Theorem mhmismgmhm 18761

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 18711	. . . 4 ⊢ (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
2		mndmgm 18711	. . . 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 2737	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		eqid 2737	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
8		eqid 2737	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
9		eqid 2737	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
10		eqid 2737	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
11		eqid 2737	. . 3 ⊢ (0g‘𝑆) = (0g‘𝑆)
12	6, 7, 8, 9, 10, 11	ismhm 18755	. 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 18666	. 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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ⟶wf 6496  ‘cfv 6500  (class class class)co 7369  Basecbs 17181  +_gcplusg 17222  0_gc0g 17404  Mgmcmgm 18608   MgmHom cmgmhm 18660  Mndcmnd 18704   MndHom cmhm 18751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8777  df-mgmhm 18662  df-sgrp 18689  df-mnd 18705  df-mhm 18753

This theorem is referenced by:  rhmisrnghm  20462