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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem32 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 39607. (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))) |
lcfrlem31.xi | ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ≠ 𝑄) |
Ref | Expression |
---|---|
lcfrlem32 | ⊢ (𝜑 → 𝐶 ≠ (0g‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem17.ne | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
2 | lcfrlem17.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | lcfrlem17.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | lcfrlem17.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | lcfrlem17.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
6 | lcfrlem17.p | . . . . 5 ⊢ + = (+g‘𝑈) | |
7 | lcfrlem17.z | . . . . 5 ⊢ 0 = (0g‘𝑈) | |
8 | lcfrlem17.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
9 | lcfrlem17.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
10 | lcfrlem17.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | 10 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = (0g‘𝐷)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
12 | lcfrlem17.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
13 | 12 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = (0g‘𝐷)) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
14 | lcfrlem17.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
15 | 14 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = (0g‘𝐷)) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
16 | 1 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = (0g‘𝐷)) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
17 | lcfrlem22.b | . . . . 5 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
18 | lcfrlem24.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑈) | |
19 | lcfrlem24.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑈) | |
20 | lcfrlem24.q | . . . . 5 ⊢ 𝑄 = (0g‘𝑆) | |
21 | lcfrlem24.r | . . . . 5 ⊢ 𝑅 = (Base‘𝑆) | |
22 | lcfrlem24.j | . . . . 5 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))))) | |
23 | lcfrlem24.ib | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
24 | 23 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = (0g‘𝐷)) → 𝐼 ∈ 𝐵) |
25 | lcfrlem24.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑈) | |
26 | lcfrlem25.d | . . . . 5 ⊢ 𝐷 = (LDual‘𝑈) | |
27 | lcfrlem28.jn | . . . . . 6 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) | |
28 | 27 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = (0g‘𝐷)) → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) |
29 | lcfrlem29.i | . . . . 5 ⊢ 𝐹 = (invr‘𝑆) | |
30 | lcfrlem30.m | . . . . 5 ⊢ − = (-g‘𝐷) | |
31 | lcfrlem30.c | . . . . 5 ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) | |
32 | lcfrlem31.xi | . . . . . 6 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ≠ 𝑄) | |
33 | 32 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = (0g‘𝐷)) → ((𝐽‘𝑋)‘𝐼) ≠ 𝑄) |
34 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = (0g‘𝐷)) → 𝐶 = (0g‘𝐷)) | |
35 | 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 33, 34 | lcfrlem31 39595 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = (0g‘𝐷)) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
36 | 35 | ex 413 | . . 3 ⊢ (𝜑 → (𝐶 = (0g‘𝐷) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
37 | 36 | necon3d 2964 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) → 𝐶 ≠ (0g‘𝐷))) |
38 | 1, 37 | mpd 15 | 1 ⊢ (𝜑 → 𝐶 ≠ (0g‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 ∖ cdif 3883 ∩ cin 3885 {csn 4561 {cpr 4563 ↦ cmpt 5156 ‘cfv 6426 ℩crio 7223 (class class class)co 7267 Basecbs 16922 +gcplusg 16972 .rcmulr 16973 Scalarcsca 16975 ·𝑠 cvsca 16976 0gc0g 17160 -gcsg 18589 invrcinvr 19923 LSpanclspn 20243 LSAtomsclsa 36996 LKerclk 37107 LDualcld 37145 HLchlt 37372 LHypclh 38006 DVecHcdvh 39100 ocHcoch 39369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-riotaBAD 36975 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-tpos 8029 df-undef 8076 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-n0 12244 df-z 12330 df-uz 12593 df-fz 13250 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-sca 16988 df-vsca 16989 df-0g 17162 df-mre 17305 df-mrc 17306 df-acs 17308 df-proset 18023 df-poset 18041 df-plt 18058 df-lub 18074 df-glb 18075 df-join 18076 df-meet 18077 df-p0 18153 df-p1 18154 df-lat 18160 df-clat 18227 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-grp 18590 df-minusg 18591 df-sbg 18592 df-subg 18762 df-cntz 18933 df-oppg 18960 df-lsm 19251 df-cmn 19398 df-abl 19399 df-mgp 19731 df-ur 19748 df-ring 19795 df-oppr 19872 df-dvdsr 19893 df-unit 19894 df-invr 19924 df-dvr 19935 df-drng 20003 df-lmod 20135 df-lss 20204 df-lsp 20244 df-lvec 20375 df-lsatoms 36998 df-lshyp 36999 df-lcv 37041 df-lfl 37080 df-lkr 37108 df-ldual 37146 df-oposet 37198 df-ol 37200 df-oml 37201 df-covers 37288 df-ats 37289 df-atl 37320 df-cvlat 37344 df-hlat 37373 df-llines 37520 df-lplanes 37521 df-lvols 37522 df-lines 37523 df-psubsp 37525 df-pmap 37526 df-padd 37818 df-lhyp 38010 df-laut 38011 df-ldil 38126 df-ltrn 38127 df-trl 38181 df-tgrp 38765 df-tendo 38777 df-edring 38779 df-dveca 39025 df-disoa 39051 df-dvech 39101 df-dib 39161 df-dic 39195 df-dih 39251 df-doch 39370 df-djh 39417 |
This theorem is referenced by: lcfrlem34 39598 |
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