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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem39 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 41296. Eliminate 𝐽. (Contributed by NM, 11-Mar-2015.) |
Ref | Expression |
---|---|
lcfrlem38.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfrlem38.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfrlem38.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfrlem38.p | ⊢ + = (+g‘𝑈) |
lcfrlem38.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcfrlem38.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfrlem38.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcfrlem38.q | ⊢ 𝑄 = (LSubSp‘𝐷) |
lcfrlem38.c | ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcfrlem38.e | ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) |
lcfrlem38.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfrlem38.g | ⊢ (𝜑 → 𝐺 ∈ 𝑄) |
lcfrlem38.gs | ⊢ (𝜑 → 𝐺 ⊆ 𝐶) |
lcfrlem38.xe | ⊢ (𝜑 → 𝑋 ∈ 𝐸) |
lcfrlem38.ye | ⊢ (𝜑 → 𝑌 ∈ 𝐸) |
lcfrlem38.z | ⊢ 0 = (0g‘𝑈) |
lcfrlem38.x | ⊢ (𝜑 → 𝑋 ≠ 0 ) |
lcfrlem38.y | ⊢ (𝜑 → 𝑌 ≠ 0 ) |
lcfrlem38.sp | ⊢ 𝑁 = (LSpan‘𝑈) |
lcfrlem38.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
lcfrlem38.b | ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) |
lcfrlem38.i | ⊢ (𝜑 → 𝐼 ∈ 𝐵) |
lcfrlem38.n | ⊢ (𝜑 → 𝐼 ≠ 0 ) |
Ref | Expression |
---|---|
lcfrlem39 | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem38.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcfrlem38.o | . 2 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | lcfrlem38.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lcfrlem38.p | . 2 ⊢ + = (+g‘𝑈) | |
5 | lcfrlem38.f | . 2 ⊢ 𝐹 = (LFnl‘𝑈) | |
6 | lcfrlem38.l | . 2 ⊢ 𝐿 = (LKer‘𝑈) | |
7 | lcfrlem38.d | . 2 ⊢ 𝐷 = (LDual‘𝑈) | |
8 | lcfrlem38.q | . 2 ⊢ 𝑄 = (LSubSp‘𝐷) | |
9 | lcfrlem38.c | . 2 ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
10 | lcfrlem38.e | . 2 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) | |
11 | lcfrlem38.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | lcfrlem38.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑄) | |
13 | lcfrlem38.gs | . 2 ⊢ (𝜑 → 𝐺 ⊆ 𝐶) | |
14 | lcfrlem38.xe | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
15 | lcfrlem38.ye | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐸) | |
16 | lcfrlem38.z | . 2 ⊢ 0 = (0g‘𝑈) | |
17 | lcfrlem38.x | . 2 ⊢ (𝜑 → 𝑋 ≠ 0 ) | |
18 | lcfrlem38.y | . 2 ⊢ (𝜑 → 𝑌 ≠ 0 ) | |
19 | lcfrlem38.sp | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
20 | lcfrlem38.ne | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
21 | lcfrlem38.b | . 2 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
22 | lcfrlem38.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
23 | lcfrlem38.n | . 2 ⊢ (𝜑 → 𝐼 ≠ 0 ) | |
24 | eqid 2726 | . 2 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
25 | eqid 2726 | . 2 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
26 | eqid 2726 | . 2 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
27 | eqid 2726 | . 2 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
28 | oveq1 7422 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑗( ·𝑠 ‘𝑈)𝑥) = (𝑘( ·𝑠 ‘𝑈)𝑥)) | |
29 | 28 | oveq2d 7431 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥)) = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥))) |
30 | 29 | eqeq2d 2737 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑣 = (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥)) ↔ 𝑣 = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥)))) |
31 | 30 | rexbidv 3169 | . . . . 5 ⊢ (𝑗 = 𝑘 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥)))) |
32 | 31 | cbvriotavw 7381 | . . . 4 ⊢ (℩𝑗 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥))) = (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥))) |
33 | 32 | mpteq2i 5250 | . . 3 ⊢ (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑗 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥)))) = (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥)))) |
34 | 33 | mpteq2i 5250 | . 2 ⊢ (𝑥 ∈ ((Base‘𝑈) ∖ { 0 }) ↦ (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑗 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·𝑠 ‘𝑈)𝑥))))) = (𝑥 ∈ ((Base‘𝑈) ∖ { 0 }) ↦ (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·𝑠 ‘𝑈)𝑥))))) |
35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 34 | lcfrlem38 41291 | 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 3420 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 {csn 4625 {cpr 4627 ∪ ciun 4995 ↦ cmpt 5228 ‘cfv 6545 ℩crio 7370 (class class class)co 7415 Basecbs 17207 +gcplusg 17260 Scalarcsca 17263 ·𝑠 cvsca 17264 0gc0g 17448 LSubSpclss 20903 LSpanclspn 20943 LFnlclfn 38767 LKerclk 38795 LDualcld 38833 HLchlt 39060 LHypclh 39695 DVecHcdvh 40789 ocHcoch 41058 |
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 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-riotaBAD 38663 |
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 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3968 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4908 df-int 4949 df-iun 4997 df-iin 4998 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8848 df-en 8966 df-dom 8967 df-sdom 8968 df-fin 8969 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-n0 12518 df-z 12604 df-uz 12868 df-fz 13532 df-struct 17143 df-sets 17160 df-slot 17178 df-ndx 17190 df-base 17208 df-ress 17237 df-plusg 17273 df-mulr 17274 df-sca 17276 df-vsca 17277 df-0g 17450 df-mre 17593 df-mrc 17594 df-acs 17596 df-proset 18314 df-poset 18332 df-plt 18349 df-lub 18365 df-glb 18366 df-join 18367 df-meet 18368 df-p0 18444 df-p1 18445 df-lat 18451 df-clat 18518 df-mgm 18627 df-sgrp 18706 df-mnd 18722 df-submnd 18768 df-grp 18925 df-minusg 18926 df-sbg 18927 df-subg 19112 df-cntz 19306 df-oppg 19335 df-lsm 19629 df-cmn 19775 df-abl 19776 df-mgp 20113 df-rng 20131 df-ur 20160 df-ring 20213 df-oppr 20311 df-dvdsr 20334 df-unit 20335 df-invr 20365 df-dvr 20378 df-nzr 20490 df-rlreg 20667 df-domn 20668 df-drng 20704 df-lmod 20833 df-lss 20904 df-lsp 20944 df-lvec 21076 df-lsatoms 38686 df-lshyp 38687 df-lcv 38729 df-lfl 38768 df-lkr 38796 df-ldual 38834 df-oposet 38886 df-ol 38888 df-oml 38889 df-covers 38976 df-ats 38977 df-atl 39008 df-cvlat 39032 df-hlat 39061 df-llines 39209 df-lplanes 39210 df-lvols 39211 df-lines 39212 df-psubsp 39214 df-pmap 39215 df-padd 39507 df-lhyp 39699 df-laut 39700 df-ldil 39815 df-ltrn 39816 df-trl 39870 df-tgrp 40454 df-tendo 40466 df-edring 40468 df-dveca 40714 df-disoa 40740 df-dvech 40790 df-dib 40850 df-dic 40884 df-dih 40940 df-doch 41059 df-djh 41106 |
This theorem is referenced by: lcfrlem40 41293 |
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