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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 37245 and mapdcnv11N 37265. 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 37464 | . . . . 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 37465 | . . . . . 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 37462 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠})) |
25 | 23, 24 | eleqtrd 2732 | . . . . 5 ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{𝑠})) |
26 | 22, 25 | elind 3831 | . . . 4 ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠}))) |
27 | 1, 5, 9 | lcdlvec 37197 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ LVec) |
28 | 1, 5, 9 | lcdlmod 37198 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LMod) |
29 | 1, 2, 3, 5, 15, 8, 9, 13 | hdmapcl 37439 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷) |
30 | 10 | eldifad 3619 | . . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷) |
31 | 15, 18 | lmodvacl 18925 | . . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) |
32 | 28, 29, 30, 31 | syl3anc 1366 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) |
33 | eqid 2651 | . . . . . . . . . . . . . 14 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
34 | 15, 33, 6 | lspsncl 19025 | . . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶)) |
35 | 28, 30, 34 | syl2anc 694 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶)) |
36 | 1, 7, 5, 33, 9 | mapdrn2 37257 | . . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶)) |
37 | 35, 36 | eleqtrrd 2733 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ ran 𝑀) |
38 | 1, 7, 9, 37 | mapdcnvid2 37263 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(◡𝑀‘(𝐿‘{𝑠}))) = (𝐿‘{𝑠})) |
39 | 12, 38 | eqtr4d 2688 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝑀‘(◡𝑀‘(𝐿‘{𝑠})))) |
40 | eqid 2651 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
41 | 1, 2, 9 | dvhlmod 36716 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod) |
42 | 3, 40, 4 | lspsncl 19025 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
43 | 41, 11, 42 | syl2anc 694 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
44 | 1, 7, 2, 40, 9, 37 | mapdcnvcl 37258 | . . . . . . . . . 10 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{𝑠})) ∈ (LSubSp‘𝑈)) |
45 | 1, 2, 40, 7, 9, 43, 44 | mapd11 37245 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(◡𝑀‘(𝐿‘{𝑠}))) ↔ (𝑁‘{𝑣}) = (◡𝑀‘(𝐿‘{𝑠})))) |
46 | 39, 45 | mpbid 222 | . . . . . . . 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 37459 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
48 | 46, 47 | eqnetrrd 2891 | . . . . . . 7 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{𝑠})) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
49 | 15, 33, 6 | lspsncl 19025 | . . . . . . . . . . 11 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶)) |
50 | 28, 32, 49 | syl2anc 694 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶)) |
51 | 50, 36 | eleqtrrd 2733 | . . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀) |
52 | 1, 7, 9, 37, 51 | mapdcnv11N 37265 | . . . . . . . 8 ⊢ (𝜑 → ((◡𝑀‘(𝐿‘{𝑠})) = (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
53 | 52 | necon3bid 2867 | . . . . . . 7 ⊢ (𝜑 → ((◡𝑀‘(𝐿‘{𝑠})) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
54 | 48, 53 | mpbid 222 | . . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) |
55 | 54 | necomd 2878 | . . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠})) |
56 | 15, 16, 6, 27, 32, 30, 55 | lspdisj2 19175 | . . . 4 ⊢ (𝜑 → ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})) = {𝑄}) |
57 | 26, 56 | eleqtrd 2732 | . . 3 ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ {𝑄}) |
58 | elsni 4227 | . . 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 37461 | . . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉) |
61 | 1, 2, 3, 5, 15, 8, 9, 60 | hdmapcl 37439 | . . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷) |
62 | eqid 2651 | . . . 4 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
63 | 15, 16, 62 | lmodsubeq0 18970 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷 ∧ (𝑆‘𝑡) ∈ 𝐷) → ((𝑠(-g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡))) |
64 | 28, 30, 61, 63 | syl3anc 1366 | . 2 ⊢ (𝜑 → ((𝑠(-g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡))) |
65 | 59, 64 | mpbid 222 | 1 ⊢ (𝜑 → 𝑠 = (𝑆‘𝑡)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∖ cdif 3604 ∩ cin 3606 {csn 4210 ◡ccnv 5142 ran crn 5144 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 0gc0g 16147 -gcsg 17471 LModclmod 18911 LSubSpclss 18980 LSpanclspn 19019 HLchlt 34955 LHypclh 35588 DVecHcdvh 36684 LCDualclcd 37192 mapdcmpd 37230 HDMapchdma 37399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-riotaBAD 34557 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-ot 4219 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-undef 7444 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-0g 16149 df-mre 16293 df-mrc 16294 df-acs 16296 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-cntz 17796 df-oppg 17822 df-lsm 18097 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-drng 18797 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lvec 19151 df-lsatoms 34581 df-lshyp 34582 df-lcv 34624 df-lfl 34663 df-lkr 34691 df-ldual 34729 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-lplanes 35103 df-lvols 35104 df-lines 35105 df-psubsp 35107 df-pmap 35108 df-padd 35400 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 df-trl 35764 df-tgrp 36348 df-tendo 36360 df-edring 36362 df-dveca 36608 df-disoa 36635 df-dvech 36685 df-dib 36745 df-dic 36779 df-dih 36835 df-doch 36954 df-djh 37001 df-lcdual 37193 df-mapd 37231 df-hvmap 37363 df-hdmap1 37400 df-hdmap 37401 |
This theorem is referenced by: hdmaprnlem10N 37468 |
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