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Mirrors > Home > ILE Home > Th. List > mhmfmhm | GIF version |
Description: The function fulfilling the conditions of mhmmnd 13008 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 13008 | . 2 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
9 | fof 5450 | . . . 4 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) | |
10 | 7, 9 | syl 14 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
11 | 2 | 3expb 1205 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
12 | 11 | ralrimivva 2569 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
13 | eqid 2187 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
14 | 2, 3, 4, 5, 6, 7, 1, 13 | mhmid 13007 | . . 3 ⊢ (𝜑 → (𝐹‘(0g‘𝐺)) = (0g‘𝐻)) |
15 | 10, 12, 14 | 3jca 1178 | . 2 ⊢ (𝜑 → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝐺)) = (0g‘𝐻))) |
16 | eqid 2187 | . . 3 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
17 | 3, 4, 5, 6, 13, 16 | ismhm 12872 | . 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 1183 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐺 MndHom 𝐻)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 979 = wceq 1363 ∈ wcel 2158 ∀wral 2465 ⟶wf 5224 –onto→wfo 5226 ‘cfv 5228 (class class class)co 5888 Basecbs 12475 +gcplusg 12550 0gc0g 12722 Mndcmnd 12836 MndHom cmhm 12868 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fo 5234 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-map 6663 df-inn 8933 df-2 8991 df-ndx 12478 df-slot 12479 df-base 12481 df-plusg 12563 df-0g 12724 df-mgm 12793 df-sgrp 12826 df-mnd 12837 df-mhm 12870 |
This theorem is referenced by: (None) |
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