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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 7174 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐴 · 𝑋)) = ((1 / 𝐴) · (𝐵 · 𝑌))) |
3 | cvsdiveqd.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℂVec) | |
4 | 3 | cvsclm 23732 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
5 | cvsdiveqd.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | cvsdiveqd.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
7 | 5, 6 | clmsscn 23685 | . . . . . . 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 3970 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
11 | cvsdiveqd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) | |
12 | 10, 11 | recid2d 11414 | . . . 4 ⊢ (𝜑 → ((1 / 𝐴) · 𝐴) = 1) |
13 | 12 | oveq1d 7173 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐴) · 𝑋) = (1 · 𝑋)) |
14 | 5 | clm1 23679 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘𝐹)) |
15 | 4, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 1 = (1r‘𝐹)) |
16 | 5 | clmring 23676 | . . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring) |
17 | eqid 2823 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
18 | 6, 17 | ringidcl 19320 | . . . . . . 7 ⊢ (𝐹 ∈ Ring → (1r‘𝐹) ∈ 𝐾) |
19 | 4, 16, 18 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐹) ∈ 𝐾) |
20 | 15, 19 | eqeltrd 2915 | . . . . 5 ⊢ (𝜑 → 1 ∈ 𝐾) |
21 | 5, 6 | cvsdivcl 23739 | . . . . 5 ⊢ ((𝑊 ∈ ℂVec ∧ (1 ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0)) → (1 / 𝐴) ∈ 𝐾) |
22 | 3, 20, 9, 11, 21 | syl13anc 1368 | . . . 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 23695 | . . . 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 1368 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐴) · 𝑋) = ((1 / 𝐴) · (𝐴 · 𝑋))) |
28 | 24, 25 | clmvs1 23699 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (1 · 𝑋) = 𝑋) |
29 | 4, 23, 28 | syl2anc 586 | . . 3 ⊢ (𝜑 → (1 · 𝑋) = 𝑋) |
30 | 13, 27, 29 | 3eqtr3d 2866 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐴 · 𝑋)) = 𝑋) |
31 | cvsdiveqd.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐾) | |
32 | 8, 31 | sseldd 3970 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
33 | 32, 10, 11 | divrec2d 11422 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐴) = ((1 / 𝐴) · 𝐵)) |
34 | 33 | oveq1d 7173 | . . 3 ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑌) = (((1 / 𝐴) · 𝐵) · 𝑌)) |
35 | cvsdiveqd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
36 | 24, 5, 25, 6 | clmvsass 23695 | . . . 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 1368 | . . 3 ⊢ (𝜑 → (((1 / 𝐴) · 𝐵) · 𝑌) = ((1 / 𝐴) · (𝐵 · 𝑌))) |
38 | 34, 37 | eqtr2d 2859 | . 2 ⊢ (𝜑 → ((1 / 𝐴) · (𝐵 · 𝑌)) = ((𝐵 / 𝐴) · 𝑌)) |
39 | 2, 30, 38 | 3eqtr3d 2866 | 1 ⊢ (𝜑 → 𝑋 = ((𝐵 / 𝐴) · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ⊆ wss 3938 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 · cmul 10544 / cdiv 11299 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 1rcur 19253 Ringcrg 19299 ℂModcclm 23668 ℂVecccvs 23729 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-subg 18278 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-subrg 19535 df-lmod 19638 df-lvec 19877 df-cnfld 20548 df-clm 23669 df-cvs 23730 |
This theorem is referenced by: ttgcontlem1 26673 |
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