hdmaprnlem8N - Mathbox for Norm Megill

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Theorem hdmaprnlem8N 42515

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 42251	. 2 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		hdmaprnlem1.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		hdmaprnlem1.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2769	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8		eqid 2769	. . 3 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
9	1, 6, 3	dvhlmod 41769	. . . 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 42511	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
25	10, 7, 11	lspsncl 21072	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑡 ∈ 𝑉) → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
26	9, 24, 25	syl2anc 595	. . 3 ⊢ (𝜑 → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
27	1, 5, 6, 7, 2, 8, 3, 26	mapdcl2 42315	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) ∈ (LSubSp‘𝐶))
28	14	eldifad 3925	. . . 4 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
29	19, 12	lspsnid 21088	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠}))
30	4, 28, 29	syl2anc 595	. . 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 42512	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
32	30, 31	eleqtrrd 2872	. 2 ⊢ (𝜑 → 𝑠 ∈ (𝑀‘(𝑁‘{𝑡})))
33	1, 6, 10, 2, 19, 13, 3, 24	hdmapcl 42489	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
34	19, 12	lspsnid 21088	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝑆‘𝑡) ∈ (𝐿‘{(𝑆‘𝑡)}))
35	4, 33, 34	syl2anc 595	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ (𝐿‘{(𝑆‘𝑡)}))
36	1, 6, 10, 11, 2, 12, 5, 13, 3, 24	hdmap10 42499	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{(𝑆‘𝑡)}))
37	35, 36	eleqtrrd 2872	. 2 ⊢ (𝜑 → (𝑆‘𝑡) ∈ (𝑀‘(𝑁‘{𝑡})))
38		eqid 2769	. . 3 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
39	38, 8	lssvsubcl 21039	. 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 851	1 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝑀‘(𝑁‘{𝑡})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∖ cdif 3910 {csn 4591 ‘cfv 6533 (class class class)co 7408 Basecbs 17265 +_gcplusg 17306 0_gc0g 17488 -_gcsg 18998 LModclmod 20955 LSubSpclss 21026 LSpanclspn 21066 HLchlt 40009 LHypclh 40643 DVecHcdvh 41737 LCDualclcd 42245 mapdcmpd 42283 HDMapchdma 42451

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-riotaBAD 39612

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-undef 8265 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12501 df-z 12588 df-uz 12859 df-fz 13532 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-0g 17490 df-mre 17634 df-mrc 17635 df-acs 17637 df-proset 18346 df-poset 18365 df-plt 18380 df-lub 18396 df-glb 18397 df-join 18398 df-meet 18399 df-p0 18475 df-p1 18476 df-lat 18484 df-clat 18551 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-submnd 18838 df-grp 18999 df-minusg 19000 df-sbg 19001 df-subg 19185 df-cntz 19383 df-oppg 19412 df-lsm 19702 df-cmn 19848 df-abl 19849 df-mgp 20213 df-rng 20227 df-ur 20260 df-ring 20313 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-nzr 20592 df-rlreg 20775 df-domn 20776 df-drng 20811 df-lmod 20957 df-lss 21027 df-lsp 21067 df-lvec 21198 df-lsatoms 39635 df-lshyp 39636 df-lcv 39678 df-lfl 39717 df-lkr 39745 df-ldual 39783 df-oposet 39835 df-ol 39837 df-oml 39838 df-covers 39925 df-ats 39926 df-atl 39957 df-cvlat 39981 df-hlat 40010 df-llines 40157 df-lplanes 40158 df-lvols 40159 df-lines 40160 df-psubsp 40162 df-pmap 40163 df-padd 40455 df-lhyp 40647 df-laut 40648 df-ldil 40763 df-ltrn 40764 df-trl 40818 df-tgrp 41402 df-tendo 41414 df-edring 41416 df-dveca 41662 df-disoa 41688 df-dvech 41738 df-dib 41798 df-dic 41832 df-dih 41888 df-doch 42007 df-djh 42054 df-lcdual 42246 df-mapd 42284 df-hvmap 42416 df-hdmap1 42452 df-hdmap 42453

This theorem is referenced by: hdmaprnlem9N 42516

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