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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 21104 | . . . . . . . 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 3963 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
| 8 | eldifsni 4790 | . . . . . . . 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 2737 | . . . . . . . 8 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
| 13 | eqid 2737 | . . . . . . . 8 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
| 14 | lvecinv.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝐹) | |
| 15 | 10, 11, 12, 13, 14 | drnginvrl 20756 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → ((𝐼‘𝐴)(.r‘𝐹)𝐴) = (1r‘𝐹)) |
| 16 | 5, 7, 9, 15 | syl3anc 1373 | . . . . . 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 21105 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
| 19 | 2, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
| 20 | 10, 11, 14 | drnginvrcl 20753 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → (𝐼‘𝐴) ∈ 𝐾) |
| 21 | 5, 7, 9, 20 | syl3anc 1373 | . . . . . 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 20885 | . . . . . 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 1374 | . . . . 5 ⊢ (𝜑 → (((𝐼‘𝐴)(.r‘𝐹)𝐴) · 𝑌) = ((𝐼‘𝐴) · (𝐴 · 𝑌))) |
| 27 | 23, 3, 24, 13 | lmodvs1 20888 | . . . . . 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 2785 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝐴) · (𝐴 · 𝑌)) = 𝑌) |
| 30 | 1, 29 | sylan9eqr 2799 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (𝐴 · 𝑌)) → ((𝐼‘𝐴) · 𝑋) = 𝑌) |
| 31 | 10, 11, 12, 13, 14 | drnginvrr 20757 | . . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → (𝐴(.r‘𝐹)(𝐼‘𝐴)) = (1r‘𝐹)) |
| 32 | 5, 7, 9, 31 | syl3anc 1373 | . . . . . 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 20885 | . . . . . 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 1374 | . . . . 5 ⊢ (𝜑 → ((𝐴(.r‘𝐹)(𝐼‘𝐴)) · 𝑋) = (𝐴 · ((𝐼‘𝐴) · 𝑋))) |
| 37 | 23, 3, 24, 13 | lmodvs1 20888 | . . . . . 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 2786 | . . . 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 2797 | . . 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 2744 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 1rcur 20178 invrcinvr 20387 DivRingcdr 20729 LModclmod 20858 LVecclvec 21101 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-drng 20731 df-lmod 20860 df-lvec 21102 |
| This theorem is referenced by: lspexch 21131 prjspersym 42617 |
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