Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > rspsnel | Structured version Visualization version GIF version |
Description: Membership in a principal ideal. Analogous to lspsnel 20180. (Contributed by Thierry Arnoux, 15-Jan-2024.) |
Ref | Expression |
---|---|
rspsnel.1 | ⊢ 𝐵 = (Base‘𝑅) |
rspsnel.2 | ⊢ · = (.r‘𝑅) |
rspsnel.3 | ⊢ 𝐾 = (RSpan‘𝑅) |
Ref | Expression |
---|---|
rspsnel | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥 ∈ 𝐵 𝐼 = (𝑥 · 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmlmod 20388 | . . 3 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
2 | simpr 484 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
3 | rspsnel.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | eleqtrdi 2849 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑅)) |
5 | eqid 2738 | . . . 4 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅)) | |
6 | eqid 2738 | . . . 4 ⊢ (Base‘(Scalar‘(ringLMod‘𝑅))) = (Base‘(Scalar‘(ringLMod‘𝑅))) | |
7 | rlmbas 20378 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
8 | rspsnel.2 | . . . . 5 ⊢ · = (.r‘𝑅) | |
9 | rlmvsca 20385 | . . . . 5 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
10 | 8, 9 | eqtri 2766 | . . . 4 ⊢ · = ( ·𝑠 ‘(ringLMod‘𝑅)) |
11 | rspsnel.3 | . . . . 5 ⊢ 𝐾 = (RSpan‘𝑅) | |
12 | rspval 20376 | . . . . 5 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
13 | 11, 12 | eqtri 2766 | . . . 4 ⊢ 𝐾 = (LSpan‘(ringLMod‘𝑅)) |
14 | 5, 6, 7, 10, 13 | lspsnel 20180 | . . 3 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝑋 ∈ (Base‘𝑅)) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))𝐼 = (𝑥 · 𝑋))) |
15 | 1, 4, 14 | syl2an2r 681 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))𝐼 = (𝑥 · 𝑋))) |
16 | rlmsca 20383 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
17 | 16 | adantr 480 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 = (Scalar‘(ringLMod‘𝑅))) |
18 | 17 | fveq2d 6760 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
19 | 3, 18 | eqtr2id 2792 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (Base‘(Scalar‘(ringLMod‘𝑅))) = 𝐵) |
20 | 19 | rexeqdv 3340 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (∃𝑥 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))𝐼 = (𝑥 · 𝑋) ↔ ∃𝑥 ∈ 𝐵 𝐼 = (𝑥 · 𝑋))) |
21 | 15, 20 | bitrd 278 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥 ∈ 𝐵 𝐼 = (𝑥 · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∃wrex 3064 {csn 4558 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 .rcmulr 16889 Scalarcsca 16891 ·𝑠 cvsca 16892 Ringcrg 19698 LModclmod 20038 LSpanclspn 20148 ringLModcrglmod 20346 RSpancrsp 20348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-mgp 19636 df-ur 19653 df-ring 19700 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-sra 20349 df-rgmod 20350 df-rsp 20352 |
This theorem is referenced by: lsmsnpridl 31488 isprmidlc 31525 |
Copyright terms: Public domain | W3C validator |