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Mirrors > Home > MPE Home > Th. List > isclmi | Structured version Visualization version GIF version |
Description: Reverse direction of isclm 22672. (Contributed by Mario Carneiro, 30-Oct-2015.) |
Ref | Expression |
---|---|
clm0.f | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
isclmi | ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1054 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ LMod) | |
2 | simp2 1055 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂfld ↾s 𝐾)) | |
3 | eqid 2610 | . . . . . . 7 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾) | |
4 | 3 | subrgbas 18612 | . . . . . 6 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 = (Base‘(ℂfld ↾s 𝐾))) |
5 | 4 | 3ad2ant3 1077 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘(ℂfld ↾s 𝐾))) |
6 | 2 | fveq2d 6107 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) = (Base‘(ℂfld ↾s 𝐾))) |
7 | 5, 6 | eqtr4d 2647 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘𝐹)) |
8 | 7 | oveq2d 6565 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (ℂfld ↾s 𝐾) = (ℂfld ↾s (Base‘𝐹))) |
9 | 2, 8 | eqtrd 2644 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂfld ↾s (Base‘𝐹))) |
10 | simp3 1056 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 ∈ (SubRing‘ℂfld)) | |
11 | 7, 10 | eqeltrrd 2689 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) ∈ (SubRing‘ℂfld)) |
12 | clm0.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
13 | eqid 2610 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
14 | 12, 13 | isclm 22672 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘ℂfld))) |
15 | 1, 9, 11, 14 | syl3anbrc 1239 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 Scalarcsca 15771 SubRingcsubrg 18599 LModclmod 18686 ℂfldccnfld 19567 ℂModcclm 22670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-subg 17414 df-ring 18372 df-subrg 18601 df-clm 22671 |
This theorem is referenced by: zlmclm 22720 cnstrcvs 22749 cncvs 22753 recvs 22754 qcvs 22755 zclmncvs 22756 |
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