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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 7439 | . . . 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 21122 | . . . . . . . 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 3975 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
8 | eldifsni 4795 | . . . . . . . 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 2735 | . . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
13 | eqid 2735 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
14 | lvecinv.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝐹) | |
15 | 10, 11, 12, 13, 14 | drnginvrl 20773 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → ((𝐼‘𝐴)(.r‘𝐹)𝐴) = (1r‘𝐹)) |
16 | 5, 7, 9, 15 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → ((𝐼‘𝐴)(.r‘𝐹)𝐴) = (1r‘𝐹)) |
17 | 16 | oveq1d 7446 | . . . . 5 ⊢ (𝜑 → (((𝐼‘𝐴)(.r‘𝐹)𝐴) · 𝑌) = ((1r‘𝐹) · 𝑌)) |
18 | lveclmod 21123 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
19 | 2, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
20 | 10, 11, 14 | drnginvrcl 20770 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → (𝐼‘𝐴) ∈ 𝐾) |
21 | 5, 7, 9, 20 | syl3anc 1370 | . . . . . 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 20902 | . . . . . 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 1371 | . . . . 5 ⊢ (𝜑 → (((𝐼‘𝐴)(.r‘𝐹)𝐴) · 𝑌) = ((𝐼‘𝐴) · (𝐴 · 𝑌))) |
27 | 23, 3, 24, 13 | lmodvs1 20905 | . . . . . 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 2783 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝐴) · (𝐴 · 𝑌)) = 𝑌) |
30 | 1, 29 | sylan9eqr 2797 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (𝐴 · 𝑌)) → ((𝐼‘𝐴) · 𝑋) = 𝑌) |
31 | 10, 11, 12, 13, 14 | drnginvrr 20774 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → (𝐴(.r‘𝐹)(𝐼‘𝐴)) = (1r‘𝐹)) |
32 | 5, 7, 9, 31 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → (𝐴(.r‘𝐹)(𝐼‘𝐴)) = (1r‘𝐹)) |
33 | 32 | oveq1d 7446 | . . . . 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 20902 | . . . . . 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 1371 | . . . . 5 ⊢ (𝜑 → ((𝐴(.r‘𝐹)(𝐼‘𝐴)) · 𝑋) = (𝐴 · ((𝐼‘𝐴) · 𝑋))) |
37 | 23, 3, 24, 13 | lmodvs1 20905 | . . . . . 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 2784 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝐴 · ((𝐼‘𝐴) · 𝑋))) |
40 | oveq2 7439 | . . . 4 ⊢ (((𝐼‘𝐴) · 𝑋) = 𝑌 → (𝐴 · ((𝐼‘𝐴) · 𝑋)) = (𝐴 · 𝑌)) | |
41 | 39, 40 | sylan9eq 2795 | . . 3 ⊢ ((𝜑 ∧ ((𝐼‘𝐴) · 𝑋) = 𝑌) → 𝑋 = (𝐴 · 𝑌)) |
42 | 30, 41 | impbida 801 | . 2 ⊢ (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ ((𝐼‘𝐴) · 𝑋) = 𝑌)) |
43 | eqcom 2742 | . 2 ⊢ (((𝐼‘𝐴) · 𝑋) = 𝑌 ↔ 𝑌 = ((𝐼‘𝐴) · 𝑋)) | |
44 | 42, 43 | bitrdi 287 | 1 ⊢ (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ 𝑌 = ((𝐼‘𝐴) · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 1rcur 20199 invrcinvr 20404 DivRingcdr 20746 LModclmod 20875 LVecclvec 21119 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-drng 20748 df-lmod 20877 df-lvec 21120 |
This theorem is referenced by: lspexch 21149 prjspersym 42594 |
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