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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem1 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 41568. 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 6908 | . 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 2735 | . . . 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 21123 | . . . . 5 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | lcfrlem1.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
13 | eqid 2735 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
14 | lcfrlem1.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐷) | |
15 | 4 | lvecdrng 21122 | . . . . . . . 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 39046 | . . . . . . . 8 ⊢ ((𝑈 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ (Base‘𝑆)) |
20 | 9, 17, 18, 19 | syl3anc 1370 | . . . . . . 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 20770 | . . . . . . 7 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
25 | 16, 20, 21, 24 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
26 | 4, 13, 3, 6 | lflcl 39046 | . . . . . . 7 ⊢ ((𝑈 ∈ LVec ∧ 𝐸 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
27 | 9, 12, 18, 26 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
28 | lcfrlem1.q | . . . . . . 7 ⊢ × = (.r‘𝑆) | |
29 | 4, 13, 28 | lmodmcl 20888 | . . . . . 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 1370 | . . . . 5 ⊢ (𝜑 → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆)) |
31 | 6, 4, 13, 7, 14, 11, 30, 17 | ldualvscl 39121 | . . . 4 ⊢ (𝜑 → (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺) ∈ 𝐹) |
32 | 3, 4, 5, 6, 7, 8, 11, 12, 31, 18 | ldualvsubval 39139 | . . 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 39120 | . . . . 5 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)))) |
34 | eqid 2735 | . . . . . . . . 9 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
35 | 13, 22, 28, 34, 23 | drnginvrr 20774 | . . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1r‘𝑆)) |
36 | 16, 20, 21, 35 | syl3anc 1370 | . . . . . . 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 20883 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring) |
39 | 11, 38 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring) |
40 | 13, 28 | ringass 20271 | . . . . . . 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 1371 | . . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)))) |
42 | 13, 28, 34 | ringlidm 20283 | . . . . . . 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 2783 | . . . . 5 ⊢ (𝜑 → ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))) = (𝐸‘𝑋)) |
45 | 33, 44 | eqtrd 2775 | . . . 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 20884 | . . . . 5 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Grp) |
48 | 11, 47 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Grp) |
49 | 13, 22, 5 | grpsubid 19055 | . . . 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 2779 | . 2 ⊢ (𝜑 → ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) = 0 ) |
52 | 2, 51 | eqtrid 2787 | 1 ⊢ (𝜑 → (𝐻‘𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 Grpcgrp 18964 -gcsg 18966 1rcur 20199 Ringcrg 20251 invrcinvr 20404 DivRingcdr 20746 LModclmod 20875 LVecclvec 21119 LFnlclfn 39039 LDualcld 39105 |
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-tp 4636 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-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 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-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 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 df-lfl 39040 df-ldual 39106 |
This theorem is referenced by: lcfrlem3 41527 |
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