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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 41595 | . . . . 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 41833 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷) | 
| 12 | hdmaprnlem1.se | . . . . . . 7 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) | |
| 13 | 12 | eldifad 3962 | . . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷) | 
| 14 | hdmaprnlem1.a | . . . . . . 7 ⊢ ✚ = (+g‘𝐶) | |
| 15 | 1, 14 | lmodvacl 20874 | . . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) | 
| 16 | 6, 11, 13, 15 | syl3anc 1372 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) | 
| 17 | eqid 2736 | . . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
| 18 | 1, 17, 2 | lspsncl 20976 | . . . . . . 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 20992 | . . . . . . 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 41594 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ LVec) | 
| 24 | hdmaprnlem1.o | . . . . . . . 8 ⊢ 0 = (0g‘𝑈) | |
| 25 | eqid 2736 | . . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
| 26 | 3, 7, 5 | dvhlmod 41113 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) | 
| 27 | hdmaprnlem1.ve | . . . . . . . . . 10 ⊢ (𝜑 → 𝑣 ∈ 𝑉) | |
| 28 | hdmaprnlem1.n | . . . . . . . . . . 11 ⊢ 𝑁 = (LSpan‘𝑈) | |
| 29 | 8, 25, 28 | lspsncl 20976 | . . . . . . . . . 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 20952 | . . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) | 
| 33 | 3, 7, 8, 24, 4, 22, 1, 9, 5, 32 | hdmapnzcl 41848 | . . . . . . 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 41852 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠})) | 
| 37 | 1, 22, 2, 23, 33, 13, 36 | lspsnne1 21120 | . . . . . 6 ⊢ (𝜑 → ¬ (𝑆‘𝑢) ∈ (𝐿‘{𝑠})) | 
| 38 | 1, 14, 17, 6, 19, 21, 11, 37 | lssvancl2 20945 | . . . . 5 ⊢ (𝜑 → ¬ ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{𝑠})) | 
| 39 | 1, 2, 6, 16, 13, 38 | lspsnne2 21121 | . . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠})) | 
| 40 | 39 | necomd 2995 | . . 3 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) | 
| 41 | 1, 17, 2 | lspsncl 20976 | . . . . . 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 41654 | . . . . 5 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶)) | 
| 44 | 42, 43 | eleqtrrd 2843 | . . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀) | 
| 45 | 3, 34, 5, 44 | mapdcnvid2 41660 | . . 3 ⊢ (𝜑 → (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) | 
| 46 | 40, 35, 45 | 3netr4d 3017 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))) | 
| 47 | 3, 34, 7, 25, 5, 44 | mapdcnvcl 41655 | . . . 4 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ∈ (LSubSp‘𝑈)) | 
| 48 | 3, 7, 25, 34, 5, 30, 47 | mapd11 41642 | . . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) = (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))) | 
| 49 | 48 | necon3bid 2984 | . 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))) | 
| 50 | 46, 49 | mpbid 232 | 1 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∖ cdif 3947 {csn 4625 ◡ccnv 5683 ran crn 5685 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 0gc0g 17485 LModclmod 20859 LSubSpclss 20930 LSpanclspn 20970 HLchlt 39352 LHypclh 39987 DVecHcdvh 41081 LCDualclcd 41589 mapdcmpd 41627 HDMapchdma 41795 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-riotaBAD 38955 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-tpos 8252 df-undef 8299 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17487 df-mre 17630 df-mrc 17631 df-acs 17633 df-proset 18341 df-poset 18360 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-p1 18472 df-lat 18478 df-clat 18545 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-cntz 19336 df-oppg 19365 df-lsm 19655 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-nzr 20514 df-rlreg 20695 df-domn 20696 df-drng 20732 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lvec 21103 df-lsatoms 38978 df-lshyp 38979 df-lcv 39021 df-lfl 39060 df-lkr 39088 df-ldual 39126 df-oposet 39178 df-ol 39180 df-oml 39181 df-covers 39268 df-ats 39269 df-atl 39300 df-cvlat 39324 df-hlat 39353 df-llines 39501 df-lplanes 39502 df-lvols 39503 df-lines 39504 df-psubsp 39506 df-pmap 39507 df-padd 39799 df-lhyp 39991 df-laut 39992 df-ldil 40107 df-ltrn 40108 df-trl 40162 df-tgrp 40746 df-tendo 40758 df-edring 40760 df-dveca 41006 df-disoa 41032 df-dvech 41082 df-dib 41142 df-dic 41176 df-dih 41232 df-doch 41351 df-djh 41398 df-lcdual 41590 df-mapd 41628 df-hvmap 41760 df-hdmap1 41796 df-hdmap 41797 | 
| This theorem is referenced by: hdmaprnlem9N 41860 hdmaprnlem3eN 41861 | 
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