Theorem mhmismgmhm 46576

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.

Step	Hyp	Ref	Expression
1		mndmgm 18632	. . . 4 ⊢ (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
2		mndmgm 18632	. . . 4 ⊢ (𝑆 ∈ Mnd → 𝑆 ∈ Mgm)
3	1, 2	anim12i 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(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) → (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))))
5	3, 4	anim12i 614	. 2 ⊢ (((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑆))) → ((𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))))
6		eqid 2733	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		eqid 2733	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
8		eqid 2733	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
9		eqid 2733	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
10		eqid 2733	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
11		eqid 2733	. . 3 ⊢ (0g‘𝑆) = (0g‘𝑆)
12	6, 7, 8, 9, 10, 11	ismhm 18673	. 2 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) ↔ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑆))))
13	6, 7, 8, 9	ismgmhm 46553	. 2 ⊢ (𝐹 ∈ (𝑅 MgmHom 𝑆) ↔ ((𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))))
14	5, 12, 13	3imtr4i 292	1 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  +_gcplusg 17197  0_gc0g 17385  Mgmcmgm 18559  Mndcmnd 18625   MndHom cmhm 18669   MgmHom cmgmhm 46547

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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-mgmhm 46549

This theorem is referenced by:  rhmisrnghm  46723