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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 | pm3.22 463 | . . 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 17910 | . . . . 5 ⊢ (𝑇 ∈ Mnd → 𝑇 ∈ Mgm) | |
5 | 4 | 3ad2ant2 1131 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑇 ∈ Mgm) |
6 | fveq2 6645 | . . . . . 6 ⊢ (𝐵 = {𝑍} → (♯‘𝐵) = (♯‘{𝑍})) | |
7 | c0snmhm.z | . . . . . . . 8 ⊢ 𝑍 = (0g‘𝑇) | |
8 | 7 | fvexi 6659 | . . . . . . 7 ⊢ 𝑍 ∈ V |
9 | hashsng 13726 | . . . . . . 7 ⊢ (𝑍 ∈ V → (♯‘{𝑍}) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{𝑍}) = 1 |
11 | 6, 10 | eqtrdi 2849 | . . . . 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 44538 | . . . 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 2799 | . . . 4 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) ∧ 𝑥 = 𝑍) → 0 = 0 ) | |
20 | 8 | snid 4561 | . . . . . 6 ⊢ 𝑍 ∈ {𝑍} |
21 | eleq2 2878 | . . . . . 6 ⊢ (𝐵 = {𝑍} → (𝑍 ∈ 𝐵 ↔ 𝑍 ∈ {𝑍})) | |
22 | 20, 21 | mpbiri 261 | . . . . 5 ⊢ (𝐵 = {𝑍} → 𝑍 ∈ 𝐵) |
23 | 22 | 3ad2ant3 1132 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑍 ∈ 𝐵) |
24 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
25 | 24, 14 | mndidcl 17918 | . . . . 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 6752 | . . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (𝐻‘𝑍) = 0 ) |
28 | 17, 27 | jca 515 | . 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (𝐻 ∈ (𝑇 MgmHom 𝑆) ∧ (𝐻‘𝑍) = 0 )) |
29 | eqid 2798 | . . 3 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
30 | eqid 2798 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
31 | 13, 24, 29, 30, 7, 14 | ismhm0 44425 | . 2 ⊢ (𝐻 ∈ (𝑇 MndHom 𝑆) ↔ ((𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd) ∧ (𝐻 ∈ (𝑇 MgmHom 𝑆) ∧ (𝐻‘𝑍) = 0 ))) |
32 | 2, 28, 31 | sylanbrc 586 | 1 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MndHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 1c1 10527 ♯chash 13686 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Mgmcmgm 17842 Mndcmnd 17903 MndHom cmhm 17946 MgmHom cmgmhm 44397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-mgmhm 44399 |
This theorem is referenced by: c0snghm 44540 |
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