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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem37 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 41284. (Contributed by NM, 8-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))) |
lcfrlem37.g | ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷)) |
lcfrlem37.gs | ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) |
lcfrlem37.e | ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) |
lcfrlem37.xe | ⊢ (𝜑 → 𝑋 ∈ 𝐸) |
lcfrlem37.ye | ⊢ (𝜑 → 𝑌 ∈ 𝐸) |
Ref | Expression |
---|---|
lcfrlem37 | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem30.c | . . . . 5 ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) | |
2 | lcfrlem25.d | . . . . . 6 ⊢ 𝐷 = (LDual‘𝑈) | |
3 | lcfrlem30.m | . . . . . 6 ⊢ − = (-g‘𝐷) | |
4 | eqid 2726 | . . . . . 6 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷) | |
5 | lcfrlem17.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | lcfrlem17.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | lcfrlem17.k | . . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | 5, 6, 7 | dvhlmod 40809 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
9 | lcfrlem37.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷)) | |
10 | lcfrlem17.o | . . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
11 | lcfrlem17.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
12 | lcfrlem17.p | . . . . . . 7 ⊢ + = (+g‘𝑈) | |
13 | lcfrlem24.t | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑈) | |
14 | lcfrlem24.s | . . . . . . 7 ⊢ 𝑆 = (Scalar‘𝑈) | |
15 | lcfrlem24.r | . . . . . . 7 ⊢ 𝑅 = (Base‘𝑆) | |
16 | lcfrlem17.z | . . . . . . 7 ⊢ 0 = (0g‘𝑈) | |
17 | eqid 2726 | . . . . . . 7 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
18 | lcfrlem24.l | . . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈) | |
19 | eqid 2726 | . . . . . . 7 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
20 | eqid 2726 | . . . . . . 7 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
21 | lcfrlem24.j | . . . . . . 7 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) | |
22 | lcfrlem37.gs | . . . . . . 7 ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) | |
23 | lcfrlem37.e | . . . . . . 7 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) | |
24 | lcfrlem37.xe | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
25 | lcfrlem17.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
26 | eldifsni 4799 | . . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
27 | 25, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
28 | eldifsn 4795 | . . . . . . . 8 ⊢ (𝑋 ∈ (𝐸 ∖ { 0 }) ↔ (𝑋 ∈ 𝐸 ∧ 𝑋 ≠ 0 )) | |
29 | 24, 27, 28 | sylanbrc 581 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐸 ∖ { 0 })) |
30 | 5, 10, 6, 11, 12, 13, 14, 15, 16, 17, 18, 2, 19, 20, 21, 7, 4, 9, 22, 23, 29 | lcfrlem16 41257 | . . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺) |
31 | eqid 2726 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
32 | lcfrlem17.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈) | |
33 | lcfrlem17.a | . . . . . . . 8 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
34 | lcfrlem17.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
35 | lcfrlem17.ne | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
36 | lcfrlem22.b | . . . . . . . 8 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
37 | lcfrlem24.q | . . . . . . . 8 ⊢ 𝑄 = (0g‘𝑆) | |
38 | lcfrlem24.ib | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
39 | lcfrlem28.jn | . . . . . . . 8 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) | |
40 | lcfrlem29.i | . . . . . . . 8 ⊢ 𝐹 = (invr‘𝑆) | |
41 | 5, 10, 6, 11, 12, 16, 32, 33, 7, 25, 34, 35, 36, 13, 14, 37, 15, 21, 38, 18, 2, 39, 40 | lcfrlem29 41270 | . . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅) |
42 | lcfrlem37.ye | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐸) | |
43 | eldifsni 4799 | . . . . . . . . . 10 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 ) | |
44 | 34, 43 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ 0 ) |
45 | eldifsn 4795 | . . . . . . . . 9 ⊢ (𝑌 ∈ (𝐸 ∖ { 0 }) ↔ (𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 )) | |
46 | 42, 44, 45 | sylanbrc 581 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝐸 ∖ { 0 })) |
47 | 5, 10, 6, 11, 12, 13, 14, 15, 16, 17, 18, 2, 19, 20, 21, 7, 4, 9, 22, 23, 46 | lcfrlem16 41257 | . . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ 𝐺) |
48 | 14, 15, 2, 31, 4, 8, 9, 41, 47 | ldualssvscl 38856 | . . . . . 6 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)) ∈ 𝐺) |
49 | 2, 3, 4, 8, 9, 30, 48 | ldualssvsubcl 38857 | . . . . 5 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) ∈ 𝐺) |
50 | 1, 49 | eqeltrid 2830 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐺) |
51 | 5, 10, 6, 11, 12, 16, 32, 33, 7, 25, 34, 35, 36, 13, 14, 37, 15, 21, 38, 18, 2, 39, 40, 3, 1 | lcfrlem36 41277 | . . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) |
52 | 2fveq3 6906 | . . . . . 6 ⊢ (𝑔 = 𝐶 → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘𝐶))) | |
53 | 52 | eleq2d 2812 | . . . . 5 ⊢ (𝑔 = 𝐶 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))) |
54 | 53 | rspcev 3608 | . . . 4 ⊢ ((𝐶 ∈ 𝐺 ∧ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
55 | 50, 51, 54 | syl2anc 582 | . . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
56 | eliun 5005 | . . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) | |
57 | 55, 56 | sylibr 233 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))) |
58 | 57, 23 | eleqtrrdi 2837 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 {crab 3419 ∖ cdif 3944 ∩ cin 3946 ⊆ wss 3947 {csn 4633 {cpr 4635 ∪ ciun 5001 ↦ cmpt 5236 ‘cfv 6554 ℩crio 7379 (class class class)co 7424 Basecbs 17213 +gcplusg 17266 .rcmulr 17267 Scalarcsca 17269 ·𝑠 cvsca 17270 0gc0g 17454 -gcsg 18930 invrcinvr 20369 LSubSpclss 20908 LSpanclspn 20948 LSAtomsclsa 38672 LFnlclfn 38755 LKerclk 38783 LDualcld 38821 HLchlt 39048 LHypclh 39683 DVecHcdvh 40777 ocHcoch 41046 |
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 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-riotaBAD 38651 |
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 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-undef 8288 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-0g 17456 df-mre 17599 df-mrc 17600 df-acs 17602 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-p1 18451 df-lat 18457 df-clat 18524 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-subg 19117 df-cntz 19311 df-oppg 19340 df-lsm 19634 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-nzr 20495 df-rlreg 20672 df-domn 20673 df-drng 20709 df-lmod 20838 df-lss 20909 df-lsp 20949 df-lvec 21081 df-lsatoms 38674 df-lshyp 38675 df-lcv 38717 df-lfl 38756 df-lkr 38784 df-ldual 38822 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-llines 39197 df-lplanes 39198 df-lvols 39199 df-lines 39200 df-psubsp 39202 df-pmap 39203 df-padd 39495 df-lhyp 39687 df-laut 39688 df-ldil 39803 df-ltrn 39804 df-trl 39858 df-tgrp 40442 df-tendo 40454 df-edring 40456 df-dveca 40702 df-disoa 40728 df-dvech 40778 df-dib 40838 df-dic 40872 df-dih 40928 df-doch 41047 df-djh 41094 |
This theorem is referenced by: lcfrlem38 41279 |
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