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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnasclg | Structured version Visualization version GIF version |
Description: The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
rnasclg.a | ⊢ 𝐴 = (algSc‘𝑊) |
rnasclg.o | ⊢ 1 = (1r‘𝑊) |
rnasclg.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
rnasclg | ⊢ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) → ran 𝐴 = (𝑁‘{ 1 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnasclg.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑊) | |
2 | eqid 2737 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2737 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
4 | eqid 2737 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | rnasclg.o | . . . 4 ⊢ 1 = (1r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | asclfval 21234 | . . 3 ⊢ 𝐴 = (𝑦 ∈ (Base‘(Scalar‘𝑊)) ↦ (𝑦( ·𝑠 ‘𝑊) 1 )) |
7 | 6 | rnmpt 5908 | . 2 ⊢ ran 𝐴 = {𝑥 ∣ ∃𝑦 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑦( ·𝑠 ‘𝑊) 1 )} |
8 | eqid 2737 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
9 | 8, 5 | ringidcl 19942 | . . 3 ⊢ (𝑊 ∈ Ring → 1 ∈ (Base‘𝑊)) |
10 | rnasclg.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
11 | 2, 3, 8, 4, 10 | lspsn 20415 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 1 ∈ (Base‘𝑊)) → (𝑁‘{ 1 }) = {𝑥 ∣ ∃𝑦 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑦( ·𝑠 ‘𝑊) 1 )}) |
12 | 9, 11 | sylan2 593 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) → (𝑁‘{ 1 }) = {𝑥 ∣ ∃𝑦 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑦( ·𝑠 ‘𝑊) 1 )}) |
13 | 7, 12 | eqtr4id 2796 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) → ran 𝐴 = (𝑁‘{ 1 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2714 ∃wrex 3071 {csn 4584 ran crn 5632 ‘cfv 6493 (class class class)co 7351 Basecbs 17042 Scalarcsca 17095 ·𝑠 cvsca 17096 1rcur 19871 Ringcrg 19917 LModclmod 20274 LSpanclspn 20384 algSccascl 21210 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-plusg 17105 df-0g 17282 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-grp 18710 df-minusg 18711 df-sbg 18712 df-mgp 19855 df-ur 19872 df-ring 19919 df-lmod 20276 df-lss 20345 df-lsp 20385 df-ascl 21213 |
This theorem is referenced by: (None) |
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