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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 | pm3.22 458 | . . 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 18729 | . . . . 5 ⊢ (𝑇 ∈ Mnd → 𝑇 ∈ Mgm) | |
5 | 4 | 3ad2ant2 1131 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑇 ∈ Mgm) |
6 | fveq2 6893 | . . . . . 6 ⊢ (𝐵 = {𝑍} → (♯‘𝐵) = (♯‘{𝑍})) | |
7 | c0snmhm.z | . . . . . . . 8 ⊢ 𝑍 = (0g‘𝑇) | |
8 | 7 | fvexi 6907 | . . . . . . 7 ⊢ 𝑍 ∈ V |
9 | hashsng 14381 | . . . . . . 7 ⊢ (𝑍 ∈ V → (♯‘{𝑍}) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{𝑍}) = 1 |
11 | 6, 10 | eqtrdi 2782 | . . . . 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 20440 | . . . 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 2727 | . . . 4 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) ∧ 𝑥 = 𝑍) → 0 = 0 ) | |
20 | 8 | snid 4659 | . . . . . 6 ⊢ 𝑍 ∈ {𝑍} |
21 | eleq2 2815 | . . . . . 6 ⊢ (𝐵 = {𝑍} → (𝑍 ∈ 𝐵 ↔ 𝑍 ∈ {𝑍})) | |
22 | 20, 21 | mpbiri 257 | . . . . 5 ⊢ (𝐵 = {𝑍} → 𝑍 ∈ 𝐵) |
23 | 22 | 3ad2ant3 1132 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑍 ∈ 𝐵) |
24 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
25 | 24, 14 | mndidcl 18737 | . . . . 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 7008 | . . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (𝐻‘𝑍) = 0 ) |
28 | 17, 27 | jca 510 | . 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (𝐻 ∈ (𝑇 MgmHom 𝑆) ∧ (𝐻‘𝑍) = 0 )) |
29 | eqid 2726 | . . 3 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
30 | eqid 2726 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
31 | 13, 24, 29, 30, 7, 14 | ismhm0 18775 | . 2 ⊢ (𝐻 ∈ (𝑇 MndHom 𝑆) ↔ ((𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝐻 ∈ (𝑇 MgmHom 𝑆) ∧ (𝐻‘𝑍) = 0 ))) |
32 | 2, 28, 31 | sylanbrc 581 | 1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MndHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 Vcvv 3462 {csn 4623 ↦ cmpt 5228 ‘cfv 6546 (class class class)co 7416 1c1 11150 ♯chash 14342 Basecbs 17208 +gcplusg 17261 0gc0g 17449 Mgmcmgm 18626 MgmHom cmgmhm 18678 Mndcmnd 18722 MndHom cmhm 18766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-oadd 8492 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-dju 9937 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-hash 14343 df-0g 17451 df-mgm 18628 df-mgmhm 18680 df-sgrp 18707 df-mnd 18723 df-mhm 18768 |
This theorem is referenced by: c0snghm 20442 |
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