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Theorem c0mgm 41699
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 17123 . . 3 (𝑇 ∈ Mnd → 𝑇 ∈ Mgm)
21anim2i 591 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
3 eqid 2610 . . . . . . 7 (Base‘𝑇) = (Base‘𝑇)
4 c0mhm.0 . . . . . . 7 0 = (0g𝑇)
53, 4mndidcl 17131 . . . . . 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 6292 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻:𝐵⟶(Base‘𝑇))
105ancli 572 . . . . . . . 8 (𝑇 ∈ Mnd → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
1110adantl 481 . . . . . . 7 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 0 ∈ (Base‘𝑇)))
12 eqid 2610 . . . . . . . 8 (+g𝑇) = (+g𝑇)
133, 12, 4mndlid 17134 . . . . . . 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 2611 . . . . . . 7 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑎) → 0 = 0 )
18 simprl 790 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
196adantr 480 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 0 ∈ (Base‘𝑇))
2016, 17, 18, 19fvmptd 6197 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑎) = 0 )
21 eqidd 2611 . . . . . . 7 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = 𝑏) → 0 = 0 )
22 simprr 792 . . . . . . 7 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
2316, 21, 22, 19fvmptd 6197 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻𝑏) = 0 )
2420, 23oveq12d 6567 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → ((𝐻𝑎)(+g𝑇)(𝐻𝑏)) = ( 0 (+g𝑇) 0 ))
25 eqidd 2611 . . . . . 6 ((((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) ∧ 𝑥 = (𝑎(+g𝑆)𝑏)) → 0 = 0 )
26 c0mhm.b . . . . . . . . 9 𝐵 = (Base‘𝑆)
27 eqid 2610 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
2826, 27mgmcl 17068 . . . . . . . 8 ((𝑆 ∈ Mgm ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
29283expb 1258 . . . . . . 7 ((𝑆 ∈ Mgm ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
3029adantlr 747 . . . . . 6 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑆)𝑏) ∈ 𝐵)
3116, 25, 30, 19fvmptd 6197 . . . . 5 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = 0 )
3215, 24, 313eqtr4rd 2655 . . . 4 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) ∧ (𝑎𝐵𝑏𝐵)) → (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
3332ralrimivva 2954 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))
349, 33jca 553 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏))))
3526, 3, 27, 12ismgmhm 41573 . 2 (𝐻 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐻:𝐵⟶(Base‘𝑇) ∧ ∀𝑎𝐵𝑏𝐵 (𝐻‘(𝑎(+g𝑆)𝑏)) = ((𝐻𝑎)(+g𝑇)(𝐻𝑏)))))
362, 34, 35sylanbrc 695 1 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Mgmcmgm 17063  Mndcmnd 17117   MgmHom cmgmhm 41567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mgmhm 41569
This theorem is referenced by:  c0rnghm  41703
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