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 2820 | . . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝑈) = ((subringAlg ‘𝐸)‘𝑈) | |
3 | 2 | sralmod 19955 | . . . 4 ⊢ (𝑈 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝑈) ∈ LMod) |
4 | 3 | 3ad2ant3 1130 | . . 3 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘𝑈) ∈ LMod) |
5 | 1, 4 | eqeltrid 2916 | . 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 2820 | . . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
9 | 8 | subrgss 19532 | . . . . . 6 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸)) |
10 | 7, 9 | srasca 19949 | . . . . 5 ⊢ (𝑈 ∈ (SubRing‘𝐸) → (𝐸 ↾s 𝑈) = (Scalar‘𝐴)) |
11 | 6, 10 | syl5eq 2867 | . . . 4 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝐹 = (Scalar‘𝐴)) |
12 | 11 | 3ad2ant3 1130 | . . 3 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐹 = (Scalar‘𝐴)) |
13 | simp2 1132 | . . 3 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐹 ∈ DivRing) | |
14 | 12, 13 | eqeltrrd 2913 | . 2 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → (Scalar‘𝐴) ∈ DivRing) |
15 | eqid 2820 | . . 3 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴) | |
16 | 15 | islvec 19872 | . 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 1082 = wceq 1536 ∈ wcel 2113 ‘cfv 6352 (class class class)co 7153 Basecbs 16479 ↾s cress 16480 Scalarcsca 16564 DivRingcdr 19498 SubRingcsubrg 19527 LModclmod 19630 LVecclvec 19870 subringAlg csra 19936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-mulr 16575 df-sca 16577 df-vsca 16578 df-ip 16579 df-0g 16711 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-grp 18102 df-subg 18272 df-mgp 19236 df-ur 19248 df-ring 19295 df-subrg 19529 df-lmod 19632 df-lvec 19871 df-sra 19940 |
This theorem is referenced by: srafldlvec 31015 drgextgsum 31021 rgmoddim 31032 fedgmullem1 31049 fedgmullem2 31050 fedgmul 31051 fldextsralvec 31069 extdgcl 31070 extdggt0 31071 |
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