hdmaprnlem8N - Mathbox for Norm Megill

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Theorem hdmaprnlem8N 42729

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 42465	. 2 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		hdmaprnlem1.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		hdmaprnlem1.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2760	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8		eqid 2760	. . 3 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
9	1, 6, 3	dvhlmod 41983	. . . 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 42725	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
25	10, 7, 11	lspsncl 21161	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑡 ∈ 𝑉) → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
26	9, 24, 25	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
27	1, 5, 6, 7, 2, 8, 3, 26	mapdcl2 42529	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) ∈ (LSubSp‘𝐶))
28	14	eldifad 3911	. . . 4 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
29	19, 12	lspsnid 21177	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠}))
30	4, 28, 29	syl2anc 596	. . 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 42726	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
32	30, 31	eleqtrrd 2863	. 2 ⊢ (𝜑 → 𝑠 ∈ (𝑀‘(𝑁‘{𝑡})))
33	1, 6, 10, 2, 19, 13, 3, 24	hdmapcl 42703	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
34	19, 12	lspsnid 21177	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝑆‘𝑡) ∈ (𝐿‘{(𝑆‘𝑡)}))
35	4, 33, 34	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ (𝐿‘{(𝑆‘𝑡)}))
36	1, 6, 10, 11, 2, 12, 5, 13, 3, 24	hdmap10 42713	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{(𝑆‘𝑡)}))
37	35, 36	eleqtrrd 2863	. 2 ⊢ (𝜑 → (𝑆‘𝑡) ∈ (𝑀‘(𝑁‘{𝑡})))
38		eqid 2760	. . 3 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
39	38, 8	lssvsubcl 21128	. 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 852	1 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝑀‘(𝑁‘{𝑡})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∖ cdif 3896 {csn 4584 ‘cfv 6533 (class class class)co 7413 Basecbs 17301 +_gcplusg 17342 0_gc0g 17524 -_gcsg 19059 LModclmod 21044 LSubSpclss 21115 LSpanclspn 21155 HLchlt 40223 LHypclh 40857 DVecHcdvh 41951 LCDualclcd 42459 mapdcmpd 42497 HDMapchdma 42665

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-riotaBAD 39826

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 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 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-tpos 8224 df-undef 8271 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12529 df-z 12616 df-uz 12888 df-fz 13562 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-sca 17358 df-vsca 17359 df-0g 17526 df-mre 17670 df-mrc 17671 df-acs 17673 df-proset 18382 df-poset 18401 df-plt 18416 df-lub 18432 df-glb 18433 df-join 18434 df-meet 18435 df-p0 18511 df-p1 18512 df-lat 18520 df-clat 18587 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-grp 19060 df-minusg 19061 df-sbg 19062 df-subg 19246 df-cntz 19444 df-oppg 19473 df-lsm 19763 df-cmn 19909 df-abl 19910 df-mgp 20274 df-rng 20288 df-ur 20321 df-ring 20374 df-oppr 20478 df-dvdsr 20498 df-unit 20499 df-invr 20529 df-dvr 20542 df-nzr 20673 df-rlreg 20856 df-domn 20857 df-drng 20892 df-lmod 21046 df-lss 21116 df-lsp 21156 df-lvec 21287 df-lsatoms 39849 df-lshyp 39850 df-lcv 39892 df-lfl 39931 df-lkr 39959 df-ldual 39997 df-oposet 40049 df-ol 40051 df-oml 40052 df-covers 40139 df-ats 40140 df-atl 40171 df-cvlat 40195 df-hlat 40224 df-llines 40371 df-lplanes 40372 df-lvols 40373 df-lines 40374 df-psubsp 40376 df-pmap 40377 df-padd 40669 df-lhyp 40861 df-laut 40862 df-ldil 40977 df-ltrn 40978 df-trl 41032 df-tgrp 41616 df-tendo 41628 df-edring 41630 df-dveca 41876 df-disoa 41902 df-dvech 41952 df-dib 42012 df-dic 42046 df-dih 42102 df-doch 42221 df-djh 42268 df-lcdual 42460 df-mapd 42498 df-hvmap 42630 df-hdmap1 42666 df-hdmap 42667

This theorem is used by: hdmaprnlem9N 42730

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