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 2821 | . . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝑈) = ((subringAlg ‘𝐸)‘𝑈) | |
3 | 2 | sralmod 19942 | . . . 4 ⊢ (𝑈 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝑈) ∈ LMod) |
4 | 3 | 3ad2ant3 1131 | . . 3 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘𝑈) ∈ LMod) |
5 | 1, 4 | eqeltrid 2917 | . 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 2821 | . . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
9 | 8 | subrgss 19519 | . . . . . 6 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸)) |
10 | 7, 9 | srasca 19936 | . . . . 5 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (𝐸 ↾s 𝑈) = (Scalar‘𝐴)) |
11 | 6, 10 | syl5eq 2868 | . . . 4 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝐹 = (Scalar‘𝐴)) |
12 | 11 | 3ad2ant3 1131 | . . 3 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐹 = (Scalar‘𝐴)) |
13 | simp2 1133 | . . 3 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐹 ∈ DivRing) | |
14 | 12, 13 | eqeltrrd 2914 | . 2 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → (Scalar‘𝐴) ∈ DivRing) |
15 | eqid 2821 | . . 3 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴) | |
16 | 15 | islvec 19859 | . 2 ⊢ (𝐴 ∈ LVec ↔ (𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ DivRing)) |
17 | 5, 14, 16 | sylanbrc 585 | 1 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 ↾s cress 16467 Scalarcsca 16551 DivRingcdr 19485 SubRingcsubrg 19514 LModclmod 19617 LVecclvec 19857 subringAlg csra 19923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-ip 16566 df-0g 16698 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-grp 18089 df-subg 18259 df-mgp 19223 df-ur 19235 df-ring 19282 df-subrg 19516 df-lmod 19619 df-lvec 19858 df-sra 19927 |
This theorem is referenced by: srafldlvec 31001 drgextgsum 31007 rgmoddim 31018 fedgmullem1 31035 fedgmullem2 31036 fedgmul 31037 fldextsralvec 31055 extdgcl 31056 extdggt0 31057 |
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