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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 18213 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 18213 | . 2 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
9 | fof 6565 | . . . 4 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) | |
10 | 7, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
11 | 2 | 3expb 1117 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
12 | 11 | ralrimivva 3156 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
13 | eqid 2798 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
14 | 2, 3, 4, 5, 6, 7, 1, 13 | mhmid 18212 | . . 3 ⊢ (𝜑 → (𝐹‘(0g‘𝐺)) = (0g‘𝐻)) |
15 | 10, 12, 14 | 3jca 1125 | . 2 ⊢ (𝜑 → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝐺)) = (0g‘𝐻))) |
16 | eqid 2798 | . . 3 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
17 | 3, 4, 5, 6, 13, 16 | ismhm 17950 | . 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 1341 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐺 MndHom 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⟶wf 6320 –onto→wfo 6322 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Mndcmnd 17903 MndHom cmhm 17946 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 |
This theorem is referenced by: (None) |
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