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Mirrors > Home > MPE Home > Th. List > cvsdiv | Structured version Visualization version GIF version |
Description: Division of the scalar ring of a complex vector space. (Contributed by Thierry Arnoux, 22-May-2019.) |
Ref | Expression |
---|---|
cvsdiv.f | ⊢ 𝐹 = (Scalar‘𝑊) |
cvsdiv.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
cvsdiv | ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝐴(/r‘𝐹)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 472 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → 𝑊 ∈ ℂVec) | |
2 | 1 | cvsclm 22734 | . . . 4 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → 𝑊 ∈ ℂMod) |
3 | cvsdiv.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | cvsdiv.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
5 | 3, 4 | clmsubrg 22674 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld)) |
6 | 2, 5 | syl 17 | . . 3 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → 𝐾 ∈ (SubRing‘ℂfld)) |
7 | simpr1 1060 | . . 3 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → 𝐴 ∈ 𝐾) | |
8 | simpr2 1061 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → 𝐵 ∈ 𝐾) | |
9 | simpr3 1062 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → 𝐵 ≠ 0) | |
10 | eldifsn 4260 | . . . . 5 ⊢ (𝐵 ∈ (𝐾 ∖ {0}) ↔ (𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) | |
11 | 8, 9, 10 | sylanbrc 695 | . . . 4 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → 𝐵 ∈ (𝐾 ∖ {0})) |
12 | 3, 4 | cvsunit 22739 | . . . . . 6 ⊢ (𝑊 ∈ ℂVec → (𝐾 ∖ {0}) = (Unit‘𝐹)) |
13 | 1, 12 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐾 ∖ {0}) = (Unit‘𝐹)) |
14 | 3, 4 | clmsca 22673 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾)) |
15 | 2, 14 | syl 17 | . . . . . 6 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → 𝐹 = (ℂfld ↾s 𝐾)) |
16 | 15 | fveq2d 6107 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (Unit‘𝐹) = (Unit‘(ℂfld ↾s 𝐾))) |
17 | 13, 16 | eqtrd 2644 | . . . 4 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐾 ∖ {0}) = (Unit‘(ℂfld ↾s 𝐾))) |
18 | 11, 17 | eleqtrd 2690 | . . 3 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → 𝐵 ∈ (Unit‘(ℂfld ↾s 𝐾))) |
19 | eqid 2610 | . . . 4 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾) | |
20 | cnflddiv 19595 | . . . 4 ⊢ / = (/r‘ℂfld) | |
21 | eqid 2610 | . . . 4 ⊢ (Unit‘(ℂfld ↾s 𝐾)) = (Unit‘(ℂfld ↾s 𝐾)) | |
22 | eqid 2610 | . . . 4 ⊢ (/r‘(ℂfld ↾s 𝐾)) = (/r‘(ℂfld ↾s 𝐾)) | |
23 | 19, 20, 21, 22 | subrgdv 18620 | . . 3 ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ (Unit‘(ℂfld ↾s 𝐾))) → (𝐴 / 𝐵) = (𝐴(/r‘(ℂfld ↾s 𝐾))𝐵)) |
24 | 6, 7, 18, 23 | syl3anc 1318 | . 2 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝐴(/r‘(ℂfld ↾s 𝐾))𝐵)) |
25 | 15 | fveq2d 6107 | . . 3 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (/r‘𝐹) = (/r‘(ℂfld ↾s 𝐾))) |
26 | 25 | oveqd 6566 | . 2 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴(/r‘𝐹)𝐵) = (𝐴(/r‘(ℂfld ↾s 𝐾))𝐵)) |
27 | 24, 26 | eqtr4d 2647 | 1 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝐴(/r‘𝐹)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 ‘cfv 5804 (class class class)co 6549 0cc0 9815 / cdiv 10563 Basecbs 15695 ↾s cress 15696 Scalarcsca 15771 Unitcui 18462 /rcdvr 18505 SubRingcsubrg 18599 ℂfldccnfld 19567 ℂModcclm 22670 ℂVecccvs 22731 |
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-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-minusg 17249 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-invr 18495 df-dvr 18506 df-drng 18572 df-subrg 18601 df-lvec 18924 df-cnfld 19568 df-clm 22671 df-cvs 22732 |
This theorem is referenced by: cvsdivcl 22741 |
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