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Mirrors > Home > MPE Home > Th. List > rsp0 | Structured version Visualization version GIF version |
Description: The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
Ref | Expression |
---|---|
rspcl.k | ⊢ 𝐾 = (RSpan‘𝑅) |
rsp0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
rsp0 | ⊢ (𝑅 ∈ Ring → (𝐾‘{ 0 }) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmlmod 19699 | . 2 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
2 | rsp0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
3 | rlm0 19691 | . . . 4 ⊢ (0g‘𝑅) = (0g‘(ringLMod‘𝑅)) | |
4 | 2, 3 | eqtri 2802 | . . 3 ⊢ 0 = (0g‘(ringLMod‘𝑅)) |
5 | rspcl.k | . . . 4 ⊢ 𝐾 = (RSpan‘𝑅) | |
6 | rspval 19687 | . . . 4 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
7 | 5, 6 | eqtri 2802 | . . 3 ⊢ 𝐾 = (LSpan‘(ringLMod‘𝑅)) |
8 | 4, 7 | lspsn0 19502 | . 2 ⊢ ((ringLMod‘𝑅) ∈ LMod → (𝐾‘{ 0 }) = { 0 }) |
9 | 1, 8 | syl 17 | 1 ⊢ (𝑅 ∈ Ring → (𝐾‘{ 0 }) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 {csn 4441 ‘cfv 6188 0gc0g 16569 Ringcrg 19020 LModclmod 19356 LSpanclspn 19465 ringLModcrglmod 19663 RSpancrsp 19665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-ip 16439 df-0g 16571 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-grp 17894 df-subg 18060 df-mgp 18963 df-ur 18975 df-ring 19022 df-subrg 19256 df-lmod 19358 df-lss 19426 df-lsp 19466 df-sra 19666 df-rgmod 19667 df-rsp 19669 |
This theorem is referenced by: lpi0 19741 |
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