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Theorem c0mgm 42434
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 17508 . . 3 (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
21anim2i 603 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3 eqid 2771 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
4 c0mhm.0 . . . . . . 7 0 = (0g𝑇)
53, 4mndidcl 17516 . . . . . 6 (𝑇 ∈ Mnd → 0 ∈ (Base‘𝑇))
65adantl 467 . . . . 5 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 0 ∈ (Base‘𝑇))
76adantr 466 . . . 4 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ 𝑥𝐵) → 0 ∈ (Base‘𝑇))
8 c0mhm.h . . . 4 𝐻 = (𝑥𝐵0 )
97, 8fmptd 6529 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
105ancli 538 . . . . . . . 8 (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
1110adantl 467 . . . . . . 7 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
12 eqid 2771 . . . . . . . 8 (+g𝑇) = (+g𝑇)
133, 12, 4mndlid 17519 . . . . . . 7 ((𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)) → ( 0 (+g𝑇) 0 ) = 0 )
1411, 13syl 17 . . . . . 6 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → ( 0 (+g𝑇) 0 ) = 0 )
1514adantr 466 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ( 0 (+g𝑇) 0 ) = 0 )
168a1i 11 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝐻 = (𝑥𝐵0 ))
17 eqidd 2772 . . . . . . 7 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
18 simprl 754 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
196adantr 466 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 0 ∈ (Base‘𝑇))
2016, 17, 18, 19fvmptd 6432 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑎) = 0 )
21 eqidd 2772 . . . . . . 7 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
22 simprr 756 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
2316, 21, 22, 19fvmptd 6432 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑏) = 0 )
2420, 23oveq12d 6814 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) = ( 0 (+g𝑇) 0 ))
25 eqidd 2772 . . . . . 6 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = (𝑎(+g𝑆)𝑏)) → 0 = 0 )
26 c0mhm.b . . . . . . . . 9 𝐵 = (Base‘𝑆)
27 eqid 2771 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
2826, 27mgmcl 17453 . . . . . . . 8 ((𝑆 ∈ Mgm ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
29283expb 1113 . . . . . . 7 ((𝑆 ∈ Mgm ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
3029adantlr 694 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
3116, 25, 30, 19fvmptd 6432 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = 0 )
3215, 24, 313eqtr4rd 2816 . . . 4 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
3332ralrimivva 3120 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
349, 33jca 501 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏))))
3526, 3, 27, 12ismgmhm 42308 . 2 (𝐻 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))))
362, 34, 35sylanbrc 572 1 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  cmpt 4864  wf 6026  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Mgmcmgm 17448  Mndcmnd 17502   MgmHom cmgmhm 42302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-map 8015  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mgmhm 42304
This theorem is referenced by:  c0rnghm  42438
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