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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcf1o | Structured version Visualization version GIF version |
Description: Define a function 𝐽 that provides a bijection from nonzero vectors 𝑉 to nonzero functionals with closed kernels 𝐶. (Contributed by NM, 22-Feb-2015.) |
Ref | Expression |
---|---|
lcf1o.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcf1o.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcf1o.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcf1o.v | ⊢ 𝑉 = (Base‘𝑈) |
lcf1o.a | ⊢ + = (+g‘𝑈) |
lcf1o.t | ⊢ · = ( ·𝑠 ‘𝑈) |
lcf1o.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lcf1o.r | ⊢ 𝑅 = (Base‘𝑆) |
lcf1o.z | ⊢ 0 = (0g‘𝑈) |
lcf1o.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcf1o.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcf1o.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcf1o.q | ⊢ 𝑄 = (0g‘𝐷) |
lcf1o.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcf1o.j | ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) |
lcflo.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
lcf1o | ⊢ (𝜑 → 𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcf1o.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcf1o.o | . 2 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | lcf1o.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lcf1o.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
5 | lcf1o.a | . 2 ⊢ + = (+g‘𝑈) | |
6 | lcf1o.t | . 2 ⊢ · = ( ·𝑠 ‘𝑈) | |
7 | lcf1o.s | . 2 ⊢ 𝑆 = (Scalar‘𝑈) | |
8 | lcf1o.r | . 2 ⊢ 𝑅 = (Base‘𝑆) | |
9 | lcf1o.z | . 2 ⊢ 0 = (0g‘𝑈) | |
10 | lcf1o.f | . 2 ⊢ 𝐹 = (LFnl‘𝑈) | |
11 | lcf1o.l | . 2 ⊢ 𝐿 = (LKer‘𝑈) | |
12 | lcf1o.d | . 2 ⊢ 𝐷 = (LDual‘𝑈) | |
13 | lcf1o.q | . 2 ⊢ 𝑄 = (0g‘𝐷) | |
14 | lcf1o.c | . 2 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
15 | lcf1o.j | . . 3 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) | |
16 | oveq1 7430 | . . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑤 + (𝑘 · 𝑥)) = (𝑧 + (𝑘 · 𝑥))) | |
17 | 16 | eqeq2d 2736 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑧 + (𝑘 · 𝑥)))) |
18 | 17 | cbvrexvw 3225 | . . . . . . . . 9 ⊢ (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑧 + (𝑘 · 𝑥))) |
19 | oveq1 7430 | . . . . . . . . . . . 12 ⊢ (𝑘 = 𝑙 → (𝑘 · 𝑥) = (𝑙 · 𝑥)) | |
20 | 19 | oveq2d 7439 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑙 → (𝑧 + (𝑘 · 𝑥)) = (𝑧 + (𝑙 · 𝑥))) |
21 | 20 | eqeq2d 2736 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑙 → (𝑣 = (𝑧 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑧 + (𝑙 · 𝑥)))) |
22 | 21 | rexbidv 3168 | . . . . . . . . 9 ⊢ (𝑘 = 𝑙 → (∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑧 + (𝑘 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑧 + (𝑙 · 𝑥)))) |
23 | 18, 22 | bitrid 282 | . . . . . . . 8 ⊢ (𝑘 = 𝑙 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑧 + (𝑙 · 𝑥)))) |
24 | 23 | cbvriotavw 7389 | . . . . . . 7 ⊢ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑧 + (𝑙 · 𝑥))) |
25 | eqeq1 2729 | . . . . . . . . 9 ⊢ (𝑣 = 𝑢 → (𝑣 = (𝑧 + (𝑙 · 𝑥)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑥)))) | |
26 | 25 | rexbidv 3168 | . . . . . . . 8 ⊢ (𝑣 = 𝑢 → (∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑧 + (𝑙 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) |
27 | 26 | riotabidv 7381 | . . . . . . 7 ⊢ (𝑣 = 𝑢 → (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑧 + (𝑙 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) |
28 | 24, 27 | eqtrid 2777 | . . . . . 6 ⊢ (𝑣 = 𝑢 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) |
29 | 28 | cbvmptv 5265 | . . . . 5 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) |
30 | sneq 4642 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
31 | 30 | fveq2d 6904 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → ( ⊥ ‘{𝑥}) = ( ⊥ ‘{𝑦})) |
32 | oveq2 7431 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑙 · 𝑥) = (𝑙 · 𝑦)) | |
33 | 32 | oveq2d 7439 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑧 + (𝑙 · 𝑥)) = (𝑧 + (𝑙 · 𝑦))) |
34 | 33 | eqeq2d 2736 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑢 = (𝑧 + (𝑙 · 𝑥)) ↔ 𝑢 = (𝑧 + (𝑙 · 𝑦)))) |
35 | 31, 34 | rexeqbidv 3330 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)) ↔ ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))) |
36 | 35 | riotabidv 7381 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥))) = (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦)))) |
37 | 36 | mpteq2dv 5254 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑥})𝑢 = (𝑧 + (𝑙 · 𝑥)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))) |
38 | 29, 37 | eqtrid 2777 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))) |
39 | 38 | cbvmptv 5265 | . . 3 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) = (𝑦 ∈ (𝑉 ∖ { 0 }) ↦ (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))) |
40 | 15, 39 | eqtri 2753 | . 2 ⊢ 𝐽 = (𝑦 ∈ (𝑉 ∖ { 0 }) ↦ (𝑢 ∈ 𝑉 ↦ (℩𝑙 ∈ 𝑅 ∃𝑧 ∈ ( ⊥ ‘{𝑦})𝑢 = (𝑧 + (𝑙 · 𝑦))))) |
41 | lcflo.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
42 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 40, 41 | lcfrlem9 41197 | 1 ⊢ (𝜑 → 𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 {crab 3418 ∖ cdif 3943 {csn 4632 ↦ cmpt 5235 –1-1-onto→wf1o 6552 ‘cfv 6553 ℩crio 7378 (class class class)co 7423 Basecbs 17208 +gcplusg 17261 Scalarcsca 17264 ·𝑠 cvsca 17265 0gc0g 17449 LFnlclfn 38703 LKerclk 38731 LDualcld 38769 HLchlt 38996 LHypclh 39631 DVecHcdvh 40725 ocHcoch 40994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-riotaBAD 38599 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7689 df-om 7876 df-1st 8002 df-2nd 8003 df-tpos 8240 df-undef 8287 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-n0 12520 df-z 12606 df-uz 12870 df-fz 13534 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-0g 17451 df-proset 18315 df-poset 18333 df-plt 18350 df-lub 18366 df-glb 18367 df-join 18368 df-meet 18369 df-p0 18445 df-p1 18446 df-lat 18452 df-clat 18519 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19112 df-cntz 19306 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-drng 20666 df-lmod 20785 df-lss 20856 df-lsp 20896 df-lvec 21028 df-lsatoms 38622 df-lshyp 38623 df-lfl 38704 df-lkr 38732 df-ldual 38770 df-oposet 38822 df-ol 38824 df-oml 38825 df-covers 38912 df-ats 38913 df-atl 38944 df-cvlat 38968 df-hlat 38997 df-llines 39145 df-lplanes 39146 df-lvols 39147 df-lines 39148 df-psubsp 39150 df-pmap 39151 df-padd 39443 df-lhyp 39635 df-laut 39636 df-ldil 39751 df-ltrn 39752 df-trl 39806 df-tgrp 40390 df-tendo 40402 df-edring 40404 df-dveca 40650 df-disoa 40676 df-dvech 40726 df-dib 40786 df-dic 40820 df-dih 40876 df-doch 40995 df-djh 41042 |
This theorem is referenced by: lcfrlem13 41202 hvmap1o 41410 |
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