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Theorem c0mgm 20397
Description: The constant mapping to zero is a magma homomorphism into a monoid. Remark: Instead of the assumption that T is a monoid, it would be sufficient that T is a magma with a right or left identity. (Contributed by AV, 17-Apr-2020.)
Hypotheses
Ref Expression
c0mhm.b 𝐵 = (Base‘𝑆)
c0mhm.0 0 = (0g𝑇)
c0mhm.h 𝐻 = (𝑥𝐵0 )
Assertion
Ref Expression
c0mgm ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑆   𝑥,𝑇   𝑥, 0
Allowed substitution hint:   𝐻(𝑥)

Proof of Theorem c0mgm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndmgm 18700 . . 3 (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
21anim2i 616 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3 eqid 2728 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
4 c0mhm.0 . . . . . . 7 0 = (0g𝑇)
53, 4mndidcl 18708 . . . . . 6 (𝑇 ∈ Mnd → 0 ∈ (Base‘𝑇))
65adantl 481 . . . . 5 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 0 ∈ (Base‘𝑇))
76adantr 480 . . . 4 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑇))
8 c0mhm.h . . . 4 𝐻 = (𝑥𝐵0 )
97, 8fmptd 7124 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
105ancli 548 . . . . . . . 8 (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
1110adantl 481 . . . . . . 7 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
12 eqid 2728 . . . . . . . 8 (+g𝑇) = (+g𝑇)
133, 12, 4mndlid 18713 . . . . . . 7 ((𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)) → ( 0 (+g𝑇) 0 ) = 0 )
1411, 13syl 17 . . . . . 6 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → ( 0 (+g𝑇) 0 ) = 0 )
1514adantr 480 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ( 0 (+g𝑇) 0 ) = 0 )
168a1i 11 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝐻 = (𝑥𝐵0 ))
17 eqidd 2729 . . . . . . 7 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
18 simprl 770 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
196adantr 480 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 0 ∈ (Base‘𝑇))
2016, 17, 18, 19fvmptd 7012 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑎) = 0 )
21 eqidd 2729 . . . . . . 7 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
22 simprr 772 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
2316, 21, 22, 19fvmptd 7012 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑏) = 0 )
2420, 23oveq12d 7438 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) = ( 0 (+g𝑇) 0 ))
25 eqidd 2729 . . . . . 6 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = (𝑎(+g𝑆)𝑏)) → 0 = 0 )
26 c0mhm.b . . . . . . . . 9 𝐵 = (Base‘𝑆)
27 eqid 2728 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
2826, 27mgmcl 18602 . . . . . . . 8 ((𝑆 ∈ Mgm ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
29283expb 1118 . . . . . . 7 ((𝑆 ∈ Mgm ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
3029adantlr 714 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
3116, 25, 30, 19fvmptd 7012 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = 0 )
3215, 24, 313eqtr4rd 2779 . . . 4 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
3332ralrimivva 3197 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
349, 33jca 511 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏))))
3526, 3, 27, 12ismgmhm 18655 . 2 (𝐻 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))))
362, 34, 35sylanbrc 582 1 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3058  cmpt 5231  wf 6544  cfv 6548  (class class class)co 7420  Basecbs 17179  +gcplusg 17232  0gc0g 17420  Mgmcmgm 18597   MgmHom cmgmhm 18649  Mndcmnd 18693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8846  df-0g 17422  df-mgm 18599  df-mgmhm 18651  df-sgrp 18678  df-mnd 18694
This theorem is referenced by:  c0rnghm  20471
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