Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 18152 | . . . 4 ⊢ (𝑅 ∈ Mnd → 𝑅 ∈ Mgm) | |
2 | mndmgm 18152 | . . . 4 ⊢ (𝑆 ∈ Mnd → 𝑆 ∈ Mgm) | |
3 | 1, 2 | anim12i 616 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (𝑅 ∈ Mgm ∧ 𝑆 ∈ Mgm)) |
4 | 3simpa 1150 | . . 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 616 | . 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 2734 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2734 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
8 | eqid 2734 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
9 | eqid 2734 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
10 | eqid 2734 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
11 | eqid 2734 | . . 3 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
12 | 6, 7, 8, 9, 10, 11 | ismhm 18192 | . 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 44964 | . 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 295 | 1 ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 +gcplusg 16767 0gc0g 16916 Mgmcmgm 18084 Mndcmnd 18145 MndHom cmhm 18188 MgmHom cmgmhm 44958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-map 8499 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-mgmhm 44960 |
This theorem is referenced by: rhmisrnghm 45105 |
Copyright terms: Public domain | W3C validator |