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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 1130 | . 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → (𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd)) |
3 | simp1 1134 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑆 ∈ Mnd) | |
4 | mndmgm 18700 | . . . . 5 ⊢ (𝑇 ∈ Mnd → 𝑇 ∈ Mgm) | |
5 | 4 | 3ad2ant2 1132 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑇 ∈ Mgm) |
6 | fveq2 6897 | . . . . . 6 ⊢ (𝐵 = {𝑍} → (♯‘𝐵) = (♯‘{𝑍})) | |
7 | c0snmhm.z | . . . . . . . 8 ⊢ 𝑍 = (0g‘𝑇) | |
8 | 7 | fvexi 6911 | . . . . . . 7 ⊢ 𝑍 ∈ V |
9 | hashsng 14360 | . . . . . . 7 ⊢ (𝑍 ∈ V → (♯‘{𝑍}) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{𝑍}) = 1 |
11 | 6, 10 | eqtrdi 2784 | . . . . 5 ⊢ (𝐵 = {𝑍} → (♯‘𝐵) = 1) |
12 | 11 | 3ad2ant3 1133 | . . . 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 20400 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆)) |
17 | 3, 5, 12, 16 | syl3anc 1369 | . . 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 2729 | . . . 4 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) ∧ 𝑥 = 𝑍) → 0 = 0 ) | |
20 | 8 | snid 4665 | . . . . . 6 ⊢ 𝑍 ∈ {𝑍} |
21 | eleq2 2818 | . . . . . 6 ⊢ (𝐵 = {𝑍} → (𝑍 ∈ 𝐵 ↔ 𝑍 ∈ {𝑍})) | |
22 | 20, 21 | mpbiri 258 | . . . . 5 ⊢ (𝐵 = {𝑍} → 𝑍 ∈ 𝐵) |
23 | 22 | 3ad2ant3 1133 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝑍 ∈ 𝐵) |
24 | eqid 2728 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
25 | 24, 14 | mndidcl 18708 | . . . . 5 ⊢ (𝑆 ∈ Mnd → 0 ∈ (Base‘𝑆)) |
26 | 25 | 3ad2ant1 1131 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 0 ∈ (Base‘𝑆)) |
27 | 18, 19, 23, 26 | fvmptd 7012 | . . 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 2728 | . . 3 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
30 | eqid 2728 | . . 3 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
31 | 13, 24, 29, 30, 7, 14 | ismhm0 18746 | . 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 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3471 {csn 4629 ↦ cmpt 5231 ‘cfv 6548 (class class class)co 7420 1c1 11139 ♯chash 14321 Basecbs 17179 +gcplusg 17232 0gc0g 17420 Mgmcmgm 18597 MgmHom cmgmhm 18649 Mndcmnd 18693 MndHom cmhm 18737 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-oadd 8490 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-hash 14322 df-0g 17422 df-mgm 18599 df-mgmhm 18651 df-sgrp 18678 df-mnd 18694 df-mhm 18739 |
This theorem is referenced by: c0snghm 20402 |
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