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Mirrors > Home > MPE Home > Th. List > cvsmuleqdivd | 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) |
cvsmuleqdivd.1 | ⊢ (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑌)) |
Ref | Expression |
---|---|
cvsmuleqdivd | ⊢ (𝜑 → 𝑋 = ((𝐵 / 𝐴) · 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvsmuleqdivd.1 | . . 3 ⊢ (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑌)) | |
2 | 1 | oveq2d 7420 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐴 · 𝑋)) = ((1 / 𝐴) · (𝐵 · 𝑌))) |
3 | cvsdiveqd.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℂVec) | |
4 | 3 | cvsclm 24624 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
5 | cvsdiveqd.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | cvsdiveqd.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
7 | 5, 6 | clmsscn 24577 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ) |
8 | 4, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐾 ⊆ ℂ) |
9 | cvsdiveqd.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
10 | 8, 9 | sseldd 3982 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
11 | cvsdiveqd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) | |
12 | 10, 11 | recid2d 11982 | . . . 4 ⊢ (𝜑 → ((1 / 𝐴) · 𝐴) = 1) |
13 | 12 | oveq1d 7419 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐴) · 𝑋) = (1 · 𝑋)) |
14 | 5 | clm1 24571 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘𝐹)) |
15 | 4, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 1 = (1r‘𝐹)) |
16 | 5 | clmring 24568 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring) |
17 | eqid 2733 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
18 | 6, 17 | ringidcl 20073 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
19 | 4, 16, 18 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐹) ∈ 𝐾) |
20 | 15, 19 | eqeltrd 2834 | . . . . 5 ⊢ (𝜑 → 1 ∈ 𝐾) |
21 | 5, 6 | cvsdivcl 24631 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (1 ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0)) → (1 / 𝐴) ∈ 𝐾) |
22 | 3, 20, 9, 11, 21 | syl13anc 1373 | . . . 4 ⊢ (𝜑 → (1 / 𝐴) ∈ 𝐾) |
23 | cvsdiveqd.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
24 | cvsdiveqd.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
25 | cvsdiveqd.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
26 | 24, 5, 25, 6 | clmvsass 24587 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ ((1 / 𝐴) ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (((1 / 𝐴) · 𝐴) · 𝑋) = ((1 / 𝐴) · (𝐴 · 𝑋))) |
27 | 4, 22, 9, 23, 26 | syl13anc 1373 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐴) · 𝑋) = ((1 / 𝐴) · (𝐴 · 𝑋))) |
28 | 24, 25 | clmvs1 24591 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (1 · 𝑋) = 𝑋) |
29 | 4, 23, 28 | syl2anc 585 | . . 3 ⊢ (𝜑 → (1 · 𝑋) = 𝑋) |
30 | 13, 27, 29 | 3eqtr3d 2781 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐴 · 𝑋)) = 𝑋) |
31 | cvsdiveqd.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
32 | 8, 31 | sseldd 3982 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
33 | 32, 10, 11 | divrec2d 11990 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐴) = ((1 / 𝐴) · 𝐵)) |
34 | 33 | oveq1d 7419 | . . 3 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑌) = (((1 / 𝐴) · 𝐵) · 𝑌)) |
35 | cvsdiveqd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
36 | 24, 5, 25, 6 | clmvsass 24587 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ ((1 / 𝐴) ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉)) → (((1 / 𝐴) · 𝐵) · 𝑌) = ((1 / 𝐴) · (𝐵 · 𝑌))) |
37 | 4, 22, 31, 35, 36 | syl13anc 1373 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐵) · 𝑌) = ((1 / 𝐴) · (𝐵 · 𝑌))) |
38 | 34, 37 | eqtr2d 2774 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐵 · 𝑌)) = ((𝐵 / 𝐴) · 𝑌)) |
39 | 2, 30, 38 | 3eqtr3d 2781 | 1 ⊢ (𝜑 → 𝑋 = ((𝐵 / 𝐴) · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ⊆ wss 3947 ‘cfv 6540 (class class class)co 7404 ℂcc 11104 0cc0 11106 1c1 11107 · cmul 11111 / cdiv 11867 Basecbs 17140 Scalarcsca 17196 ·𝑠 cvsca 17197 1rcur 19996 Ringcrg 20047 ℂModcclm 24560 ℂVecccvs 24621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-subg 18997 df-cmn 19643 df-mgp 19980 df-ur 19997 df-ring 20049 df-cring 20050 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 df-dvr 20204 df-drng 20306 df-subrg 20349 df-lmod 20461 df-lvec 20702 df-cnfld 20930 df-clm 24561 df-cvs 24622 |
This theorem is referenced by: ttgcontlem1 28122 |
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