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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem33 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 41295. (Contributed by NM, 10-Mar-2015.) |
Ref | Expression |
---|---|
lcfrlem17.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfrlem17.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfrlem17.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfrlem17.v | ⊢ 𝑉 = (Base‘𝑈) |
lcfrlem17.p | ⊢ + = (+g‘𝑈) |
lcfrlem17.z | ⊢ 0 = (0g‘𝑈) |
lcfrlem17.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lcfrlem17.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
lcfrlem17.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfrlem17.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lcfrlem17.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
lcfrlem17.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
lcfrlem22.b | ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) |
lcfrlem24.t | ⊢ · = ( ·𝑠 ‘𝑈) |
lcfrlem24.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lcfrlem24.q | ⊢ 𝑄 = (0g‘𝑆) |
lcfrlem24.r | ⊢ 𝑅 = (Base‘𝑆) |
lcfrlem24.j | ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) |
lcfrlem24.ib | ⊢ (𝜑 → 𝐼 ∈ 𝐵) |
lcfrlem24.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfrlem25.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcfrlem28.jn | ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) |
lcfrlem29.i | ⊢ 𝐹 = (invr‘𝑆) |
lcfrlem30.m | ⊢ − = (-g‘𝐷) |
lcfrlem30.c | ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) |
lcfrlem33.xi | ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) = 𝑄) |
Ref | Expression |
---|---|
lcfrlem33 | ⊢ (𝜑 → 𝐶 ≠ (0g‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem30.c | . . 3 ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) | |
2 | lcfrlem33.xi | . . . . . . . . 9 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) = 𝑄) | |
3 | 2 | oveq2d 7430 | . . . . . . . 8 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼)) = ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)𝑄)) |
4 | lcfrlem17.h | . . . . . . . . . . 11 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | lcfrlem17.u | . . . . . . . . . . 11 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | lcfrlem17.k | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 4, 5, 6 | dvhlmod 40820 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LMod) |
8 | lcfrlem24.s | . . . . . . . . . . 11 ⊢ 𝑆 = (Scalar‘𝑈) | |
9 | 8 | lmodring 20838 | . . . . . . . . . 10 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring) |
10 | 7, 9 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Ring) |
11 | 4, 5, 6 | dvhlvec 40819 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LVec) |
12 | 8 | lvecdrng 21077 | . . . . . . . . . . 11 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing) |
13 | 11, 12 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ DivRing) |
14 | lcfrlem17.o | . . . . . . . . . . . 12 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
15 | lcfrlem17.v | . . . . . . . . . . . 12 ⊢ 𝑉 = (Base‘𝑈) | |
16 | lcfrlem17.p | . . . . . . . . . . . 12 ⊢ + = (+g‘𝑈) | |
17 | lcfrlem24.t | . . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝑈) | |
18 | lcfrlem24.r | . . . . . . . . . . . 12 ⊢ 𝑅 = (Base‘𝑆) | |
19 | lcfrlem17.z | . . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑈) | |
20 | eqid 2726 | . . . . . . . . . . . 12 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
21 | lcfrlem24.l | . . . . . . . . . . . 12 ⊢ 𝐿 = (LKer‘𝑈) | |
22 | lcfrlem25.d | . . . . . . . . . . . 12 ⊢ 𝐷 = (LDual‘𝑈) | |
23 | eqid 2726 | . . . . . . . . . . . 12 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
24 | eqid 2726 | . . . . . . . . . . . 12 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
25 | lcfrlem24.j | . . . . . . . . . . . 12 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) | |
26 | lcfrlem17.y | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
27 | 4, 14, 5, 15, 16, 17, 8, 18, 19, 20, 21, 22, 23, 24, 25, 6, 26 | lcfrlem10 41262 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈)) |
28 | lcfrlem17.a | . . . . . . . . . . . . 13 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
29 | lcfrlem17.n | . . . . . . . . . . . . . 14 ⊢ 𝑁 = (LSpan‘𝑈) | |
30 | lcfrlem17.x | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
31 | lcfrlem17.ne | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
32 | lcfrlem22.b | . . . . . . . . . . . . . 14 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
33 | 4, 14, 5, 15, 16, 19, 29, 28, 6, 30, 26, 31, 32 | lcfrlem22 41274 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
34 | 15, 28, 7, 33 | lsatssv 38707 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ⊆ 𝑉) |
35 | lcfrlem24.ib | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
36 | 34, 35 | sseldd 3980 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
37 | 8, 18, 15, 20 | lflcl 38773 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅) |
38 | 7, 27, 36, 37 | syl3anc 1368 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅) |
39 | lcfrlem28.jn | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) | |
40 | lcfrlem24.q | . . . . . . . . . . 11 ⊢ 𝑄 = (0g‘𝑆) | |
41 | lcfrlem29.i | . . . . . . . . . . 11 ⊢ 𝐹 = (invr‘𝑆) | |
42 | 18, 40, 41 | drnginvrcl 20725 | . . . . . . . . . 10 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅) |
43 | 13, 38, 39, 42 | syl3anc 1368 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅) |
44 | eqid 2726 | . . . . . . . . . 10 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
45 | 18, 44, 40 | ringrz 20267 | . . . . . . . . 9 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅) → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)𝑄) = 𝑄) |
46 | 10, 43, 45 | syl2anc 582 | . . . . . . . 8 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)𝑄) = 𝑄) |
47 | 3, 46 | eqtrd 2766 | . . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼)) = 𝑄) |
48 | 47 | oveq1d 7429 | . . . . . 6 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)) = (𝑄( ·𝑠 ‘𝐷)(𝐽‘𝑌))) |
49 | eqid 2726 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
50 | 20, 8, 40, 22, 49, 23, 7, 27 | ldual0vs 38869 | . . . . . 6 ⊢ (𝜑 → (𝑄( ·𝑠 ‘𝐷)(𝐽‘𝑌)) = (0g‘𝐷)) |
51 | 48, 50 | eqtrd 2766 | . . . . 5 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)) = (0g‘𝐷)) |
52 | 51 | oveq2d 7430 | . . . 4 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) = ((𝐽‘𝑋) − (0g‘𝐷))) |
53 | 22, 7 | ldualgrp 38855 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ Grp) |
54 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
55 | 4, 14, 5, 15, 16, 17, 8, 18, 19, 20, 21, 22, 23, 24, 25, 6, 30 | lcfrlem10 41262 | . . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LFnl‘𝑈)) |
56 | 20, 22, 54, 7, 55 | ldualelvbase 38836 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (Base‘𝐷)) |
57 | lcfrlem30.m | . . . . . 6 ⊢ − = (-g‘𝐷) | |
58 | 54, 23, 57 | grpsubid1 19013 | . . . . 5 ⊢ ((𝐷 ∈ Grp ∧ (𝐽‘𝑋) ∈ (Base‘𝐷)) → ((𝐽‘𝑋) − (0g‘𝐷)) = (𝐽‘𝑋)) |
59 | 53, 56, 58 | syl2anc 582 | . . . 4 ⊢ (𝜑 → ((𝐽‘𝑋) − (0g‘𝐷)) = (𝐽‘𝑋)) |
60 | 52, 59 | eqtrd 2766 | . . 3 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) = (𝐽‘𝑋)) |
61 | 1, 60 | eqtrid 2778 | . 2 ⊢ (𝜑 → 𝐶 = (𝐽‘𝑋)) |
62 | 4, 14, 5, 15, 16, 17, 8, 18, 19, 20, 21, 22, 23, 24, 25, 6, 30 | lcfrlem13 41265 | . . 3 ⊢ (𝜑 → (𝐽‘𝑋) ∈ ({𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∖ {(0g‘𝐷)})) |
63 | eldifsni 4790 | . . 3 ⊢ ((𝐽‘𝑋) ∈ ({𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∖ {(0g‘𝐷)}) → (𝐽‘𝑋) ≠ (0g‘𝐷)) | |
64 | 62, 63 | syl 17 | . 2 ⊢ (𝜑 → (𝐽‘𝑋) ≠ (0g‘𝐷)) |
65 | 61, 64 | eqnetrd 2998 | 1 ⊢ (𝜑 → 𝐶 ≠ (0g‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 {crab 3420 ∖ cdif 3944 ∩ cin 3946 {csn 4624 {cpr 4626 ↦ cmpt 5227 ‘cfv 6544 ℩crio 7369 (class class class)co 7414 Basecbs 17206 +gcplusg 17259 .rcmulr 17260 Scalarcsca 17262 ·𝑠 cvsca 17263 0gc0g 17447 Grpcgrp 18921 -gcsg 18923 Ringcrg 20210 invrcinvr 20363 DivRingcdr 20701 LModclmod 20830 LSpanclspn 20942 LVecclvec 21074 LSAtomsclsa 38683 LFnlclfn 38766 LKerclk 38794 LDualcld 38832 HLchlt 39059 LHypclh 39694 DVecHcdvh 40788 ocHcoch 41057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-riotaBAD 38662 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-iin 4997 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7867 df-1st 7993 df-2nd 7994 df-tpos 8231 df-undef 8278 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-n0 12517 df-z 12603 df-uz 12867 df-fz 13531 df-struct 17142 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-mulr 17273 df-sca 17275 df-vsca 17276 df-0g 17449 df-mre 17592 df-mrc 17593 df-acs 17595 df-proset 18313 df-poset 18331 df-plt 18348 df-lub 18364 df-glb 18365 df-join 18366 df-meet 18367 df-p0 18443 df-p1 18444 df-lat 18450 df-clat 18517 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-subg 19111 df-cntz 19305 df-oppg 19334 df-lsm 19628 df-cmn 19774 df-abl 19775 df-mgp 20112 df-rng 20130 df-ur 20159 df-ring 20212 df-oppr 20310 df-dvdsr 20333 df-unit 20334 df-invr 20364 df-dvr 20377 df-drng 20703 df-lmod 20832 df-lss 20903 df-lsp 20943 df-lvec 21075 df-lsatoms 38685 df-lshyp 38686 df-lcv 38728 df-lfl 38767 df-lkr 38795 df-ldual 38833 df-oposet 38885 df-ol 38887 df-oml 38888 df-covers 38975 df-ats 38976 df-atl 39007 df-cvlat 39031 df-hlat 39060 df-llines 39208 df-lplanes 39209 df-lvols 39210 df-lines 39211 df-psubsp 39213 df-pmap 39214 df-padd 39506 df-lhyp 39698 df-laut 39699 df-ldil 39814 df-ltrn 39815 df-trl 39869 df-tgrp 40453 df-tendo 40465 df-edring 40467 df-dveca 40713 df-disoa 40739 df-dvech 40789 df-dib 40849 df-dic 40883 df-dih 40939 df-doch 41058 df-djh 41105 |
This theorem is referenced by: lcfrlem34 41286 |
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