmhmb 18972 . . 3 ((𝑆 ∈ Grp ∧ 𝑈 ∈ Grp) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈))
85, 6, 7syl2anr 597 . 2 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈))
94, 8eleqtrrd 2841 1 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpHom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ccom 5635  (class class class)co 7351   MndHom cmhm 18553  Grpcgrp 18702   GrpHom cghm 18958
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 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
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 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-map 8725  df-0g 17277  df-mgm 18451  df-sgrp 18500  df-mnd 18511  df-mhm 18555  df-grp 18705  df-ghm 18959
This theorem is referenced by:  gimco  19011  rhmco  20118  lmhmco  20451  lmhmvsca  20453  frgpcyg  20927  nmoco  24047  nghmco  24048  rnghmco  46100
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