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Theorem c0mgm 45186
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 18213 . . 3 (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
21anim2i 620 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3 eqid 2739 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
4 c0mhm.0 . . . . . . 7 0 = (0g𝑇)
53, 4mndidcl 18221 . . . . . 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 6953 . . 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 2739 . . . . . . . 8 (+g𝑇) = (+g𝑇)
133, 12, 4mndlid 18226 . . . . . . 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 2740 . . . . . . 7 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
18 simprl 771 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
196adantr 484 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 0 ∈ (Base‘𝑇))
2016, 17, 18, 19fvmptd 6847 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑎) = 0 )
21 eqidd 2740 . . . . . . 7 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
22 simprr 773 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
2316, 21, 22, 19fvmptd 6847 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑏) = 0 )
2420, 23oveq12d 7253 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) = ( 0 (+g𝑇) 0 ))
25 eqidd 2740 . . . . . 6 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = (𝑎(+g𝑆)𝑏)) → 0 = 0 )
26 c0mhm.b . . . . . . . . 9 𝐵 = (Base‘𝑆)
27 eqid 2739 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
2826, 27mgmcl 18150 . . . . . . . 8 ((𝑆 ∈ Mgm ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
29283expb 1122 . . . . . . 7 ((𝑆 ∈ Mgm ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
3029adantlr 715 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
3116, 25, 30, 19fvmptd 6847 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = 0 )
3215, 24, 313eqtr4rd 2790 . . . 4 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
3332ralrimivva 3115 . . 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 45056 . 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 1543  wcel 2112  wral 3064  cmpt 5152  wf 6397  cfv 6401  (class class class)co 7235  Basecbs 16793  +gcplusg 16835  0gc0g 16977  Mgmcmgm 18145  Mndcmnd 18206   MgmHom cmgmhm 45050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5209  ax-nul 5216  ax-pow 5275  ax-pr 5339  ax-un 7545
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5153  df-id 5472  df-xp 5575  df-rel 5576  df-cnv 5577  df-co 5578  df-dm 5579  df-rn 5580  df-res 5581  df-ima 5582  df-iota 6359  df-fun 6403  df-fn 6404  df-f 6405  df-fv 6409  df-riota 7192  df-ov 7238  df-oprab 7239  df-mpo 7240  df-map 8534  df-0g 16979  df-mgm 18147  df-sgrp 18196  df-mnd 18207  df-mgmhm 45052
This theorem is referenced by:  c0rnghm  45190
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