Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > c0snmhm | Structured version Visualization version GIF version |
Description: The constant mapping to zero is a monoid homomorphism from the trivial monoid (consisting of the zero only) to any monoid. (Contributed by AV, 17-Apr-2020.) |
Ref | Expression |
---|---|
zrrhm.b | ⊢ 𝐵 = (Base‘𝑇) |
zrrhm.0 | ⊢ 0 = (0g‘𝑆) |
zrrhm.h | ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) |
c0snmhm.z | ⊢ 𝑍 = (0g‘𝑇) |
Ref | Expression |
---|---|
c0snmhm | ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MndHom 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.22 462 | . . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd)) | |
2 | 1 | 3adant3 1128 | . 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd)) |
3 | simp1 1132 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑆 ∈ Mnd) | |
4 | mndmgm 17912 | . . . . 5 ⊢ (𝑇 ∈ Mnd → 𝑇 ∈ Mgm) | |
5 | 4 | 3ad2ant2 1130 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑇 ∈ Mgm) |
6 | fveq2 6664 | . . . . . 6 ⊢ (𝐵 = {𝑍} → (♯‘𝐵) = (♯‘{𝑍})) | |
7 | c0snmhm.z | . . . . . . . 8 ⊢ 𝑍 = (0g‘𝑇) | |
8 | 7 | fvexi 6678 | . . . . . . 7 ⊢ 𝑍 ∈ V |
9 | hashsng 13724 | . . . . . . 7 ⊢ (𝑍 ∈ V → (♯‘{𝑍}) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{𝑍}) = 1 |
11 | 6, 10 | syl6eq 2872 | . . . . 5 ⊢ (𝐵 = {𝑍} → (♯‘𝐵) = 1) |
12 | 11 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (♯‘𝐵) = 1) |
13 | zrrhm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑇) | |
14 | zrrhm.0 | . . . . 5 ⊢ 0 = (0g‘𝑆) | |
15 | zrrhm.h | . . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) | |
16 | 13, 14, 15 | c0snmgmhm 44179 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆)) |
17 | 3, 5, 12, 16 | syl3anc 1367 | . . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MgmHom 𝑆)) |
18 | 15 | a1i 11 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )) |
19 | eqidd 2822 | . . . 4 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) ∧ 𝑥 = 𝑍) → 0 = 0 ) | |
20 | 8 | snid 4594 | . . . . . 6 ⊢ 𝑍 ∈ {𝑍} |
21 | eleq2 2901 | . . . . . 6 ⊢ (𝐵 = {𝑍} → (𝑍 ∈ 𝐵 ↔ 𝑍 ∈ {𝑍})) | |
22 | 20, 21 | mpbiri 260 | . . . . 5 ⊢ (𝐵 = {𝑍} → 𝑍 ∈ 𝐵) |
23 | 22 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑍 ∈ 𝐵) |
24 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
25 | 24, 14 | mndidcl 17920 | . . . . 5 ⊢ (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆)) |
26 | 25 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 0 ∈ (Base‘𝑆)) |
27 | 18, 19, 23, 26 | fvmptd 6769 | . . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (𝐻‘𝑍) = 0 ) |
28 | 17, 27 | jca 514 | . 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (𝐻 ∈ (𝑇 MgmHom 𝑆) ∧ (𝐻‘𝑍) = 0 )) |
29 | eqid 2821 | . . 3 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
30 | eqid 2821 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
31 | 13, 24, 29, 30, 7, 14 | ismhm0 44066 | . 2 ⊢ (𝐻 ∈ (𝑇 MndHom 𝑆) ↔ ((𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝐻 ∈ (𝑇 MgmHom 𝑆) ∧ (𝐻‘𝑍) = 0 ))) |
32 | 2, 28, 31 | sylanbrc 585 | 1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MndHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {csn 4560 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 1c1 10532 ♯chash 13684 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Mgmcmgm 17844 Mndcmnd 17905 MndHom cmhm 17948 MgmHom cmgmhm 44038 |
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-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-mgmhm 44040 |
This theorem is referenced by: c0snghm 44181 |
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