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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 18983 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 18983 | . 2 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
9 | fof 6805 | . . . 4 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) | |
10 | 7, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
11 | 2 | 3expb 1120 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
12 | 11 | ralrimivva 3200 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
13 | eqid 2732 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
14 | 2, 3, 4, 5, 6, 7, 1, 13 | mhmid 18982 | . . 3 ⊢ (𝜑 → (𝐹‘(0g‘𝐺)) = (0g‘𝐻)) |
15 | 10, 12, 14 | 3jca 1128 | . 2 ⊢ (𝜑 → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝐺)) = (0g‘𝐻))) |
16 | eqid 2732 | . . 3 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
17 | 3, 4, 5, 6, 13, 16 | ismhm 18707 | . 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 1344 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐺 MndHom 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⟶wf 6539 –onto→wfo 6541 ‘cfv 6543 (class class class)co 7411 Basecbs 17148 +gcplusg 17201 0gc0g 17389 Mndcmnd 18659 MndHom cmhm 18703 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-map 8824 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 |
This theorem is referenced by: (None) |
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