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Mirrors > Home > MPE Home > Th. List > lvecinv | Structured version Visualization version GIF version |
Description: Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.) |
Ref | Expression |
---|---|
lvecinv.v | ⊢ 𝑉 = (Base‘𝑊) |
lvecinv.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lvecinv.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lvecinv.k | ⊢ 𝐾 = (Base‘𝐹) |
lvecinv.o | ⊢ 0 = (0g‘𝐹) |
lvecinv.i | ⊢ 𝐼 = (invr‘𝐹) |
lvecinv.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lvecinv.a | ⊢ (𝜑 → 𝐴 ∈ (𝐾 ∖ { 0 })) |
lvecinv.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lvecinv.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lvecinv | ⊢ (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ 𝑌 = ((𝐼‘𝐴) · 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7158 | . . . 4 ⊢ (𝑋 = (𝐴 · 𝑌) → ((𝐼‘𝐴) · 𝑋) = ((𝐼‘𝐴) · (𝐴 · 𝑌))) | |
2 | lvecinv.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
3 | lvecinv.f | . . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | 3 | lvecdrng 19871 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing) |
5 | 2, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ DivRing) |
6 | lvecinv.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ∖ { 0 })) | |
7 | 6 | eldifad 3948 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
8 | eldifsni 4716 | . . . . . . . 8 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) → 𝐴 ≠ 0 ) | |
9 | 6, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 0 ) |
10 | lvecinv.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹) | |
11 | lvecinv.o | . . . . . . . 8 ⊢ 0 = (0g‘𝐹) | |
12 | eqid 2821 | . . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
13 | eqid 2821 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
14 | lvecinv.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝐹) | |
15 | 10, 11, 12, 13, 14 | drnginvrl 19515 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → ((𝐼‘𝐴)(.r‘𝐹)𝐴) = (1r‘𝐹)) |
16 | 5, 7, 9, 15 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → ((𝐼‘𝐴)(.r‘𝐹)𝐴) = (1r‘𝐹)) |
17 | 16 | oveq1d 7165 | . . . . 5 ⊢ (𝜑 → (((𝐼‘𝐴)(.r‘𝐹)𝐴) · 𝑌) = ((1r‘𝐹) · 𝑌)) |
18 | lveclmod 19872 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
19 | 2, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
20 | 10, 11, 14 | drnginvrcl 19513 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → (𝐼‘𝐴) ∈ 𝐾) |
21 | 5, 7, 9, 20 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → (𝐼‘𝐴) ∈ 𝐾) |
22 | lvecinv.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
23 | lvecinv.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
24 | lvecinv.t | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊) | |
25 | 23, 3, 24, 10, 12 | lmodvsass 19653 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ ((𝐼‘𝐴) ∈ 𝐾 ∧ 𝐴 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉)) → (((𝐼‘𝐴)(.r‘𝐹)𝐴) · 𝑌) = ((𝐼‘𝐴) · (𝐴 · 𝑌))) |
26 | 19, 21, 7, 22, 25 | syl13anc 1368 | . . . . 5 ⊢ (𝜑 → (((𝐼‘𝐴)(.r‘𝐹)𝐴) · 𝑌) = ((𝐼‘𝐴) · (𝐴 · 𝑌))) |
27 | 23, 3, 24, 13 | lmodvs1 19656 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1r‘𝐹) · 𝑌) = 𝑌) |
28 | 19, 22, 27 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ((1r‘𝐹) · 𝑌) = 𝑌) |
29 | 17, 26, 28 | 3eqtr3d 2864 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝐴) · (𝐴 · 𝑌)) = 𝑌) |
30 | 1, 29 | sylan9eqr 2878 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (𝐴 · 𝑌)) → ((𝐼‘𝐴) · 𝑋) = 𝑌) |
31 | 10, 11, 12, 13, 14 | drnginvrr 19516 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → (𝐴(.r‘𝐹)(𝐼‘𝐴)) = (1r‘𝐹)) |
32 | 5, 7, 9, 31 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → (𝐴(.r‘𝐹)(𝐼‘𝐴)) = (1r‘𝐹)) |
33 | 32 | oveq1d 7165 | . . . . 5 ⊢ (𝜑 → ((𝐴(.r‘𝐹)(𝐼‘𝐴)) · 𝑋) = ((1r‘𝐹) · 𝑋)) |
34 | lvecinv.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
35 | 23, 3, 24, 10, 12 | lmodvsass 19653 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∈ 𝐾 ∧ (𝐼‘𝐴) ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝐴(.r‘𝐹)(𝐼‘𝐴)) · 𝑋) = (𝐴 · ((𝐼‘𝐴) · 𝑋))) |
36 | 19, 7, 21, 34, 35 | syl13anc 1368 | . . . . 5 ⊢ (𝜑 → ((𝐴(.r‘𝐹)(𝐼‘𝐴)) · 𝑋) = (𝐴 · ((𝐼‘𝐴) · 𝑋))) |
37 | 23, 3, 24, 13 | lmodvs1 19656 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((1r‘𝐹) · 𝑋) = 𝑋) |
38 | 19, 34, 37 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ((1r‘𝐹) · 𝑋) = 𝑋) |
39 | 33, 36, 38 | 3eqtr3rd 2865 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝐴 · ((𝐼‘𝐴) · 𝑋))) |
40 | oveq2 7158 | . . . 4 ⊢ (((𝐼‘𝐴) · 𝑋) = 𝑌 → (𝐴 · ((𝐼‘𝐴) · 𝑋)) = (𝐴 · 𝑌)) | |
41 | 39, 40 | sylan9eq 2876 | . . 3 ⊢ ((𝜑 ∧ ((𝐼‘𝐴) · 𝑋) = 𝑌) → 𝑋 = (𝐴 · 𝑌)) |
42 | 30, 41 | impbida 799 | . 2 ⊢ (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ ((𝐼‘𝐴) · 𝑋) = 𝑌)) |
43 | eqcom 2828 | . 2 ⊢ (((𝐼‘𝐴) · 𝑋) = 𝑌 ↔ 𝑌 = ((𝐼‘𝐴) · 𝑋)) | |
44 | 42, 43 | syl6bb 289 | 1 ⊢ (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ 𝑌 = ((𝐼‘𝐴) · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3933 {csn 4561 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 .rcmulr 16560 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 1rcur 19245 invrcinvr 19415 DivRingcdr 19496 LModclmod 19628 LVecclvec 19868 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-drng 19498 df-lmod 19630 df-lvec 19869 |
This theorem is referenced by: lspexch 19895 prjspersym 39250 |
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