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| Mirrors > Home > MPE Home > Th. List > recvsOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of recvs 25179 as of 23-Nov-2024. (Contributed by AV, 22-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| recvs.r | ⊢ 𝑅 = (ringLMod‘ℝfld) |
| Ref | Expression |
|---|---|
| recvsOLD | ⊢ 𝑅 ∈ ℂVec |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | refld 21637 | . . . . . 6 ⊢ ℝfld ∈ Field | |
| 2 | fldidom 20771 | . . . . . . 7 ⊢ (ℝfld ∈ Field → ℝfld ∈ IDomn) | |
| 3 | isidom 20725 | . . . . . . . 8 ⊢ (ℝfld ∈ IDomn ↔ (ℝfld ∈ CRing ∧ ℝfld ∈ Domn)) | |
| 4 | crngring 20242 | . . . . . . . . 9 ⊢ (ℝfld ∈ CRing → ℝfld ∈ Ring) | |
| 5 | 4 | adantr 480 | . . . . . . . 8 ⊢ ((ℝfld ∈ CRing ∧ ℝfld ∈ Domn) → ℝfld ∈ Ring) |
| 6 | 3, 5 | sylbi 217 | . . . . . . 7 ⊢ (ℝfld ∈ IDomn → ℝfld ∈ Ring) |
| 7 | 2, 6 | syl 17 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld ∈ Ring) |
| 8 | 1, 7 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Ring |
| 9 | rlmlmod 21210 | . . . . 5 ⊢ (ℝfld ∈ Ring → (ringLMod‘ℝfld) ∈ LMod) | |
| 10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ (ringLMod‘ℝfld) ∈ LMod |
| 11 | rlmsca 21205 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld = (Scalar‘(ringLMod‘ℝfld))) | |
| 12 | 1, 11 | ax-mp 5 | . . . . 5 ⊢ ℝfld = (Scalar‘(ringLMod‘ℝfld)) |
| 13 | df-refld 21623 | . . . . 5 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
| 14 | 12, 13 | eqtr3i 2767 | . . . 4 ⊢ (Scalar‘(ringLMod‘ℝfld)) = (ℂfld ↾s ℝ) |
| 15 | resubdrg 21626 | . . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
| 16 | 15 | simpli 483 | . . . 4 ⊢ ℝ ∈ (SubRing‘ℂfld) |
| 17 | eqid 2737 | . . . . 5 ⊢ (Scalar‘(ringLMod‘ℝfld)) = (Scalar‘(ringLMod‘ℝfld)) | |
| 18 | 17 | isclmi 25110 | . . . 4 ⊢ (((ringLMod‘ℝfld) ∈ LMod ∧ (Scalar‘(ringLMod‘ℝfld)) = (ℂfld ↾s ℝ) ∧ ℝ ∈ (SubRing‘ℂfld)) → (ringLMod‘ℝfld) ∈ ℂMod) |
| 19 | 10, 14, 16, 18 | mp3an 1463 | . . 3 ⊢ (ringLMod‘ℝfld) ∈ ℂMod |
| 20 | 15 | simpri 485 | . . . 4 ⊢ ℝfld ∈ DivRing |
| 21 | rlmlvec 21211 | . . . 4 ⊢ (ℝfld ∈ DivRing → (ringLMod‘ℝfld) ∈ LVec) | |
| 22 | 20, 21 | ax-mp 5 | . . 3 ⊢ (ringLMod‘ℝfld) ∈ LVec |
| 23 | 19, 22 | elini 4199 | . 2 ⊢ (ringLMod‘ℝfld) ∈ (ℂMod ∩ LVec) |
| 24 | recvs.r | . 2 ⊢ 𝑅 = (ringLMod‘ℝfld) | |
| 25 | df-cvs 25157 | . 2 ⊢ ℂVec = (ℂMod ∩ LVec) | |
| 26 | 23, 24, 25 | 3eltr4i 2854 | 1 ⊢ 𝑅 ∈ ℂVec |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 ↾s cress 17274 Scalarcsca 17300 Ringcrg 20230 CRingccrg 20231 SubRingcsubrg 20569 Domncdomn 20692 IDomncidom 20693 DivRingcdr 20729 Fieldcfield 20730 LModclmod 20858 LVecclvec 21101 ringLModcrglmod 21171 ℂfldccnfld 21364 ℝfldcrefld 21622 ℂModcclm 25095 ℂVecccvs 25156 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-nzr 20513 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-domn 20695 df-idom 20696 df-drng 20731 df-field 20732 df-lmod 20860 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-refld 21623 df-clm 25096 df-cvs 25157 |
| This theorem is referenced by: (None) |
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