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 7161 | . 2 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) = ((𝐵 / 𝐴) · ((𝐴 / 𝐵) · 𝑌))) |
3 | cvsdiveqd.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℂVec) | |
4 | 3 | cvsclm 23657 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
5 | cvsdiveqd.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | cvsdiveqd.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
7 | 5, 6 | clmsscn 23610 | . . . . . . 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 3965 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
11 | cvsdiveqd.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
12 | 8, 11 | sseldd 3965 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | cvsdiveqd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 0) | |
14 | cvsdiveqd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) | |
15 | 10, 12, 13, 14 | divcan6d 11423 | . . . 4 ⊢ (𝜑 → ((𝐵 / 𝐴) · (𝐴 / 𝐵)) = 1) |
16 | 15 | oveq1d 7160 | . . 3 ⊢ (𝜑 → (((𝐵 / 𝐴) · (𝐴 / 𝐵)) · 𝑌) = (1 · 𝑌)) |
17 | 5, 6 | cvsdivcl 23664 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐵 ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0)) → (𝐵 / 𝐴) ∈ 𝐾) |
18 | 3, 9, 11, 14, 17 | syl13anc 1364 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ 𝐾) |
19 | 5, 6 | cvsdivcl 23664 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾) |
20 | 3, 11, 9, 13, 19 | syl13anc 1364 | . . . 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 23620 | . . . 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 1364 | . . 3 ⊢ (𝜑 → (((𝐵 / 𝐴) · (𝐴 / 𝐵)) · 𝑌) = ((𝐵 / 𝐴) · ((𝐴 / 𝐵) · 𝑌))) |
26 | 22, 23 | clmvs1 23624 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑌 ∈ 𝑉) → (1 · 𝑌) = 𝑌) |
27 | 4, 21, 26 | syl2anc 584 | . . 3 ⊢ (𝜑 → (1 · 𝑌) = 𝑌) |
28 | 16, 25, 27 | 3eqtr3d 2861 | . 2 ⊢ (𝜑 → ((𝐵 / 𝐴) · ((𝐴 / 𝐵) · 𝑌)) = 𝑌) |
29 | 2, 28 | eqtrd 2853 | 1 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 · cmul 10530 / cdiv 11285 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 ℂModcclm 23593 ℂVecccvs 23654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-subrg 19462 df-lmod 19565 df-lvec 19804 df-cnfld 20474 df-clm 23594 df-cvs 23655 |
This theorem is referenced by: ttgcontlem1 26598 |
Copyright terms: Public domain | W3C validator |