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Theorem ismhm0 41805
Description: Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.)
Hypotheses
Ref Expression
ismhm0.b |- B = ( Base ` S )
ismhm0.c |- C = ( Base ` T )
ismhm0.p |- .+ = ( +g ` S )
ismhm0.q |- .+^ = ( +g ` T )
ismhm0.z |- .0. = ( 0g ` S )
ismhm0.y |- Y = ( 0g ` T )
Assertion
Ref Expression
ismhm0 |- ( F e. ( S MndHom T ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( F e. ( S MgmHom T ) /\ ( F ` .0. ) = Y ) ) )
Proof of Theorem ismhm0
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismhm0.b . . 3 |- B = ( Base ` S )
2 ismhm0.c . . 3 |- C = ( Base ` T )
3 ismhm0.p . . 3 |- .+ = ( +g ` S )
4 ismhm0.q . . 3 |- .+^ = ( +g ` T )
5 ismhm0.z . . 3 |- .0. = ( 0g ` S )
6 ismhm0.y . . 3 |- Y = ( 0g ` T )
71, 2, 3, 4, 5, 6ismhm 17337 . 2 |- ( F e. ( S MndHom T ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) /\ ( F ` .0. ) = Y ) ) )
8 df-3an 1039 . . . 4 |- ( ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) /\ ( F ` .0. ) = Y ) <-> ( ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) ) /\ ( F ` .0. ) = Y ) )
9 mndmgm 17300 . . . . . . . 8 |- ( S e. Mnd -> S e. Mgm )
10 mndmgm 17300 . . . . . . . 8 |- ( T e. Mnd -> T e. Mgm )
119, 10anim12i 590 . . . . . . 7 |- ( ( S e. Mnd /\ T e. Mnd ) -> ( S e. Mgm /\ T e. Mgm ) )
1211biantrurd 529 . . . . . 6 |- ( ( S e. Mnd /\ T e. Mnd ) -> ( ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) ) <-> ( ( S e. Mgm /\ T e. Mgm ) /\ ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) ) ) ) )
131, 2, 3, 4ismgmhm 41783 . . . . . 6 |- ( F e. ( S MgmHom T ) <-> ( ( S e. Mgm /\ T e. Mgm ) /\ ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) ) ) )
1412, 13syl6bbr 278 . . . . 5 |- ( ( S e. Mnd /\ T e. Mnd ) -> ( ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) ) <-> F e. ( S MgmHom T ) ) )
1514anbi1d 741 . . . 4 |- ( ( S e. Mnd /\ T e. Mnd ) -> ( ( ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) ) /\ ( F ` .0. ) = Y ) <-> ( F e. ( S MgmHom T ) /\ ( F ` .0. ) = Y ) ) )
168, 15syl5bb 272 . . 3 |- ( ( S e. Mnd /\ T e. Mnd ) -> ( ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) /\ ( F ` .0. ) = Y ) <-> ( F e. ( S MgmHom T ) /\ ( F ` .0. ) = Y ) ) )
1716pm5.32i 669 . 2 |- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) /\ ( F ` .0. ) = Y ) ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( F e. ( S MgmHom T ) /\ ( F ` .0. ) = Y ) ) )
187, 17bitri 264 1 |- ( F e. ( S MndHom T ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( F e. ( S MgmHom T ) /\ ( F ` .0. ) = Y ) ) )
Colors of variables: wff setvar class
Syntax hints:   <-> wb 196   /\ 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:  c0snmhm  41915
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