hdmaprnlem8N - Mathbox for Norm Megill

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Theorem hdmaprnlem8N 42316

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 42052	. 2 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		hdmaprnlem1.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		hdmaprnlem1.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2737	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8		eqid 2737	. . 3 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
9	1, 6, 3	dvhlmod 41570	. . . 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 42312	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
25	10, 7, 11	lspsncl 20963	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑡 ∈ 𝑉) → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
26	9, 24, 25	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
27	1, 5, 6, 7, 2, 8, 3, 26	mapdcl2 42116	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) ∈ (LSubSp‘𝐶))
28	14	eldifad 3902	. . . 4 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
29	19, 12	lspsnid 20979	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠}))
30	4, 28, 29	syl2anc 585	. . 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 42313	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
32	30, 31	eleqtrrd 2840	. 2 ⊢ (𝜑 → 𝑠 ∈ (𝑀‘(𝑁‘{𝑡})))
33	1, 6, 10, 2, 19, 13, 3, 24	hdmapcl 42290	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
34	19, 12	lspsnid 20979	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝑆‘𝑡) ∈ (𝐿‘{(𝑆‘𝑡)}))
35	4, 33, 34	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ (𝐿‘{(𝑆‘𝑡)}))
36	1, 6, 10, 11, 2, 12, 5, 13, 3, 24	hdmap10 42300	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{(𝑆‘𝑡)}))
37	35, 36	eleqtrrd 2840	. 2 ⊢ (𝜑 → (𝑆‘𝑡) ∈ (𝑀‘(𝑁‘{𝑡})))
38		eqid 2737	. . 3 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
39	38, 8	lssvsubcl 20930	. 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 839	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 1542 ∈ wcel 2114 ∖ cdif 3887 {csn 4568 ‘cfv 6492 (class class class)co 7360 Basecbs 17170 +_gcplusg 17211 0_gc0g 17393 -_gcsg 18902 LModclmod 20846 LSubSpclss 20917 LSpanclspn 20957 HLchlt 39810 LHypclh 40444 DVecHcdvh 41538 LCDualclcd 42046 mapdcmpd 42084 HDMapchdma 42252

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-riotaBAD 39413

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-ot 4577 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8169 df-undef 8216 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-0g 17395 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18251 df-poset 18270 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18389 df-clat 18456 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-cntz 19283 df-oppg 19312 df-lsm 19602 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-nzr 20481 df-rlreg 20662 df-domn 20663 df-drng 20699 df-lmod 20848 df-lss 20918 df-lsp 20958 df-lvec 21090 df-lsatoms 39436 df-lshyp 39437 df-lcv 39479 df-lfl 39518 df-lkr 39546 df-ldual 39584 df-oposet 39636 df-ol 39638 df-oml 39639 df-covers 39726 df-ats 39727 df-atl 39758 df-cvlat 39782 df-hlat 39811 df-llines 39958 df-lplanes 39959 df-lvols 39960 df-lines 39961 df-psubsp 39963 df-pmap 39964 df-padd 40256 df-lhyp 40448 df-laut 40449 df-ldil 40564 df-ltrn 40565 df-trl 40619 df-tgrp 41203 df-tendo 41215 df-edring 41217 df-dveca 41463 df-disoa 41489 df-dvech 41539 df-dib 41599 df-dic 41633 df-dih 41689 df-doch 41808 df-djh 41855 df-lcdual 42047 df-mapd 42085 df-hvmap 42217 df-hdmap1 42253 df-hdmap 42254

This theorem is referenced by: hdmaprnlem9N 42317

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