Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rveccmod | Structured version Visualization version GIF version |
Description: Real vector spaces are subcomplex modules (elemental version). (Contributed by BJ, 6-Jan-2024.) |
Ref | Expression |
---|---|
bj-rveccmod | ⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-rvecmod 35445 | . 2 ⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ LMod) | |
2 | df-refld 20791 | . . 3 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
3 | 2 | a1i 11 | . 2 ⊢ (𝑉 ∈ ℝ-Vec → ℝfld = (ℂfld ↾s ℝ)) |
4 | resubdrg 20794 | . . . 4 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
5 | 4 | simpli 483 | . . 3 ⊢ ℝ ∈ (SubRing‘ℂfld) |
6 | 5 | a1i 11 | . 2 ⊢ (𝑉 ∈ ℝ-Vec → ℝ ∈ (SubRing‘ℂfld)) |
7 | bj-rvecrr 35447 | . . . 4 ⊢ (𝑉 ∈ ℝ-Vec → (Scalar‘𝑉) = ℝfld) | |
8 | 7 | eqcomd 2745 | . . 3 ⊢ (𝑉 ∈ ℝ-Vec → ℝfld = (Scalar‘𝑉)) |
9 | rebase 20792 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑉 ∈ ℝ-Vec → ℝ = (Base‘ℝfld)) |
11 | 8, 10 | bj-isclm 35441 | . 2 ⊢ (𝑉 ∈ ℝ-Vec → (𝑉 ∈ ℂMod ↔ (𝑉 ∈ LMod ∧ ℝfld = (ℂfld ↾s ℝ) ∧ ℝ ∈ (SubRing‘ℂfld)))) |
12 | 1, 3, 6, 11 | mpbir3and 1340 | 1 ⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2109 ‘cfv 6430 (class class class)co 7268 ℝcr 10854 Basecbs 16893 ↾s cress 16922 Scalarcsca 16946 DivRingcdr 19972 SubRingcsubrg 20001 LModclmod 20104 ℂfldccnfld 20578 ℝfldcrefld 20790 ℂModcclm 24206 ℝ-Veccrrvec 35442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-addf 10934 ax-mulf 10935 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-tpos 8026 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-fz 13222 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-0g 17133 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561 df-minusg 18562 df-subg 18733 df-cmn 19369 df-mgp 19702 df-ur 19719 df-ring 19766 df-cring 19767 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-dvr 19906 df-drng 19974 df-subrg 20003 df-cnfld 20579 df-refld 20791 df-clm 24207 df-bj-rvec 35443 |
This theorem is referenced by: bj-rvecsscmod 35453 |
Copyright terms: Public domain | W3C validator |