Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cvsdiveqd | Structured version Visualization version GIF version |
Description: An equality involving ratios in a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.) |
Ref | Expression |
---|---|
cvsdiveqd.v | ⊢ 𝑉 = (Base‘𝑊) |
cvsdiveqd.t | ⊢ · = ( ·𝑠 ‘𝑊) |
cvsdiveqd.f | ⊢ 𝐹 = (Scalar‘𝑊) |
cvsdiveqd.k | ⊢ 𝐾 = (Base‘𝐹) |
cvsdiveqd.w | ⊢ (𝜑 → 𝑊 ∈ ℂVec) |
cvsdiveqd.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
cvsdiveqd.b | ⊢ (𝜑 → 𝐵 ∈ 𝐾) |
cvsdiveqd.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
cvsdiveqd.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
cvsdiveqd.1 | ⊢ (𝜑 → 𝐴 ≠ 0) |
cvsdiveqd.2 | ⊢ (𝜑 → 𝐵 ≠ 0) |
cvsdiveqd.3 | ⊢ (𝜑 → 𝑋 = ((𝐴 / 𝐵) · 𝑌)) |
Ref | Expression |
---|---|
cvsdiveqd | ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvsdiveqd.3 | . . 3 ⊢ (𝜑 → 𝑋 = ((𝐴 / 𝐵) · 𝑌)) | |
2 | 1 | oveq2d 7229 | . 2 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) = ((𝐵 / 𝐴) · ((𝐴 / 𝐵) · 𝑌))) |
3 | cvsdiveqd.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℂVec) | |
4 | 3 | cvsclm 24023 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
5 | cvsdiveqd.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | cvsdiveqd.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
7 | 5, 6 | clmsscn 23976 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ) |
8 | 4, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐾 ⊆ ℂ) |
9 | cvsdiveqd.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
10 | 8, 9 | sseldd 3902 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
11 | cvsdiveqd.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
12 | 8, 11 | sseldd 3902 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | cvsdiveqd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 0) | |
14 | cvsdiveqd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) | |
15 | 10, 12, 13, 14 | divcan6d 11627 | . . . 4 ⊢ (𝜑 → ((𝐵 / 𝐴) · (𝐴 / 𝐵)) = 1) |
16 | 15 | oveq1d 7228 | . . 3 ⊢ (𝜑 → (((𝐵 / 𝐴) · (𝐴 / 𝐵)) · 𝑌) = (1 · 𝑌)) |
17 | 5, 6 | cvsdivcl 24030 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐵 ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0)) → (𝐵 / 𝐴) ∈ 𝐾) |
18 | 3, 9, 11, 14, 17 | syl13anc 1374 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ 𝐾) |
19 | 5, 6 | cvsdivcl 24030 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾) |
20 | 3, 11, 9, 13, 19 | syl13anc 1374 | . . . 4 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ 𝐾) |
21 | cvsdiveqd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
22 | cvsdiveqd.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
23 | cvsdiveqd.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
24 | 22, 5, 23, 6 | clmvsass 23986 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ ((𝐵 / 𝐴) ∈ 𝐾 ∧ (𝐴 / 𝐵) ∈ 𝐾 ∧ 𝑌 ∈ 𝑉)) → (((𝐵 / 𝐴) · (𝐴 / 𝐵)) · 𝑌) = ((𝐵 / 𝐴) · ((𝐴 / 𝐵) · 𝑌))) |
25 | 4, 18, 20, 21, 24 | syl13anc 1374 | . . 3 ⊢ (𝜑 → (((𝐵 / 𝐴) · (𝐴 / 𝐵)) · 𝑌) = ((𝐵 / 𝐴) · ((𝐴 / 𝐵) · 𝑌))) |
26 | 22, 23 | clmvs1 23990 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑌 ∈ 𝑉) → (1 · 𝑌) = 𝑌) |
27 | 4, 21, 26 | syl2anc 587 | . . 3 ⊢ (𝜑 → (1 · 𝑌) = 𝑌) |
28 | 16, 25, 27 | 3eqtr3d 2785 | . 2 ⊢ (𝜑 → ((𝐵 / 𝐴) · ((𝐴 / 𝐵) · 𝑌)) = 𝑌) |
29 | 2, 28 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ⊆ wss 3866 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 0cc0 10729 1c1 10730 · cmul 10734 / cdiv 11489 Basecbs 16760 Scalarcsca 16805 ·𝑠 cvsca 16806 ℂModcclm 23959 ℂVecccvs 24020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 df-minusg 18369 df-subg 18540 df-cmn 19172 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-drng 19769 df-subrg 19798 df-lmod 19901 df-lvec 20140 df-cnfld 20364 df-clm 23960 df-cvs 24021 |
This theorem is referenced by: ttgcontlem1 26976 |
Copyright terms: Public domain | W3C validator |