hdmaprnlem8N - Mathbox for Norm Megill

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Theorem hdmaprnlem8N 42441

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 42177	. 2 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		hdmaprnlem1.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		hdmaprnlem1.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2761	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8		eqid 2761	. . 3 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
9	1, 6, 3	dvhlmod 41695	. . . 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 42437	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
25	10, 7, 11	lspsncl 21032	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑡 ∈ 𝑉) → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
26	9, 24, 25	syl2anc 593	. . 3 ⊢ (𝜑 → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
27	1, 5, 6, 7, 2, 8, 3, 26	mapdcl2 42241	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) ∈ (LSubSp‘𝐶))
28	14	eldifad 3914	. . . 4 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
29	19, 12	lspsnid 21048	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠}))
30	4, 28, 29	syl2anc 593	. . 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 42438	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
32	30, 31	eleqtrrd 2864	. 2 ⊢ (𝜑 → 𝑠 ∈ (𝑀‘(𝑁‘{𝑡})))
33	1, 6, 10, 2, 19, 13, 3, 24	hdmapcl 42415	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
34	19, 12	lspsnid 21048	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝑆‘𝑡) ∈ (𝐿‘{(𝑆‘𝑡)}))
35	4, 33, 34	syl2anc 593	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ (𝐿‘{(𝑆‘𝑡)}))
36	1, 6, 10, 11, 2, 12, 5, 13, 3, 24	hdmap10 42425	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{(𝑆‘𝑡)}))
37	35, 36	eleqtrrd 2864	. 2 ⊢ (𝜑 → (𝑆‘𝑡) ∈ (𝑀‘(𝑁‘{𝑡})))
38		eqid 2761	. . 3 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
39	38, 8	lssvsubcl 20999	. 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 849	1 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝑀‘(𝑁‘{𝑡})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∖ cdif 3899 {csn 4579 ‘cfv 6516 (class class class)co 7391 Basecbs 17236 +_gcplusg 17277 0_gc0g 17459 -_gcsg 18968 LModclmod 20915 LSubSpclss 20986 LSpanclspn 21026 HLchlt 39935 LHypclh 40569 DVecHcdvh 41663 LCDualclcd 42171 mapdcmpd 42209 HDMapchdma 42377

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-riotaBAD 39538

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-om 7842 df-1st 7965 df-2nd 7966 df-tpos 8200 df-undef 8247 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-n0 12476 df-z 12563 df-uz 12834 df-fz 13507 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-sca 17293 df-vsca 17294 df-0g 17461 df-mre 17605 df-mrc 17606 df-acs 17608 df-proset 18317 df-poset 18336 df-plt 18351 df-lub 18367 df-glb 18368 df-join 18369 df-meet 18370 df-p0 18446 df-p1 18447 df-lat 18455 df-clat 18522 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18969 df-minusg 18970 df-sbg 18971 df-subg 19156 df-cntz 19348 df-oppg 19377 df-lsm 19667 df-cmn 19813 df-abl 19814 df-mgp 20178 df-rng 20190 df-ur 20219 df-ring 20272 df-oppr 20373 df-dvdsr 20393 df-unit 20394 df-invr 20424 df-dvr 20437 df-nzr 20550 df-rlreg 20731 df-domn 20732 df-drng 20768 df-lmod 20917 df-lss 20987 df-lsp 21027 df-lvec 21158 df-lsatoms 39561 df-lshyp 39562 df-lcv 39604 df-lfl 39643 df-lkr 39671 df-ldual 39709 df-oposet 39761 df-ol 39763 df-oml 39764 df-covers 39851 df-ats 39852 df-atl 39883 df-cvlat 39907 df-hlat 39936 df-llines 40083 df-lplanes 40084 df-lvols 40085 df-lines 40086 df-psubsp 40088 df-pmap 40089 df-padd 40381 df-lhyp 40573 df-laut 40574 df-ldil 40689 df-ltrn 40690 df-trl 40744 df-tgrp 41328 df-tendo 41340 df-edring 41342 df-dveca 41588 df-disoa 41614 df-dvech 41664 df-dib 41724 df-dic 41758 df-dih 41814 df-doch 41933 df-djh 41980 df-lcdual 42172 df-mapd 42210 df-hvmap 42342 df-hdmap1 42378 df-hdmap 42379

This theorem is referenced by: hdmaprnlem9N 42442

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