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Mirrors > Home > ILE Home > Th. List > mhmfmhm | GIF version |
Description: The function fulfilling the conditions of mhmmnd 12869 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 12869 | . 2 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
9 | fof 5434 | . . . 4 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) | |
10 | 7, 9 | syl 14 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
11 | 2 | 3expb 1204 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
12 | 11 | ralrimivva 2559 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
13 | eqid 2177 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
14 | 2, 3, 4, 5, 6, 7, 1, 13 | mhmid 12868 | . . 3 ⊢ (𝜑 → (𝐹‘(0g‘𝐺)) = (0g‘𝐻)) |
15 | 10, 12, 14 | 3jca 1177 | . 2 ⊢ (𝜑 → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝐺)) = (0g‘𝐻))) |
16 | eqid 2177 | . . 3 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
17 | 3, 4, 5, 6, 13, 16 | ismhm 12743 | . 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 1182 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐺 MndHom 𝐻)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ⟶wf 5208 –onto→wfo 5210 ‘cfv 5212 (class class class)co 5869 Basecbs 12445 +gcplusg 12518 0gc0g 12653 Mndcmnd 12709 MndHom cmhm 12739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1re 7896 ax-addrcl 7899 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fo 5218 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-map 6644 df-inn 8909 df-2 8967 df-ndx 12448 df-slot 12449 df-base 12451 df-plusg 12531 df-0g 12655 df-mgm 12667 df-sgrp 12700 df-mnd 12710 df-mhm 12741 |
This theorem is referenced by: (None) |
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