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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sralvec | Structured version Visualization version GIF version |
Description: Given a sub division ring 𝐹 of a division ring 𝐸, 𝐸 may be considered as a vector space over 𝐹, which becomes the field of scalars. (Contributed by Thierry Arnoux, 24-May-2023.) |
Ref | Expression |
---|---|
sralvec.a | ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑈) |
sralvec.f | ⊢ 𝐹 = (𝐸 ↾s 𝑈) |
Ref | Expression |
---|---|
sralvec | ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sralvec.a | . . 3 ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑈) | |
2 | eqid 2726 | . . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝑈) = ((subringAlg ‘𝐸)‘𝑈) | |
3 | 2 | sralmod 21173 | . . . 4 ⊢ (𝑈 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝑈) ∈ LMod) |
4 | 3 | 3ad2ant3 1132 | . . 3 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘𝑈) ∈ LMod) |
5 | 1, 4 | eqeltrid 2830 | . 2 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LMod) |
6 | sralvec.f | . . . . 5 ⊢ 𝐹 = (𝐸 ↾s 𝑈) | |
7 | 1 | a1i 11 | . . . . . 6 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝐴 = ((subringAlg ‘𝐸)‘𝑈)) |
8 | eqid 2726 | . . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
9 | 8 | subrgss 20556 | . . . . . 6 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸)) |
10 | 7, 9 | srasca 21162 | . . . . 5 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (𝐸 ↾s 𝑈) = (Scalar‘𝐴)) |
11 | 6, 10 | eqtrid 2778 | . . . 4 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝐹 = (Scalar‘𝐴)) |
12 | 11 | 3ad2ant3 1132 | . . 3 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐹 = (Scalar‘𝐴)) |
13 | simp2 1134 | . . 3 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐹 ∈ DivRing) | |
14 | 12, 13 | eqeltrrd 2827 | . 2 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → (Scalar‘𝐴) ∈ DivRing) |
15 | eqid 2726 | . . 3 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴) | |
16 | 15 | islvec 21082 | . 2 ⊢ (𝐴 ∈ LVec ↔ (𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ DivRing)) |
17 | 5, 14, 16 | sylanbrc 581 | 1 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 ↾s cress 17242 Scalarcsca 17269 SubRingcsubrg 20551 DivRingcdr 20707 LModclmod 20836 LVecclvec 21080 subringAlg csra 21149 |
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 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-ip 17284 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-subg 19117 df-mgp 20118 df-ur 20165 df-ring 20218 df-subrg 20553 df-lmod 20838 df-lvec 21081 df-sra 21151 |
This theorem is referenced by: srafldlvec 33483 drgextgsum 33491 rgmoddimOLD 33505 fedgmullem1 33524 fedgmullem2 33525 fedgmul 33526 fldextsralvec 33544 extdgcl 33545 extdggt0 33546 |
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