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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 7362 | . . . 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 20562 | . . . . . . . 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 3921 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
8 | eldifsni 4749 | . . . . . . . 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 2736 | . . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
13 | eqid 2736 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
14 | lvecinv.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝐹) | |
15 | 10, 11, 12, 13, 14 | drnginvrl 20204 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → ((𝐼‘𝐴)(.r‘𝐹)𝐴) = (1r‘𝐹)) |
16 | 5, 7, 9, 15 | syl3anc 1371 | . . . . . 6 ⊢ (𝜑 → ((𝐼‘𝐴)(.r‘𝐹)𝐴) = (1r‘𝐹)) |
17 | 16 | oveq1d 7369 | . . . . 5 ⊢ (𝜑 → (((𝐼‘𝐴)(.r‘𝐹)𝐴) · 𝑌) = ((1r‘𝐹) · 𝑌)) |
18 | lveclmod 20563 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
19 | 2, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
20 | 10, 11, 14 | drnginvrcl 20201 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → (𝐼‘𝐴) ∈ 𝐾) |
21 | 5, 7, 9, 20 | syl3anc 1371 | . . . . . 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 20343 | . . . . . 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 1372 | . . . . 5 ⊢ (𝜑 → (((𝐼‘𝐴)(.r‘𝐹)𝐴) · 𝑌) = ((𝐼‘𝐴) · (𝐴 · 𝑌))) |
27 | 23, 3, 24, 13 | lmodvs1 20346 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1r‘𝐹) · 𝑌) = 𝑌) |
28 | 19, 22, 27 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((1r‘𝐹) · 𝑌) = 𝑌) |
29 | 17, 26, 28 | 3eqtr3d 2784 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝐴) · (𝐴 · 𝑌)) = 𝑌) |
30 | 1, 29 | sylan9eqr 2798 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (𝐴 · 𝑌)) → ((𝐼‘𝐴) · 𝑋) = 𝑌) |
31 | 10, 11, 12, 13, 14 | drnginvrr 20205 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → (𝐴(.r‘𝐹)(𝐼‘𝐴)) = (1r‘𝐹)) |
32 | 5, 7, 9, 31 | syl3anc 1371 | . . . . . 6 ⊢ (𝜑 → (𝐴(.r‘𝐹)(𝐼‘𝐴)) = (1r‘𝐹)) |
33 | 32 | oveq1d 7369 | . . . . 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 20343 | . . . . . 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 1372 | . . . . 5 ⊢ (𝜑 → ((𝐴(.r‘𝐹)(𝐼‘𝐴)) · 𝑋) = (𝐴 · ((𝐼‘𝐴) · 𝑋))) |
37 | 23, 3, 24, 13 | lmodvs1 20346 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((1r‘𝐹) · 𝑋) = 𝑋) |
38 | 19, 34, 37 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((1r‘𝐹) · 𝑋) = 𝑋) |
39 | 33, 36, 38 | 3eqtr3rd 2785 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝐴 · ((𝐼‘𝐴) · 𝑋))) |
40 | oveq2 7362 | . . . 4 ⊢ (((𝐼‘𝐴) · 𝑋) = 𝑌 → (𝐴 · ((𝐼‘𝐴) · 𝑋)) = (𝐴 · 𝑌)) | |
41 | 39, 40 | sylan9eq 2796 | . . 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 2743 | . 2 ⊢ (((𝐼‘𝐴) · 𝑋) = 𝑌 ↔ 𝑌 = ((𝐼‘𝐴) · 𝑋)) | |
44 | 42, 43 | bitrdi 286 | 1 ⊢ (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ 𝑌 = ((𝐼‘𝐴) · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ≠ wne 2942 ∖ cdif 3906 {csn 4585 ‘cfv 6494 (class class class)co 7354 Basecbs 17080 .rcmulr 17131 Scalarcsca 17133 ·𝑠 cvsca 17134 0gc0g 17318 1rcur 19909 invrcinvr 20096 DivRingcdr 20181 LModclmod 20318 LVecclvec 20559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-2nd 7919 df-tpos 8154 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-3 12214 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-plusg 17143 df-mulr 17144 df-0g 17320 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-grp 18748 df-minusg 18749 df-mgp 19893 df-ur 19910 df-ring 19962 df-oppr 20045 df-dvdsr 20066 df-unit 20067 df-invr 20097 df-drng 20183 df-lmod 20320 df-lvec 20560 |
This theorem is referenced by: lspexch 20586 prjspersym 40921 |
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