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Theorem mhmismgmhm 41806
Description: Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.)
Assertion
Ref Expression
mhmismgmhm |- ( F e. ( R MndHom S ) -> F e. ( R MgmHom S ) )
Proof of Theorem mhmismgmhm
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndmgm 17300 . . . 4 |- ( R e. Mnd -> R e. Mgm )
2 mndmgm 17300 . . . 4 |- ( S e. Mnd -> S e. Mgm )
31, 2anim12i 590 . . 3 |- ( ( R e. Mnd /\ S e. Mnd ) -> ( R e. Mgm /\ S e. Mgm ) )
4 3simpa 1058 . . 3 |- ( ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) /\ ( F ` ( 0g ` R ) ) = ( 0g ` S ) ) -> ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) ) )
53, 4anim12i 590 . 2 |- ( ( ( R e. Mnd /\ S e. Mnd ) /\ ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) /\ ( F ` ( 0g ` R ) ) = ( 0g ` S ) ) ) -> ( ( R e. Mgm /\ S e. Mgm ) /\ ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) ) ) )
6 eqid 2622 . . 3 |- ( Base ` R ) = ( Base ` R )
7 eqid 2622 . . 3 |- ( Base ` S ) = ( Base ` S )
8 eqid 2622 . . 3 |- ( +g ` R ) = ( +g ` R )
9 eqid 2622 . . 3 |- ( +g ` S ) = ( +g ` S )
10 eqid 2622 . . 3 |- ( 0g ` R ) = ( 0g ` R )
11 eqid 2622 . . 3 |- ( 0g ` S ) = ( 0g ` S )
126, 7, 8, 9, 10, 11ismhm 17337 . 2 |- ( F e. ( R MndHom S ) <-> ( ( R e. Mnd /\ S e. Mnd ) /\ ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) /\ ( F ` ( 0g ` R ) ) = ( 0g ` S ) ) ) )
136, 7, 8, 9ismgmhm 41783 . 2 |- ( F e. ( R MgmHom S ) <-> ( ( R e. Mgm /\ S e. Mgm ) /\ ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) ) ) )
145, 12, 133imtr4i 281 1 |- ( F e. ( R MndHom S ) -> F e. ( R MgmHom S ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100  Mgmcmgm 17240   Mndcmnd 17294   MndHom cmhm 17333   MgmHom cmgmhm 41777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-mgmhm 41779
This theorem is referenced by:  rhmisrnghm  41920
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