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Theorem mhmismgmhm 46186
Description: Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.)
Assertion
Ref Expression
mhmismgmhm (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆))

Proof of Theorem mhmismgmhm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndmgm 18568 . . . 4 (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
2 mndmgm 18568 . . . 4 (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
31, 2anim12i 614 . . 3 ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm))
4 3simpa 1149 . . 3 ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ (𝐹‘(0g𝑅)) = (0g𝑆)) → (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))))
53, 4anim12i 614 . 2 (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ (𝐹‘(0g𝑅)) = (0g𝑆))) → ((𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
6 eqid 2733 . . 3 (Base‘𝑅) = (Base‘𝑅)
7 eqid 2733 . . 3 (Base‘𝑆) = (Base‘𝑆)
8 eqid 2733 . . 3 (+g𝑅) = (+g𝑅)
9 eqid 2733 . . 3 (+g𝑆) = (+g𝑆)
10 eqid 2733 . . 3 (0g𝑅) = (0g𝑅)
11 eqid 2733 . . 3 (0g𝑆) = (0g𝑆)
126, 7, 8, 9, 10, 11ismhm 18608 . 2 (𝐹 ∈ (𝑅 MndHom 𝑆) ↔ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∧ (𝐹‘(0g𝑅)) = (0g𝑆))))
136, 7, 8, 9ismgmhm 46163 . 2 (𝐹 ∈ (𝑅 MgmHom 𝑆) ↔ ((𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g𝑅)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
145, 12, 133imtr4i 292 1 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wf 6493  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Mgmcmgm 18500  Mndcmnd 18561   MndHom cmhm 18604   MgmHom cmgmhm 46157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-mgmhm 46159
This theorem is referenced by:  rhmisrnghm  46304
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