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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaprnlem9N | Structured version Visualization version GIF version |
Description: Part of proof of part 12 in [Baer] p. 49 line 21, s=S(t). TODO: we seem to be going back and forth with mapd11 37660 and mapdcnv11N 37680. Use better hypotheses and/or theorems? (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) |
hdmaprnlem1.t2 | ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) |
hdmaprnlem1.p | ⊢ + = (+g‘𝑈) |
hdmaprnlem1.pt | ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)}))) |
Ref | Expression |
---|---|
hdmaprnlem9N | ⊢ (𝜑 → 𝑠 = (𝑆‘𝑡)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmaprnlem1.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmaprnlem1.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmaprnlem1.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmaprnlem1.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
5 | hdmaprnlem1.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
6 | hdmaprnlem1.l | . . . . . 6 ⊢ 𝐿 = (LSpan‘𝐶) | |
7 | hdmaprnlem1.m | . . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
8 | hdmaprnlem1.s | . . . . . 6 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
9 | hdmaprnlem1.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | hdmaprnlem1.se | . . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) | |
11 | hdmaprnlem1.ve | . . . . . 6 ⊢ (𝜑 → 𝑣 ∈ 𝑉) | |
12 | hdmaprnlem1.e | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) | |
13 | hdmaprnlem1.ue | . . . . . 6 ⊢ (𝜑 → 𝑢 ∈ 𝑉) | |
14 | hdmaprnlem1.un | . . . . . 6 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) | |
15 | hdmaprnlem1.d | . . . . . 6 ⊢ 𝐷 = (Base‘𝐶) | |
16 | hdmaprnlem1.q | . . . . . 6 ⊢ 𝑄 = (0g‘𝐶) | |
17 | hdmaprnlem1.o | . . . . . 6 ⊢ 0 = (0g‘𝑈) | |
18 | hdmaprnlem1.a | . . . . . 6 ⊢ ✚ = (+g‘𝐶) | |
19 | hdmaprnlem1.t2 | . . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) | |
20 | hdmaprnlem1.p | . . . . . 6 ⊢ + = (+g‘𝑈) | |
21 | hdmaprnlem1.pt | . . . . . 6 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)}))) | |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 | hdmaprnlem7N 37876 | . . . . 5 ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) |
23 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 | hdmaprnlem8N 37877 | . . . . . 6 ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ (𝑀‘(𝑁‘{𝑡}))) |
24 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 | hdmaprnlem4N 37874 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠})) |
25 | 23, 24 | eleqtrd 2880 | . . . . 5 ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{𝑠})) |
26 | 22, 25 | elind 3996 | . . . 4 ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠}))) |
27 | 1, 5, 9 | lcdlvec 37612 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ LVec) |
28 | 1, 5, 9 | lcdlmod 37613 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LMod) |
29 | 1, 2, 3, 5, 15, 8, 9, 13 | hdmapcl 37851 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷) |
30 | 10 | eldifad 3781 | . . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷) |
31 | 15, 18 | lmodvacl 19195 | . . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) |
32 | 28, 29, 30, 31 | syl3anc 1491 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) |
33 | eqid 2799 | . . . . . . . . . . . . . 14 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
34 | 15, 33, 6 | lspsncl 19298 | . . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶)) |
35 | 28, 30, 34 | syl2anc 580 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶)) |
36 | 1, 7, 5, 33, 9 | mapdrn2 37672 | . . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶)) |
37 | 35, 36 | eleqtrrd 2881 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ ran 𝑀) |
38 | 1, 7, 9, 37 | mapdcnvid2 37678 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(◡𝑀‘(𝐿‘{𝑠}))) = (𝐿‘{𝑠})) |
39 | 12, 38 | eqtr4d 2836 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝑀‘(◡𝑀‘(𝐿‘{𝑠})))) |
40 | eqid 2799 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
41 | 1, 2, 9 | dvhlmod 37131 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod) |
42 | 3, 40, 4 | lspsncl 19298 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
43 | 41, 11, 42 | syl2anc 580 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
44 | 1, 7, 2, 40, 9, 37 | mapdcnvcl 37673 | . . . . . . . . . 10 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{𝑠})) ∈ (LSubSp‘𝑈)) |
45 | 1, 2, 40, 7, 9, 43, 44 | mapd11 37660 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(◡𝑀‘(𝐿‘{𝑠}))) ↔ (𝑁‘{𝑣}) = (◡𝑀‘(𝐿‘{𝑠})))) |
46 | 39, 45 | mpbid 224 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑣}) = (◡𝑀‘(𝐿‘{𝑠}))) |
47 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | hdmaprnlem3N 37871 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
48 | 46, 47 | eqnetrrd 3039 | . . . . . . 7 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{𝑠})) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
49 | 15, 33, 6 | lspsncl 19298 | . . . . . . . . . . 11 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶)) |
50 | 28, 32, 49 | syl2anc 580 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶)) |
51 | 50, 36 | eleqtrrd 2881 | . . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀) |
52 | 1, 7, 9, 37, 51 | mapdcnv11N 37680 | . . . . . . . 8 ⊢ (𝜑 → ((◡𝑀‘(𝐿‘{𝑠})) = (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
53 | 52 | necon3bid 3015 | . . . . . . 7 ⊢ (𝜑 → ((◡𝑀‘(𝐿‘{𝑠})) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
54 | 48, 53 | mpbid 224 | . . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) |
55 | 54 | necomd 3026 | . . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠})) |
56 | 15, 16, 6, 27, 32, 30, 55 | lspdisj2 19448 | . . . 4 ⊢ (𝜑 → ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})) = {𝑄}) |
57 | 26, 56 | eleqtrd 2880 | . . 3 ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ {𝑄}) |
58 | elsni 4385 | . . 3 ⊢ ((𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ {𝑄} → (𝑠(-g‘𝐶)(𝑆‘𝑡)) = 𝑄) | |
59 | 57, 58 | syl 17 | . 2 ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) = 𝑄) |
60 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 | hdmaprnlem4tN 37873 | . . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉) |
61 | 1, 2, 3, 5, 15, 8, 9, 60 | hdmapcl 37851 | . . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷) |
62 | eqid 2799 | . . . 4 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
63 | 15, 16, 62 | lmodsubeq0 19240 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷 ∧ (𝑆‘𝑡) ∈ 𝐷) → ((𝑠(-g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡))) |
64 | 28, 30, 61, 63 | syl3anc 1491 | . 2 ⊢ (𝜑 → ((𝑠(-g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡))) |
65 | 59, 64 | mpbid 224 | 1 ⊢ (𝜑 → 𝑠 = (𝑆‘𝑡)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∖ cdif 3766 ∩ cin 3768 {csn 4368 ◡ccnv 5311 ran crn 5313 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 +gcplusg 16267 0gc0g 16415 -gcsg 17740 LModclmod 19181 LSubSpclss 19250 LSpanclspn 19292 HLchlt 35371 LHypclh 36005 DVecHcdvh 37099 LCDualclcd 37607 mapdcmpd 37645 HDMapchdma 37813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-riotaBAD 34974 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-ot 4377 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-tpos 7590 df-undef 7637 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-0g 16417 df-mre 16561 df-mrc 16562 df-acs 16564 df-proset 17243 df-poset 17261 df-plt 17273 df-lub 17289 df-glb 17290 df-join 17291 df-meet 17292 df-p0 17354 df-p1 17355 df-lat 17361 df-clat 17423 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-grp 17741 df-minusg 17742 df-sbg 17743 df-subg 17904 df-cntz 18062 df-oppg 18088 df-lsm 18364 df-cmn 18510 df-abl 18511 df-mgp 18806 df-ur 18818 df-ring 18865 df-oppr 18939 df-dvdsr 18957 df-unit 18958 df-invr 18988 df-dvr 18999 df-drng 19067 df-lmod 19183 df-lss 19251 df-lsp 19293 df-lvec 19424 df-lsatoms 34997 df-lshyp 34998 df-lcv 35040 df-lfl 35079 df-lkr 35107 df-ldual 35145 df-oposet 35197 df-ol 35199 df-oml 35200 df-covers 35287 df-ats 35288 df-atl 35319 df-cvlat 35343 df-hlat 35372 df-llines 35519 df-lplanes 35520 df-lvols 35521 df-lines 35522 df-psubsp 35524 df-pmap 35525 df-padd 35817 df-lhyp 36009 df-laut 36010 df-ldil 36125 df-ltrn 36126 df-trl 36180 df-tgrp 36764 df-tendo 36776 df-edring 36778 df-dveca 37024 df-disoa 37050 df-dvech 37100 df-dib 37160 df-dic 37194 df-dih 37250 df-doch 37369 df-djh 37416 df-lcdual 37608 df-mapd 37646 df-hvmap 37778 df-hdmap1 37814 df-hdmap 37815 |
This theorem is referenced by: hdmaprnlem10N 37880 |
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