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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 7447 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐴 · 𝑋)) = ((1 / 𝐴) · (𝐵 · 𝑌))) |
3 | cvsdiveqd.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℂVec) | |
4 | 3 | cvsclm 25173 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
5 | cvsdiveqd.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | cvsdiveqd.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
7 | 5, 6 | clmsscn 25126 | . . . . . . 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 3996 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
11 | cvsdiveqd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) | |
12 | 10, 11 | recid2d 12037 | . . . 4 ⊢ (𝜑 → ((1 / 𝐴) · 𝐴) = 1) |
13 | 12 | oveq1d 7446 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐴) · 𝑋) = (1 · 𝑋)) |
14 | 5 | clm1 25120 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘𝐹)) |
15 | 4, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 1 = (1r‘𝐹)) |
16 | 5 | clmring 25117 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring) |
17 | eqid 2735 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
18 | 6, 17 | ringidcl 20280 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
19 | 4, 16, 18 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐹) ∈ 𝐾) |
20 | 15, 19 | eqeltrd 2839 | . . . . 5 ⊢ (𝜑 → 1 ∈ 𝐾) |
21 | 5, 6 | cvsdivcl 25180 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (1 ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0)) → (1 / 𝐴) ∈ 𝐾) |
22 | 3, 20, 9, 11, 21 | syl13anc 1371 | . . . 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 25136 | . . . 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 1371 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐴) · 𝑋) = ((1 / 𝐴) · (𝐴 · 𝑋))) |
28 | 24, 25 | clmvs1 25140 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (1 · 𝑋) = 𝑋) |
29 | 4, 23, 28 | syl2anc 584 | . . 3 ⊢ (𝜑 → (1 · 𝑋) = 𝑋) |
30 | 13, 27, 29 | 3eqtr3d 2783 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐴 · 𝑋)) = 𝑋) |
31 | cvsdiveqd.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
32 | 8, 31 | sseldd 3996 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
33 | 32, 10, 11 | divrec2d 12045 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐴) = ((1 / 𝐴) · 𝐵)) |
34 | 33 | oveq1d 7446 | . . 3 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑌) = (((1 / 𝐴) · 𝐵) · 𝑌)) |
35 | cvsdiveqd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
36 | 24, 5, 25, 6 | clmvsass 25136 | . . . 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 1371 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐵) · 𝑌) = ((1 / 𝐴) · (𝐵 · 𝑌))) |
38 | 34, 37 | eqtr2d 2776 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐵 · 𝑌)) = ((𝐵 / 𝐴) · 𝑌)) |
39 | 2, 30, 38 | 3eqtr3d 2783 | 1 ⊢ (𝜑 → 𝑋 = ((𝐵 / 𝐴) · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 · cmul 11158 / cdiv 11918 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 1rcur 20199 Ringcrg 20251 ℂModcclm 25109 ℂVecccvs 25170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-subrg 20587 df-drng 20748 df-lmod 20877 df-lvec 21120 df-cnfld 21383 df-clm 25110 df-cvs 25171 |
This theorem is referenced by: ttgcontlem1 28914 |
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