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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for lcfr 41587. Note that 𝑋 is z in Mario's notes. (Contributed by NM, 27-Feb-2015.) |
| Ref | Expression |
|---|---|
| lcfrlem1.v | ⊢ 𝑉 = (Base‘𝑈) |
| lcfrlem1.s | ⊢ 𝑆 = (Scalar‘𝑈) |
| lcfrlem1.q | ⊢ × = (.r‘𝑆) |
| lcfrlem1.z | ⊢ 0 = (0g‘𝑆) |
| lcfrlem1.i | ⊢ 𝐼 = (invr‘𝑆) |
| lcfrlem1.f | ⊢ 𝐹 = (LFnl‘𝑈) |
| lcfrlem1.d | ⊢ 𝐷 = (LDual‘𝑈) |
| lcfrlem1.t | ⊢ · = ( ·𝑠 ‘𝐷) |
| lcfrlem1.m | ⊢ − = (-g‘𝐷) |
| lcfrlem1.u | ⊢ (𝜑 → 𝑈 ∈ LVec) |
| lcfrlem1.e | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
| lcfrlem1.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
| lcfrlem1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| lcfrlem1.n | ⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 ) |
| lcfrlem1.h | ⊢ 𝐻 = (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)) |
| Ref | Expression |
|---|---|
| lcfrlem1 | ⊢ (𝜑 → (𝐻‘𝑋) = 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lcfrlem1.h | . . 3 ⊢ 𝐻 = (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)) | |
| 2 | 1 | fveq1i 6907 | . 2 ⊢ (𝐻‘𝑋) = ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) |
| 3 | lcfrlem1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
| 4 | lcfrlem1.s | . . . 4 ⊢ 𝑆 = (Scalar‘𝑈) | |
| 5 | eqid 2737 | . . . 4 ⊢ (-g‘𝑆) = (-g‘𝑆) | |
| 6 | lcfrlem1.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑈) | |
| 7 | lcfrlem1.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑈) | |
| 8 | lcfrlem1.m | . . . 4 ⊢ − = (-g‘𝐷) | |
| 9 | lcfrlem1.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec) | |
| 10 | lveclmod 21105 | . . . . 5 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod) | |
| 11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 12 | lcfrlem1.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
| 13 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
| 14 | lcfrlem1.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐷) | |
| 15 | 4 | lvecdrng 21104 | . . . . . . . 8 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing) |
| 16 | 9, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ DivRing) |
| 17 | lcfrlem1.g | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
| 18 | lcfrlem1.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 19 | 4, 13, 3, 6 | lflcl 39065 | . . . . . . . 8 ⊢ ((𝑈 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ (Base‘𝑆)) |
| 20 | 9, 17, 18, 19 | syl3anc 1373 | . . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑋) ∈ (Base‘𝑆)) |
| 21 | lcfrlem1.n | . . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 ) | |
| 22 | lcfrlem1.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑆) | |
| 23 | lcfrlem1.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝑆) | |
| 24 | 13, 22, 23 | drnginvrcl 20753 | . . . . . . 7 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
| 25 | 16, 20, 21, 24 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
| 26 | 4, 13, 3, 6 | lflcl 39065 | . . . . . . 7 ⊢ ((𝑈 ∈ LVec ∧ 𝐸 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
| 27 | 9, 12, 18, 26 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
| 28 | lcfrlem1.q | . . . . . . 7 ⊢ × = (.r‘𝑆) | |
| 29 | 4, 13, 28 | lmodmcl 20871 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆) ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆)) |
| 30 | 11, 25, 27, 29 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆)) |
| 31 | 6, 4, 13, 7, 14, 11, 30, 17 | ldualvscl 39140 | . . . 4 ⊢ (𝜑 → (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺) ∈ 𝐹) |
| 32 | 3, 4, 5, 6, 7, 8, 11, 12, 31, 18 | ldualvsubval 39158 | . . 3 ⊢ (𝜑 → ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) = ((𝐸‘𝑋)(-g‘𝑆)((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋))) |
| 33 | 6, 3, 4, 13, 28, 7, 14, 9, 30, 17, 18 | ldualvsval 39139 | . . . . 5 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)))) |
| 34 | eqid 2737 | . . . . . . . . 9 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
| 35 | 13, 22, 28, 34, 23 | drnginvrr 20757 | . . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1r‘𝑆)) |
| 36 | 16, 20, 21, 35 | syl3anc 1373 | . . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1r‘𝑆)) |
| 37 | 36 | oveq1d 7446 | . . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((1r‘𝑆) × (𝐸‘𝑋))) |
| 38 | 4 | lmodring 20866 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring) |
| 39 | 11, 38 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring) |
| 40 | 13, 28 | ringass 20250 | . . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ ((𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆) ∧ (𝐸‘𝑋) ∈ (Base‘𝑆))) → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)))) |
| 41 | 39, 20, 25, 27, 40 | syl13anc 1374 | . . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)))) |
| 42 | 13, 28, 34 | ringlidm 20266 | . . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((1r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋)) |
| 43 | 39, 27, 42 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ((1r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋)) |
| 44 | 37, 41, 43 | 3eqtr3d 2785 | . . . . 5 ⊢ (𝜑 → ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))) = (𝐸‘𝑋)) |
| 45 | 33, 44 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = (𝐸‘𝑋)) |
| 46 | 45 | oveq2d 7447 | . . 3 ⊢ (𝜑 → ((𝐸‘𝑋)(-g‘𝑆)((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋)) = ((𝐸‘𝑋)(-g‘𝑆)(𝐸‘𝑋))) |
| 47 | 4 | lmodfgrp 20867 | . . . . 5 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Grp) |
| 48 | 11, 47 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Grp) |
| 49 | 13, 22, 5 | grpsubid 19042 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((𝐸‘𝑋)(-g‘𝑆)(𝐸‘𝑋)) = 0 ) |
| 50 | 48, 27, 49 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝐸‘𝑋)(-g‘𝑆)(𝐸‘𝑋)) = 0 ) |
| 51 | 32, 46, 50 | 3eqtrd 2781 | . 2 ⊢ (𝜑 → ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) = 0 ) |
| 52 | 2, 51 | eqtrid 2789 | 1 ⊢ (𝜑 → (𝐻‘𝑋) = 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 Grpcgrp 18951 -gcsg 18953 1rcur 20178 Ringcrg 20230 invrcinvr 20387 DivRingcdr 20729 LModclmod 20858 LVecclvec 21101 LFnlclfn 39058 LDualcld 39124 |
| 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-tp 4631 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-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 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-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 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 df-lfl 39059 df-ldual 39125 |
| This theorem is referenced by: lcfrlem3 41546 |
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