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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaprnlem3N | Structured version Visualization version GIF version |
Description: Part of proof of part 12 in [Baer] p. 49 line 15, T ≠ P. Our (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) is Baer's P, where P* = G(u'+s). (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hdmaprnlem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmaprnlem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmaprnlem1.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmaprnlem1.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmaprnlem1.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmaprnlem1.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmaprnlem1.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmaprnlem1.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmaprnlem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmaprnlem1.se | ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) |
hdmaprnlem1.ve | ⊢ (𝜑 → 𝑣 ∈ 𝑉) |
hdmaprnlem1.e | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) |
hdmaprnlem1.ue | ⊢ (𝜑 → 𝑢 ∈ 𝑉) |
hdmaprnlem1.un | ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) |
hdmaprnlem1.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmaprnlem1.q | ⊢ 𝑄 = (0g‘𝐶) |
hdmaprnlem1.o | ⊢ 0 = (0g‘𝑈) |
hdmaprnlem1.a | ⊢ ✚ = (+g‘𝐶) |
Ref | Expression |
---|---|
hdmaprnlem3N | ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmaprnlem1.d | . . . . 5 ⊢ 𝐷 = (Base‘𝐶) | |
2 | hdmaprnlem1.l | . . . . 5 ⊢ 𝐿 = (LSpan‘𝐶) | |
3 | hdmaprnlem1.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | hdmaprnlem1.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
5 | hdmaprnlem1.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 3, 4, 5 | lcdlmod 41302 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod) |
7 | hdmaprnlem1.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | hdmaprnlem1.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
9 | hdmaprnlem1.s | . . . . . . 7 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
10 | hdmaprnlem1.ue | . . . . . . 7 ⊢ (𝜑 → 𝑢 ∈ 𝑉) | |
11 | 3, 7, 8, 4, 1, 9, 5, 10 | hdmapcl 41540 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷) |
12 | hdmaprnlem1.se | . . . . . . 7 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) | |
13 | 12 | eldifad 3959 | . . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷) |
14 | hdmaprnlem1.a | . . . . . . 7 ⊢ ✚ = (+g‘𝐶) | |
15 | 1, 14 | lmodvacl 20845 | . . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) |
16 | 6, 11, 13, 15 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) |
17 | eqid 2726 | . . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
18 | 1, 17, 2 | lspsncl 20948 | . . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶)) |
19 | 6, 13, 18 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶)) |
20 | 1, 2 | lspsnid 20964 | . . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠})) |
21 | 6, 13, 20 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐿‘{𝑠})) |
22 | hdmaprnlem1.q | . . . . . . 7 ⊢ 𝑄 = (0g‘𝐶) | |
23 | 3, 4, 5 | lcdlvec 41301 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ LVec) |
24 | hdmaprnlem1.o | . . . . . . . 8 ⊢ 0 = (0g‘𝑈) | |
25 | eqid 2726 | . . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
26 | 3, 7, 5 | dvhlmod 40820 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
27 | hdmaprnlem1.ve | . . . . . . . . . 10 ⊢ (𝜑 → 𝑣 ∈ 𝑉) | |
28 | hdmaprnlem1.n | . . . . . . . . . . 11 ⊢ 𝑁 = (LSpan‘𝑈) | |
29 | 8, 25, 28 | lspsncl 20948 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
30 | 26, 27, 29 | syl2anc 582 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
31 | hdmaprnlem1.un | . . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) | |
32 | 24, 25, 26, 30, 10, 31 | lssneln0 20924 | . . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) |
33 | 3, 7, 8, 24, 4, 22, 1, 9, 5, 32 | hdmapnzcl 41555 | . . . . . . 7 ⊢ (𝜑 → (𝑆‘𝑢) ∈ (𝐷 ∖ {𝑄})) |
34 | hdmaprnlem1.m | . . . . . . . 8 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
35 | hdmaprnlem1.e | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) | |
36 | 3, 7, 8, 28, 4, 2, 34, 9, 5, 12, 27, 35, 10, 31 | hdmaprnlem1N 41559 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠})) |
37 | 1, 22, 2, 23, 33, 13, 36 | lspsnne1 21092 | . . . . . 6 ⊢ (𝜑 → ¬ (𝑆‘𝑢) ∈ (𝐿‘{𝑠})) |
38 | 1, 14, 17, 6, 19, 21, 11, 37 | lssvancl2 20917 | . . . . 5 ⊢ (𝜑 → ¬ ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{𝑠})) |
39 | 1, 2, 6, 16, 13, 38 | lspsnne2 21093 | . . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠})) |
40 | 39 | necomd 2986 | . . 3 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) |
41 | 1, 17, 2 | lspsncl 20948 | . . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶)) |
42 | 6, 16, 41 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶)) |
43 | 3, 34, 4, 17, 5 | mapdrn2 41361 | . . . . 5 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶)) |
44 | 42, 43 | eleqtrrd 2829 | . . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀) |
45 | 3, 34, 5, 44 | mapdcnvid2 41367 | . . 3 ⊢ (𝜑 → (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) |
46 | 40, 35, 45 | 3netr4d 3008 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))) |
47 | 3, 34, 7, 25, 5, 44 | mapdcnvcl 41362 | . . . 4 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ∈ (LSubSp‘𝑈)) |
48 | 3, 7, 25, 34, 5, 30, 47 | mapd11 41349 | . . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) = (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))) |
49 | 48 | necon3bid 2975 | . 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))) |
50 | 46, 49 | mpbid 231 | 1 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3944 {csn 4624 ◡ccnv 5672 ran crn 5674 ‘cfv 6544 (class class class)co 7414 Basecbs 17206 +gcplusg 17259 0gc0g 17447 LModclmod 20830 LSubSpclss 20902 LSpanclspn 20942 HLchlt 39059 LHypclh 39694 DVecHcdvh 40788 LCDualclcd 41296 mapdcmpd 41334 HDMapchdma 41502 |
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-ot 4633 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-nzr 20489 df-rlreg 20666 df-domn 20667 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 df-lcdual 41297 df-mapd 41335 df-hvmap 41467 df-hdmap1 41503 df-hdmap 41504 |
This theorem is referenced by: hdmaprnlem9N 41567 hdmaprnlem3eN 41568 |
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