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Mirrors > Home > MPE Home > Th. List > Mathboxes > c0rnghm | Structured version Visualization version GIF version |
Description: The constant mapping to zero is a nonunital ring homomorphism from any nonunital ring to the zero ring. (Contributed by AV, 17-Apr-2020.) |
Ref | Expression |
---|---|
c0mhm.b | ⊢ 𝐵 = (Base‘𝑆) |
c0mhm.0 | ⊢ 0 = (0g‘𝑇) |
c0mhm.h | ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) |
Ref | Expression |
---|---|
c0rnghm | ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RngHomo 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringssrng 44142 | . . . . . 6 ⊢ Ring ⊆ Rng | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑆 ∈ Rng → Ring ⊆ Rng) |
3 | 2 | ssdifssd 4117 | . . . 4 ⊢ (𝑆 ∈ Rng → (Ring ∖ NzRing) ⊆ Rng) |
4 | 3 | sseld 3964 | . . 3 ⊢ (𝑆 ∈ Rng → (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Rng)) |
5 | 4 | imdistani 571 | . 2 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑆 ∈ Rng ∧ 𝑇 ∈ Rng)) |
6 | rngabl 44139 | . . . . 5 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel) | |
7 | ablgrp 18903 | . . . . 5 ⊢ (𝑆 ∈ Abel → 𝑆 ∈ Grp) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Grp) |
9 | eldifi 4101 | . . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring) | |
10 | ringgrp 19294 | . . . . 5 ⊢ (𝑇 ∈ Ring → 𝑇 ∈ Grp) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Grp) |
12 | c0mhm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
13 | c0mhm.0 | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
14 | c0mhm.h | . . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) | |
15 | 12, 13, 14 | c0ghm 44173 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇)) |
16 | 8, 11, 15 | syl2an 597 | . . 3 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 GrpHom 𝑇)) |
17 | eqid 2819 | . . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
18 | eqid 2819 | . . . . . . . . 9 ⊢ (1r‘𝑇) = (1r‘𝑇) | |
19 | 17, 13, 18 | 0ring1eq0 44134 | . . . . . . . 8 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → (1r‘𝑇) = 0 ) |
20 | 19 | eqcomd 2825 | . . . . . . 7 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → 0 = (1r‘𝑇)) |
21 | 20 | mpteq2dv 5153 | . . . . . 6 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → (𝑥 ∈ 𝐵 ↦ 0 ) = (𝑥 ∈ 𝐵 ↦ (1r‘𝑇))) |
22 | 21 | adantl 484 | . . . . 5 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ 𝐵 ↦ 0 ) = (𝑥 ∈ 𝐵 ↦ (1r‘𝑇))) |
23 | 14, 22 | syl5eq 2866 | . . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 = (𝑥 ∈ 𝐵 ↦ (1r‘𝑇))) |
24 | eqid 2819 | . . . . . . 7 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
25 | 24 | rngmgp 44140 | . . . . . 6 ⊢ (𝑆 ∈ Rng → (mulGrp‘𝑆) ∈ Smgrp) |
26 | sgrpmgm 17898 | . . . . . 6 ⊢ ((mulGrp‘𝑆) ∈ Smgrp → (mulGrp‘𝑆) ∈ Mgm) | |
27 | 25, 26 | syl 17 | . . . . 5 ⊢ (𝑆 ∈ Rng → (mulGrp‘𝑆) ∈ Mgm) |
28 | eqid 2819 | . . . . . . 7 ⊢ (mulGrp‘𝑇) = (mulGrp‘𝑇) | |
29 | 28 | ringmgp 19295 | . . . . . 6 ⊢ (𝑇 ∈ Ring → (mulGrp‘𝑇) ∈ Mnd) |
30 | 9, 29 | syl 17 | . . . . 5 ⊢ (𝑇 ∈ (Ring ∖ NzRing) → (mulGrp‘𝑇) ∈ Mnd) |
31 | 24, 12 | mgpbas 19237 | . . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑆)) |
32 | 28, 18 | ringidval 19245 | . . . . . 6 ⊢ (1r‘𝑇) = (0g‘(mulGrp‘𝑇)) |
33 | eqid 2819 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ (1r‘𝑇)) = (𝑥 ∈ 𝐵 ↦ (1r‘𝑇)) | |
34 | 31, 32, 33 | c0mgm 44171 | . . . . 5 ⊢ (((mulGrp‘𝑆) ∈ Mgm ∧ (mulGrp‘𝑇) ∈ Mnd) → (𝑥 ∈ 𝐵 ↦ (1r‘𝑇)) ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇))) |
35 | 27, 30, 34 | syl2an 597 | . . . 4 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑥 ∈ 𝐵 ↦ (1r‘𝑇)) ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇))) |
36 | 23, 35 | eqeltrd 2911 | . . 3 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇))) |
37 | 16, 36 | jca 514 | . 2 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝐻 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐻 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇)))) |
38 | 24, 28 | isrnghmmul 44155 | . 2 ⊢ (𝐻 ∈ (𝑆 RngHomo 𝑇) ↔ ((𝑆 ∈ Rng ∧ 𝑇 ∈ Rng) ∧ (𝐻 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐻 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑇))))) |
39 | 5, 37, 38 | sylanbrc 585 | 1 ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RngHomo 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∖ cdif 3931 ⊆ wss 3934 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 0gc0g 16705 Mgmcmgm 17842 Smgrpcsgrp 17892 Mndcmnd 17903 Grpcgrp 18095 GrpHom cghm 18347 Abelcabl 18899 mulGrpcmgp 19231 1rcur 19243 Ringcrg 19289 NzRingcnzr 20022 MgmHom cmgmhm 44035 Rngcrng 44136 RngHomo crngh 44147 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-dju 9322 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-n0 11890 df-xnn0 11960 df-z 11974 df-uz 12236 df-fz 12885 df-hash 13683 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-ghm 18348 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-nzr 20023 df-mgmhm 44037 df-rng0 44137 df-rnghomo 44149 |
This theorem is referenced by: zrtermorngc 44263 |
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