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Theorem ghmco 18373
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 18363 . . 3 (𝐹 ∈ (𝑇 GrpHom 𝑈) → 𝐹 ∈ (𝑇 MndHom 𝑈))
2 ghmmhm 18363 . . 3 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝐺 ∈ (𝑆 MndHom 𝑇))
3 mhmco 17983 . . 3 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MndHom 𝑈))
41, 2, 3syl2an 598 . 2 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MndHom 𝑈))
5 ghmgrp1 18355 . . 3 (𝐺 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
6 ghmgrp2 18356 . . 3 (𝐹 ∈ (𝑇 GrpHom 𝑈) → 𝑈 ∈ Grp)
7 ghmmhmb 18364 . . 3 ((𝑆 ∈ Grp ∧ 𝑈 ∈ Grp) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈))
85, 6, 7syl2anr 599 . 2 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈))
94, 8eleqtrrd 2896 1 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpHom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  ccom 5527  (class class class)co 7139   MndHom cmhm 17949  Grpcgrp 18098   GrpHom cghm 18350
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 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-grp 18101  df-ghm 18351
This theorem is referenced by:  gimco  18403  rhmco  19488  lmhmco  19811  lmhmvsca  19813  frgpcyg  20268  nmoco  23346  nghmco  23347  rnghmco  44518
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