hdmaprnlem8N - Mathbox for Norm Megill

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Theorem hdmaprnlem8N 41895

Description: Part of proof of part 12 in [Baer] p. 49 line 19, s-St ∈ (Ft)* = T*. (Contributed by NM, 27-May-2015.) (New usage is discouraged.)

**Hypotheses**
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 = (0_g‘𝐶)
hdmaprnlem1.o	⊢ 0 = (0_g‘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)})))

**Assertion**
Ref	Expression
hdmaprnlem8N	⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝑀‘(𝑁‘{𝑡})))

**Proof of Theorem hdmaprnlem8N**
Step	Hyp	Ref	Expression
1		hdmaprnlem1.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmaprnlem1.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
3		hdmaprnlem1.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	lcdlmod 41631	. 2 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		hdmaprnlem1.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		hdmaprnlem1.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2731	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8		eqid 2731	. . 3 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
9	1, 6, 3	dvhlmod 41149	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		hdmaprnlem1.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
11		hdmaprnlem1.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
12		hdmaprnlem1.l	. . . . 5 ⊢ 𝐿 = (LSpan‘𝐶)
13		hdmaprnlem1.s	. . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
14		hdmaprnlem1.se	. . . . 5 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
15		hdmaprnlem1.ve	. . . . 5 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
16		hdmaprnlem1.e	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
17		hdmaprnlem1.ue	. . . . 5 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
18		hdmaprnlem1.un	. . . . 5 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
19		hdmaprnlem1.d	. . . . 5 ⊢ 𝐷 = (Base‘𝐶)
20		hdmaprnlem1.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝐶)
21		hdmaprnlem1.o	. . . . 5 ⊢ 0 = (0_g‘𝑈)
22		hdmaprnlem1.a	. . . . 5 ⊢ ✚ = (+_g‘𝐶)
23		hdmaprnlem1.t2	. . . . 5 ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))
24	1, 6, 10, 11, 2, 12, 5, 13, 3, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23	hdmaprnlem4tN 41891	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
25	10, 7, 11	lspsncl 20905	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑡 ∈ 𝑉) → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
26	9, 24, 25	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
27	1, 5, 6, 7, 2, 8, 3, 26	mapdcl2 41695	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) ∈ (LSubSp‘𝐶))
28	14	eldifad 3909	. . . 4 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
29	19, 12	lspsnid 20921	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠}))
30	4, 28, 29	syl2anc 584	. . 3 ⊢ (𝜑 → 𝑠 ∈ (𝐿‘{𝑠}))
31	1, 6, 10, 11, 2, 12, 5, 13, 3, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23	hdmaprnlem4N 41892	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
32	30, 31	eleqtrrd 2834	. 2 ⊢ (𝜑 → 𝑠 ∈ (𝑀‘(𝑁‘{𝑡})))
33	1, 6, 10, 2, 19, 13, 3, 24	hdmapcl 41869	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
34	19, 12	lspsnid 20921	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝑆‘𝑡) ∈ (𝐿‘{(𝑆‘𝑡)}))
35	4, 33, 34	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ (𝐿‘{(𝑆‘𝑡)}))
36	1, 6, 10, 11, 2, 12, 5, 13, 3, 24	hdmap10 41879	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{(𝑆‘𝑡)}))
37	35, 36	eleqtrrd 2834	. 2 ⊢ (𝜑 → (𝑆‘𝑡) ∈ (𝑀‘(𝑁‘{𝑡})))
38		eqid 2731	. . 3 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
39	38, 8	lssvsubcl 20872	. 2 ⊢ (((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑡})) ∈ (LSubSp‘𝐶)) ∧ (𝑠 ∈ (𝑀‘(𝑁‘{𝑡})) ∧ (𝑆‘𝑡) ∈ (𝑀‘(𝑁‘{𝑡})))) → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝑀‘(𝑁‘{𝑡})))
40	4, 27, 32, 37, 39	syl22anc 838	1 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝑀‘(𝑁‘{𝑡})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∖ cdif 3894 {csn 4571 ‘cfv 6476 (class class class)co 7341 Basecbs 17115 +_gcplusg 17156 0_gc0g 17338 -_gcsg 18843 LModclmod 20788 LSubSpclss 20859 LSpanclspn 20899 HLchlt 39389 LHypclh 40023 DVecHcdvh 41117 LCDualclcd 41625 mapdcmpd 41663 HDMapchdma 41831

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-riotaBAD 38992

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-ot 4580 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-undef 8198 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-n0 12377 df-z 12464 df-uz 12728 df-fz 13403 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-sca 17172 df-vsca 17173 df-0g 17340 df-mre 17483 df-mrc 17484 df-acs 17486 df-proset 18195 df-poset 18214 df-plt 18229 df-lub 18245 df-glb 18246 df-join 18247 df-meet 18248 df-p0 18324 df-p1 18325 df-lat 18333 df-clat 18400 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-grp 18844 df-minusg 18845 df-sbg 18846 df-subg 19031 df-cntz 19224 df-oppg 19253 df-lsm 19543 df-cmn 19689 df-abl 19690 df-mgp 20054 df-rng 20066 df-ur 20095 df-ring 20148 df-oppr 20250 df-dvdsr 20270 df-unit 20271 df-invr 20301 df-dvr 20314 df-nzr 20423 df-rlreg 20604 df-domn 20605 df-drng 20641 df-lmod 20790 df-lss 20860 df-lsp 20900 df-lvec 21032 df-lsatoms 39015 df-lshyp 39016 df-lcv 39058 df-lfl 39097 df-lkr 39125 df-ldual 39163 df-oposet 39215 df-ol 39217 df-oml 39218 df-covers 39305 df-ats 39306 df-atl 39337 df-cvlat 39361 df-hlat 39390 df-llines 39537 df-lplanes 39538 df-lvols 39539 df-lines 39540 df-psubsp 39542 df-pmap 39543 df-padd 39835 df-lhyp 40027 df-laut 40028 df-ldil 40143 df-ltrn 40144 df-trl 40198 df-tgrp 40782 df-tendo 40794 df-edring 40796 df-dveca 41042 df-disoa 41068 df-dvech 41118 df-dib 41178 df-dic 41212 df-dih 41268 df-doch 41387 df-djh 41434 df-lcdual 41626 df-mapd 41664 df-hvmap 41796 df-hdmap1 41832 df-hdmap 41833

This theorem is referenced by: hdmaprnlem9N 41896

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