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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 7172 | . 2 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) = ((𝐵 / 𝐴) · ((𝐴 / 𝐵) · 𝑌))) |
3 | cvsdiveqd.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℂVec) | |
4 | 3 | cvsclm 23730 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
5 | cvsdiveqd.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | cvsdiveqd.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
7 | 5, 6 | clmsscn 23683 | . . . . . . 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 3968 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
11 | cvsdiveqd.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
12 | 8, 11 | sseldd 3968 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | cvsdiveqd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 0) | |
14 | cvsdiveqd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) | |
15 | 10, 12, 13, 14 | divcan6d 11435 | . . . 4 ⊢ (𝜑 → ((𝐵 / 𝐴) · (𝐴 / 𝐵)) = 1) |
16 | 15 | oveq1d 7171 | . . 3 ⊢ (𝜑 → (((𝐵 / 𝐴) · (𝐴 / 𝐵)) · 𝑌) = (1 · 𝑌)) |
17 | 5, 6 | cvsdivcl 23737 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐵 ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0)) → (𝐵 / 𝐴) ∈ 𝐾) |
18 | 3, 9, 11, 14, 17 | syl13anc 1368 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐴) ∈ 𝐾) |
19 | 5, 6 | cvsdivcl 23737 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾) |
20 | 3, 11, 9, 13, 19 | syl13anc 1368 | . . . 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 23693 | . . . 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 1368 | . . 3 ⊢ (𝜑 → (((𝐵 / 𝐴) · (𝐴 / 𝐵)) · 𝑌) = ((𝐵 / 𝐴) · ((𝐴 / 𝐵) · 𝑌))) |
26 | 22, 23 | clmvs1 23697 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑌 ∈ 𝑉) → (1 · 𝑌) = 𝑌) |
27 | 4, 21, 26 | syl2anc 586 | . . 3 ⊢ (𝜑 → (1 · 𝑌) = 𝑌) |
28 | 16, 25, 27 | 3eqtr3d 2864 | . 2 ⊢ (𝜑 → ((𝐵 / 𝐴) · ((𝐴 / 𝐵) · 𝑌)) = 𝑌) |
29 | 2, 28 | eqtrd 2856 | 1 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 · cmul 10542 / cdiv 11297 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 ℂModcclm 23666 ℂVecccvs 23727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-subg 18276 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-subrg 19533 df-lmod 19636 df-lvec 19875 df-cnfld 20546 df-clm 23667 df-cvs 23728 |
This theorem is referenced by: ttgcontlem1 26671 |
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