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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 6557 | . . . 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 18926 | . . . . . . . 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 3552 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
8 | eldifsni 4261 | . . . . . . . 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 2610 | . . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
13 | eqid 2610 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
14 | lvecinv.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝐹) | |
15 | 10, 11, 12, 13, 14 | drnginvrl 18589 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → ((𝐼‘𝐴)(.r‘𝐹)𝐴) = (1r‘𝐹)) |
16 | 5, 7, 9, 15 | syl3anc 1318 | . . . . . 6 ⊢ (𝜑 → ((𝐼‘𝐴)(.r‘𝐹)𝐴) = (1r‘𝐹)) |
17 | 16 | oveq1d 6564 | . . . . 5 ⊢ (𝜑 → (((𝐼‘𝐴)(.r‘𝐹)𝐴) · 𝑌) = ((1r‘𝐹) · 𝑌)) |
18 | lveclmod 18927 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
19 | 2, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
20 | 10, 11, 14 | drnginvrcl 18587 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → (𝐼‘𝐴) ∈ 𝐾) |
21 | 5, 7, 9, 20 | syl3anc 1318 | . . . . . 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 18711 | . . . . . 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 1320 | . . . . 5 ⊢ (𝜑 → (((𝐼‘𝐴)(.r‘𝐹)𝐴) · 𝑌) = ((𝐼‘𝐴) · (𝐴 · 𝑌))) |
27 | 23, 3, 24, 13 | lmodvs1 18714 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1r‘𝐹) · 𝑌) = 𝑌) |
28 | 19, 22, 27 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → ((1r‘𝐹) · 𝑌) = 𝑌) |
29 | 17, 26, 28 | 3eqtr3d 2652 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝐴) · (𝐴 · 𝑌)) = 𝑌) |
30 | 1, 29 | sylan9eqr 2666 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (𝐴 · 𝑌)) → ((𝐼‘𝐴) · 𝑋) = 𝑌) |
31 | 10, 11, 12, 13, 14 | drnginvrr 18590 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → (𝐴(.r‘𝐹)(𝐼‘𝐴)) = (1r‘𝐹)) |
32 | 5, 7, 9, 31 | syl3anc 1318 | . . . . . 6 ⊢ (𝜑 → (𝐴(.r‘𝐹)(𝐼‘𝐴)) = (1r‘𝐹)) |
33 | 32 | oveq1d 6564 | . . . . 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 18711 | . . . . . 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 1320 | . . . . 5 ⊢ (𝜑 → ((𝐴(.r‘𝐹)(𝐼‘𝐴)) · 𝑋) = (𝐴 · ((𝐼‘𝐴) · 𝑋))) |
37 | 23, 3, 24, 13 | lmodvs1 18714 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((1r‘𝐹) · 𝑋) = 𝑋) |
38 | 19, 34, 37 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → ((1r‘𝐹) · 𝑋) = 𝑋) |
39 | 33, 36, 38 | 3eqtr3rd 2653 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝐴 · ((𝐼‘𝐴) · 𝑋))) |
40 | oveq2 6557 | . . . 4 ⊢ (((𝐼‘𝐴) · 𝑋) = 𝑌 → (𝐴 · ((𝐼‘𝐴) · 𝑋)) = (𝐴 · 𝑌)) | |
41 | 39, 40 | sylan9eq 2664 | . . 3 ⊢ ((𝜑 ∧ ((𝐼‘𝐴) · 𝑋) = 𝑌) → 𝑋 = (𝐴 · 𝑌)) |
42 | 30, 41 | impbida 873 | . 2 ⊢ (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ ((𝐼‘𝐴) · 𝑋) = 𝑌)) |
43 | eqcom 2617 | . 2 ⊢ (((𝐼‘𝐴) · 𝑋) = 𝑌 ↔ 𝑌 = ((𝐼‘𝐴) · 𝑋)) | |
44 | 42, 43 | syl6bb 275 | 1 ⊢ (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ 𝑌 = ((𝐼‘𝐴) · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 1rcur 18324 invrcinvr 18494 DivRingcdr 18570 LModclmod 18686 LVecclvec 18923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-drng 18572 df-lmod 18688 df-lvec 18924 |
This theorem is referenced by: lspexch 18950 |
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