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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 6940 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐴 · 𝑋)) = ((1 / 𝐴) · (𝐵 · 𝑌))) |
3 | cvsdiveqd.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℂVec) | |
4 | 3 | cvsclm 23344 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
5 | cvsdiveqd.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | cvsdiveqd.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
7 | 5, 6 | clmsscn 23297 | . . . . . . 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 3822 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
11 | cvsdiveqd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) | |
12 | 10, 11 | recid2d 11150 | . . . 4 ⊢ (𝜑 → ((1 / 𝐴) · 𝐴) = 1) |
13 | 12 | oveq1d 6939 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐴) · 𝑋) = (1 · 𝑋)) |
14 | 5 | clm1 23291 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘𝐹)) |
15 | 4, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 1 = (1r‘𝐹)) |
16 | 5 | clmring 23288 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring) |
17 | eqid 2778 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
18 | 6, 17 | ringidcl 18966 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
19 | 4, 16, 18 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐹) ∈ 𝐾) |
20 | 15, 19 | eqeltrd 2859 | . . . . 5 ⊢ (𝜑 → 1 ∈ 𝐾) |
21 | 5, 6 | cvsdivcl 23351 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (1 ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0)) → (1 / 𝐴) ∈ 𝐾) |
22 | 3, 20, 9, 11, 21 | syl13anc 1440 | . . . 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 23307 | . . . 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 1440 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐴) · 𝑋) = ((1 / 𝐴) · (𝐴 · 𝑋))) |
28 | 24, 25 | clmvs1 23311 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (1 · 𝑋) = 𝑋) |
29 | 4, 23, 28 | syl2anc 579 | . . 3 ⊢ (𝜑 → (1 · 𝑋) = 𝑋) |
30 | 13, 27, 29 | 3eqtr3d 2822 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐴 · 𝑋)) = 𝑋) |
31 | cvsdiveqd.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
32 | 8, 31 | sseldd 3822 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
33 | 32, 10, 11 | divrec2d 11158 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐴) = ((1 / 𝐴) · 𝐵)) |
34 | 33 | oveq1d 6939 | . . 3 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑌) = (((1 / 𝐴) · 𝐵) · 𝑌)) |
35 | cvsdiveqd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
36 | 24, 5, 25, 6 | clmvsass 23307 | . . . 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 1440 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐵) · 𝑌) = ((1 / 𝐴) · (𝐵 · 𝑌))) |
38 | 34, 37 | eqtr2d 2815 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐵 · 𝑌)) = ((𝐵 / 𝐴) · 𝑌)) |
39 | 2, 30, 38 | 3eqtr3d 2822 | 1 ⊢ (𝜑 → 𝑋 = ((𝐵 / 𝐴) · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ⊆ wss 3792 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 0cc0 10274 1c1 10275 · cmul 10279 / cdiv 11035 Basecbs 16266 Scalarcsca 16352 ·𝑠 cvsca 16353 1rcur 18899 Ringcrg 18945 ℂModcclm 23280 ℂVecccvs 23341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-fz 12649 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-0g 16499 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-grp 17823 df-minusg 17824 df-subg 17986 df-cmn 18592 df-mgp 18888 df-ur 18900 df-ring 18947 df-cring 18948 df-oppr 19021 df-dvdsr 19039 df-unit 19040 df-invr 19070 df-dvr 19081 df-drng 19152 df-subrg 19181 df-lmod 19268 df-lvec 19509 df-cnfld 20154 df-clm 23281 df-cvs 23342 |
This theorem is referenced by: ttgcontlem1 26251 |
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