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 39602 | . . . . 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 39840 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷) |
12 | hdmaprnlem1.se | . . . . . . 7 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) | |
13 | 12 | eldifad 3904 | . . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷) |
14 | hdmaprnlem1.a | . . . . . . 7 ⊢ ✚ = (+g‘𝐶) | |
15 | 1, 14 | lmodvacl 20135 | . . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) |
16 | 6, 11, 13, 15 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) |
17 | eqid 2740 | . . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
18 | 1, 17, 2 | lspsncl 20237 | . . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶)) |
19 | 6, 13, 18 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶)) |
20 | 1, 2 | lspsnid 20253 | . . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠})) |
21 | 6, 13, 20 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐿‘{𝑠})) |
22 | hdmaprnlem1.q | . . . . . . 7 ⊢ 𝑄 = (0g‘𝐶) | |
23 | 3, 4, 5 | lcdlvec 39601 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ LVec) |
24 | hdmaprnlem1.o | . . . . . . . 8 ⊢ 0 = (0g‘𝑈) | |
25 | eqid 2740 | . . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
26 | 3, 7, 5 | dvhlmod 39120 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
27 | hdmaprnlem1.ve | . . . . . . . . . 10 ⊢ (𝜑 → 𝑣 ∈ 𝑉) | |
28 | hdmaprnlem1.n | . . . . . . . . . . 11 ⊢ 𝑁 = (LSpan‘𝑈) | |
29 | 8, 25, 28 | lspsncl 20237 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
30 | 26, 27, 29 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
31 | hdmaprnlem1.un | . . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) | |
32 | 24, 25, 26, 30, 10, 31 | lssneln0 20212 | . . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) |
33 | 3, 7, 8, 24, 4, 22, 1, 9, 5, 32 | hdmapnzcl 39855 | . . . . . . 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 39859 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠})) |
37 | 1, 22, 2, 23, 33, 13, 36 | lspsnne1 20377 | . . . . . 6 ⊢ (𝜑 → ¬ (𝑆‘𝑢) ∈ (𝐿‘{𝑠})) |
38 | 1, 14, 17, 6, 19, 21, 11, 37 | lssvancl2 20205 | . . . . 5 ⊢ (𝜑 → ¬ ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{𝑠})) |
39 | 1, 2, 6, 16, 13, 38 | lspsnne2 20378 | . . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠})) |
40 | 39 | necomd 3001 | . . 3 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) |
41 | 1, 17, 2 | lspsncl 20237 | . . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶)) |
42 | 6, 16, 41 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶)) |
43 | 3, 34, 4, 17, 5 | mapdrn2 39661 | . . . . 5 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶)) |
44 | 42, 43 | eleqtrrd 2844 | . . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀) |
45 | 3, 34, 5, 44 | mapdcnvid2 39667 | . . 3 ⊢ (𝜑 → (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) |
46 | 40, 35, 45 | 3netr4d 3023 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))) |
47 | 3, 34, 7, 25, 5, 44 | mapdcnvcl 39662 | . . . 4 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ∈ (LSubSp‘𝑈)) |
48 | 3, 7, 25, 34, 5, 30, 47 | mapd11 39649 | . . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) = (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))) |
49 | 48 | necon3bid 2990 | . 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 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∖ cdif 3889 {csn 4567 ◡ccnv 5589 ran crn 5591 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 +gcplusg 16960 0gc0g 17148 LModclmod 20121 LSubSpclss 20191 LSpanclspn 20231 HLchlt 37360 LHypclh 37994 DVecHcdvh 39088 LCDualclcd 39596 mapdcmpd 39634 HDMapchdma 39802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-riotaBAD 36963 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-tpos 8033 df-undef 8080 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-0g 17150 df-mre 17293 df-mrc 17294 df-acs 17296 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-oppg 18948 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 df-lsatoms 36986 df-lshyp 36987 df-lcv 37029 df-lfl 37068 df-lkr 37096 df-ldual 37134 df-oposet 37186 df-ol 37188 df-oml 37189 df-covers 37276 df-ats 37277 df-atl 37308 df-cvlat 37332 df-hlat 37361 df-llines 37508 df-lplanes 37509 df-lvols 37510 df-lines 37511 df-psubsp 37513 df-pmap 37514 df-padd 37806 df-lhyp 37998 df-laut 37999 df-ldil 38114 df-ltrn 38115 df-trl 38169 df-tgrp 38753 df-tendo 38765 df-edring 38767 df-dveca 39013 df-disoa 39039 df-dvech 39089 df-dib 39149 df-dic 39183 df-dih 39239 df-doch 39358 df-djh 39405 df-lcdual 39597 df-mapd 39635 df-hvmap 39767 df-hdmap1 39803 df-hdmap 39804 |
This theorem is referenced by: hdmaprnlem9N 39867 hdmaprnlem3eN 39868 |
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