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Mirrors > Home > MPE Home > Th. List > Mathboxes > mhmismgmhm | Structured version Visualization version GIF version |
Description: Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.) |
Ref | Expression |
---|---|
mhmismgmhm | ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndmgm 17521 | . . . 4 ⊢ (𝑅 ∈ Mnd → 𝑅 ∈ Mgm) | |
2 | mndmgm 17521 | . . . 4 ⊢ (𝑆 ∈ Mnd → 𝑆 ∈ Mgm) | |
3 | 1, 2 | anim12i 591 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm)) |
4 | 3simpa 1143 | . . 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 591 | . 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 2760 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2760 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
8 | eqid 2760 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
9 | eqid 2760 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
10 | eqid 2760 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
11 | eqid 2760 | . . 3 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
12 | 6, 7, 8, 9, 10, 11 | ismhm 17558 | . 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 42311 | . 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 281 | 1 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 +gcplusg 16163 0gc0g 16322 Mgmcmgm 17461 Mndcmnd 17515 MndHom cmhm 17554 MgmHom cmgmhm 42305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-map 8027 df-sgrp 17505 df-mnd 17516 df-mhm 17556 df-mgmhm 42307 |
This theorem is referenced by: rhmisrnghm 42448 |
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