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Mirrors > Home > MPE Home > Th. List > 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 | pm3.22 459 | . . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd)) | |
2 | 1 | 3adant3 1129 | . 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd)) |
3 | simp1 1133 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑆 ∈ Mnd) | |
4 | mndmgm 18670 | . . . . 5 ⊢ (𝑇 ∈ Mnd → 𝑇 ∈ Mgm) | |
5 | 4 | 3ad2ant2 1131 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑇 ∈ Mgm) |
6 | fveq2 6882 | . . . . . 6 ⊢ (𝐵 = {𝑍} → (♯‘𝐵) = (♯‘{𝑍})) | |
7 | c0snmhm.z | . . . . . . . 8 ⊢ 𝑍 = (0g‘𝑇) | |
8 | 7 | fvexi 6896 | . . . . . . 7 ⊢ 𝑍 ∈ V |
9 | hashsng 14330 | . . . . . . 7 ⊢ (𝑍 ∈ V → (♯‘{𝑍}) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{𝑍}) = 1 |
11 | 6, 10 | eqtrdi 2780 | . . . . 5 ⊢ (𝐵 = {𝑍} → (♯‘𝐵) = 1) |
12 | 11 | 3ad2ant3 1132 | . . . 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 20360 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆)) |
17 | 3, 5, 12, 16 | syl3anc 1368 | . . 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 2725 | . . . 4 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) ∧ 𝑥 = 𝑍) → 0 = 0 ) | |
20 | 8 | snid 4657 | . . . . . 6 ⊢ 𝑍 ∈ {𝑍} |
21 | eleq2 2814 | . . . . . 6 ⊢ (𝐵 = {𝑍} → (𝑍 ∈ 𝐵 ↔ 𝑍 ∈ {𝑍})) | |
22 | 20, 21 | mpbiri 258 | . . . . 5 ⊢ (𝐵 = {𝑍} → 𝑍 ∈ 𝐵) |
23 | 22 | 3ad2ant3 1132 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑍 ∈ 𝐵) |
24 | eqid 2724 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
25 | 24, 14 | mndidcl 18678 | . . . . 5 ⊢ (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆)) |
26 | 25 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 0 ∈ (Base‘𝑆)) |
27 | 18, 19, 23, 26 | fvmptd 6996 | . . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (𝐻‘𝑍) = 0 ) |
28 | 17, 27 | jca 511 | . 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (𝐻 ∈ (𝑇 MgmHom 𝑆) ∧ (𝐻‘𝑍) = 0 )) |
29 | eqid 2724 | . . 3 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
30 | eqid 2724 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
31 | 13, 24, 29, 30, 7, 14 | ismhm0 18716 | . 2 ⊢ (𝐻 ∈ (𝑇 MndHom 𝑆) ↔ ((𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝐻 ∈ (𝑇 MgmHom 𝑆) ∧ (𝐻‘𝑍) = 0 ))) |
32 | 2, 28, 31 | sylanbrc 582 | 1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MndHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3466 {csn 4621 ↦ cmpt 5222 ‘cfv 6534 (class class class)co 7402 1c1 11108 ♯chash 14291 Basecbs 17149 +gcplusg 17202 0gc0g 17390 Mgmcmgm 18567 MgmHom cmgmhm 18619 Mndcmnd 18663 MndHom cmhm 18707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-dju 9893 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-hash 14292 df-0g 17392 df-mgm 18569 df-mgmhm 18621 df-sgrp 18648 df-mnd 18664 df-mhm 18709 |
This theorem is referenced by: c0snghm 20362 |
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