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Theorem c0mgm 44463
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 17918 . . 3 (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
21anim2i 619 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3 eqid 2824 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
4 c0mhm.0 . . . . . . 7 0 = (0g𝑇)
53, 4mndidcl 17926 . . . . . 6 (𝑇 ∈ Mnd → 0 ∈ (Base‘𝑇))
65adantl 485 . . . . 5 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 0 ∈ (Base‘𝑇))
76adantr 484 . . . 4 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑇))
8 c0mhm.h . . . 4 𝐻 = (𝑥𝐵0 )
97, 8fmptd 6869 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
105ancli 552 . . . . . . . 8 (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
1110adantl 485 . . . . . . 7 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
12 eqid 2824 . . . . . . . 8 (+g𝑇) = (+g𝑇)
133, 12, 4mndlid 17931 . . . . . . 7 ((𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)) → ( 0 (+g𝑇) 0 ) = 0 )
1411, 13syl 17 . . . . . 6 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → ( 0 (+g𝑇) 0 ) = 0 )
1514adantr 484 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ( 0 (+g𝑇) 0 ) = 0 )
168a1i 11 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝐻 = (𝑥𝐵0 ))
17 eqidd 2825 . . . . . . 7 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
18 simprl 770 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
196adantr 484 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 0 ∈ (Base‘𝑇))
2016, 17, 18, 19fvmptd 6766 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑎) = 0 )
21 eqidd 2825 . . . . . . 7 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
22 simprr 772 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
2316, 21, 22, 19fvmptd 6766 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑏) = 0 )
2420, 23oveq12d 7167 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) = ( 0 (+g𝑇) 0 ))
25 eqidd 2825 . . . . . 6 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = (𝑎(+g𝑆)𝑏)) → 0 = 0 )
26 c0mhm.b . . . . . . . . 9 𝐵 = (Base‘𝑆)
27 eqid 2824 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
2826, 27mgmcl 17855 . . . . . . . 8 ((𝑆 ∈ Mgm ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
29283expb 1117 . . . . . . 7 ((𝑆 ∈ Mgm ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
3029adantlr 714 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
3116, 25, 30, 19fvmptd 6766 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = 0 )
3215, 24, 313eqtr4rd 2870 . . . 4 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
3332ralrimivva 3186 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
349, 33jca 515 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏))))
3526, 3, 27, 12ismgmhm 44333 . 2 (𝐻 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))))
362, 34, 35sylanbrc 586 1 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  cmpt 5132  wf 6339  cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  0gc0g 16713  Mgmcmgm 17850  Mndcmnd 17911   MgmHom cmgmhm 44327
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mgmhm 44329
This theorem is referenced by:  c0rnghm  44467
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