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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem29 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 37719. Baer p. 45 line 16: "But Gx' and Gy'' are distinct points and so x' and y'' are independent elements in B. (Contributed by NM, 22-Mar-2015.) |
Ref | Expression |
---|---|
mapdpg.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdpg.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdpg.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdpg.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdpg.s | ⊢ − = (-g‘𝑈) |
mapdpg.z | ⊢ 0 = (0g‘𝑈) |
mapdpg.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdpg.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdpg.f | ⊢ 𝐹 = (Base‘𝐶) |
mapdpg.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdpg.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdpg.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdpg.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdpg.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdpg.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
mapdpg.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdpg.e | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) |
mapdpgem25.h1 | ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) |
mapdpgem25.i1 | ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) |
mapdpglem26.a | ⊢ 𝐴 = (Scalar‘𝑈) |
mapdpglem26.b | ⊢ 𝐵 = (Base‘𝐴) |
mapdpglem26.t | ⊢ · = ( ·𝑠 ‘𝐶) |
mapdpglem26.o | ⊢ 𝑂 = (0g‘𝐴) |
mapdpglem28.ve | ⊢ (𝜑 → 𝑣 ∈ 𝐵) |
mapdpglem28.u1 | ⊢ (𝜑 → ℎ = (𝑢 · 𝑖)) |
mapdpglem28.u2 | ⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) |
Ref | Expression |
---|---|
mapdpglem29 | ⊢ (𝜑 → (𝐽‘{𝐺}) ≠ (𝐽‘{𝑖})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpg.ne | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
2 | mapdpg.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | mapdpg.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | eqid 2797 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
5 | mapdpg.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
6 | mapdpg.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 2, 3, 6 | dvhlmod 37123 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
8 | mapdpg.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
9 | 8 | eldifad 3779 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
10 | mapdpg.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
11 | mapdpg.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈) | |
12 | 10, 4, 11 | lspsncl 19295 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
13 | 7, 9, 12 | syl2anc 580 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
14 | mapdpg.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
15 | 14 | eldifad 3779 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
16 | 10, 4, 11 | lspsncl 19295 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
17 | 7, 15, 16 | syl2anc 580 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
18 | 2, 3, 4, 5, 6, 13, 17 | mapd11 37652 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) = (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
19 | 18 | necon3bid 3013 | . . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ≠ (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))) |
20 | 1, 19 | mpbird 249 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) ≠ (𝑀‘(𝑁‘{𝑌}))) |
21 | mapdpg.e | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
22 | mapdpgem25.i1 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) | |
23 | 22 | simprd 490 | . . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))) |
24 | 23 | simpld 489 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖})) |
25 | 20, 21, 24 | 3netr3d 3045 | 1 ⊢ (𝜑 → (𝐽‘{𝐺}) ≠ (𝐽‘{𝑖})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 ∖ cdif 3764 {csn 4366 ‘cfv 6099 (class class class)co 6876 Basecbs 16181 Scalarcsca 16267 ·𝑠 cvsca 16268 0gc0g 16412 -gcsg 17737 LModclmod 19178 LSubSpclss 19247 LSpanclspn 19289 HLchlt 35363 LHypclh 35997 DVecHcdvh 37091 LCDualclcd 37599 mapdcmpd 37637 |
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 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-riotaBAD 34966 |
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 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-iin 4711 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-tpos 7588 df-undef 7635 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-map 8095 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-n0 11577 df-z 11663 df-uz 11927 df-fz 12577 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-sca 16280 df-vsca 16281 df-0g 16414 df-proset 17240 df-poset 17258 df-plt 17270 df-lub 17286 df-glb 17287 df-join 17288 df-meet 17289 df-p0 17351 df-p1 17352 df-lat 17358 df-clat 17420 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-submnd 17648 df-grp 17738 df-minusg 17739 df-sbg 17740 df-subg 17901 df-cntz 18059 df-lsm 18361 df-cmn 18507 df-abl 18508 df-mgp 18803 df-ur 18815 df-ring 18862 df-oppr 18936 df-dvdsr 18954 df-unit 18955 df-invr 18985 df-dvr 18996 df-drng 19064 df-lmod 19180 df-lss 19248 df-lsp 19290 df-lvec 19421 df-lsatoms 34989 df-lshyp 34990 df-lfl 35071 df-lkr 35099 df-oposet 35189 df-ol 35191 df-oml 35192 df-covers 35279 df-ats 35280 df-atl 35311 df-cvlat 35335 df-hlat 35364 df-llines 35511 df-lplanes 35512 df-lvols 35513 df-lines 35514 df-psubsp 35516 df-pmap 35517 df-padd 35809 df-lhyp 36001 df-laut 36002 df-ldil 36117 df-ltrn 36118 df-trl 36172 df-tgrp 36756 df-tendo 36768 df-edring 36770 df-dveca 37016 df-disoa 37042 df-dvech 37092 df-dib 37152 df-dic 37186 df-dih 37242 df-doch 37361 df-djh 37408 df-mapd 37638 |
This theorem is referenced by: mapdpglem30 37715 |
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