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Mirrors > Home > MPE Home > Th. List > tchclm | Structured version Visualization version GIF version |
Description: Lemma for tchcph 22844. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
tchval.n | ⊢ 𝐺 = (toℂHil‘𝑊) |
tchcph.v | ⊢ 𝑉 = (Base‘𝑊) |
tchcph.f | ⊢ 𝐹 = (Scalar‘𝑊) |
tchcph.1 | ⊢ (𝜑 → 𝑊 ∈ PreHil) |
tchcph.2 | ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) |
Ref | Expression |
---|---|
tchclm | ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tchcph.1 | . . 3 ⊢ (𝜑 → 𝑊 ∈ PreHil) | |
2 | phllmod 19794 | . . 3 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | eqid 2610 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
5 | tchcph.2 | . . . 4 ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) | |
6 | phllvec 19793 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) |
8 | tchcph.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
9 | 8 | lvecdrng 18926 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ DivRing) |
11 | 4, 5, 10 | cphsubrglem 22785 | . . 3 ⊢ (𝜑 → (𝐹 = (ℂfld ↾s (Base‘𝐹)) ∧ (Base‘𝐹) = (𝐾 ∩ ℂ) ∧ (Base‘𝐹) ∈ (SubRing‘ℂfld))) |
12 | 11 | simp1d 1066 | . 2 ⊢ (𝜑 → 𝐹 = (ℂfld ↾s (Base‘𝐹))) |
13 | 11 | simp3d 1068 | . 2 ⊢ (𝜑 → (Base‘𝐹) ∈ (SubRing‘ℂfld)) |
14 | 8, 4 | isclm 22672 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘ℂfld))) |
15 | 3, 12, 13, 14 | syl3anbrc 1239 | 1 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 Basecbs 15695 ↾s cress 15696 Scalarcsca 15771 DivRingcdr 18570 SubRingcsubrg 18599 LModclmod 18686 LVecclvec 18923 ℂfldccnfld 19567 PreHilcphl 19788 ℂModcclm 22670 toℂHilctch 22775 |
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-rep 4699 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-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 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-int 4411 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-seq 12664 df-exp 12723 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-subg 17414 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-drng 18572 df-subrg 18601 df-lvec 18924 df-cnfld 19568 df-phl 19790 df-clm 22671 |
This theorem is referenced by: tchcphlem3 22840 ipcau2 22841 tchcphlem1 22842 tchcphlem2 22843 tchcph 22844 |
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