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Mirrors > Home > MPE Home > Th. List > mhmfmhm | Structured version Visualization version GIF version |
Description: The function fulfilling the conditions of mhmmnd 18215 is a monoid homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
Ref | Expression |
---|---|
ghmgrp.f | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
ghmgrp.x | ⊢ 𝑋 = (Base‘𝐺) |
ghmgrp.y | ⊢ 𝑌 = (Base‘𝐻) |
ghmgrp.p | ⊢ + = (+g‘𝐺) |
ghmgrp.q | ⊢ ⨣ = (+g‘𝐻) |
ghmgrp.1 | ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) |
mhmmnd.3 | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
Ref | Expression |
---|---|
mhmfmhm | ⊢ (𝜑 → 𝐹 ∈ (𝐺 MndHom 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhmmnd.3 | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
2 | ghmgrp.f | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) | |
3 | ghmgrp.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
4 | ghmgrp.y | . . 3 ⊢ 𝑌 = (Base‘𝐻) | |
5 | ghmgrp.p | . . 3 ⊢ + = (+g‘𝐺) | |
6 | ghmgrp.q | . . 3 ⊢ ⨣ = (+g‘𝐻) | |
7 | ghmgrp.1 | . . 3 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) | |
8 | 2, 3, 4, 5, 6, 7, 1 | mhmmnd 18215 | . 2 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
9 | fof 6584 | . . . 4 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) | |
10 | 7, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
11 | 2 | 3expb 1116 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
12 | 11 | ralrimivva 3191 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
13 | eqid 2821 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
14 | 2, 3, 4, 5, 6, 7, 1, 13 | mhmid 18214 | . . 3 ⊢ (𝜑 → (𝐹‘(0g‘𝐺)) = (0g‘𝐻)) |
15 | 10, 12, 14 | 3jca 1124 | . 2 ⊢ (𝜑 → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝐺)) = (0g‘𝐻))) |
16 | eqid 2821 | . . 3 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
17 | 3, 4, 5, 6, 13, 16 | ismhm 17952 | . 2 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) ↔ ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝐺)) = (0g‘𝐻)))) |
18 | 1, 8, 15, 17 | syl21anbrc 1340 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐺 MndHom 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⟶wf 6345 –onto→wfo 6347 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Mndcmnd 17905 MndHom cmhm 17948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 |
This theorem is referenced by: (None) |
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