hdmaprnlem7N - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaprnlem7N		Structured version Visualization version GIF version

Theorem hdmaprnlem7N 37931

Description: Part of proof of part 12 in [Baer] p. 49 line 19, s-St ∈ G(u'+s) = P*. (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
hdmaprnlem7N	⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))

**Proof of Theorem hdmaprnlem7N**
Step	Hyp	Ref	Expression
1		hdmaprnlem1.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
2		hdmaprnlem1.a	. . 3 ⊢ ✚ = (+_g‘𝐶)
3		eqid 2826	. . 3 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
4		hdmaprnlem1.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		hdmaprnlem1.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
6		hdmaprnlem1.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7	4, 5, 6	lcdlmod 37668	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
8		lmodabl 19267	. . . 4 ⊢ (𝐶 ∈ LMod → 𝐶 ∈ Abel)
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Abel)
10		hdmaprnlem1.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
11		hdmaprnlem1.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
12		hdmaprnlem1.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
13		hdmaprnlem1.ue	. . . 4 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
14	4, 10, 11, 5, 1, 12, 6, 13	hdmapcl 37906	. . 3 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
15		hdmaprnlem1.se	. . . 4 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
16	15	eldifad 3811	. . 3 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
17		hdmaprnlem1.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
18		hdmaprnlem1.l	. . . . 5 ⊢ 𝐿 = (LSpan‘𝐶)
19		hdmaprnlem1.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
20		hdmaprnlem1.ve	. . . . 5 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
21		hdmaprnlem1.e	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
22		hdmaprnlem1.un	. . . . 5 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
23		hdmaprnlem1.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝐶)
24		hdmaprnlem1.o	. . . . 5 ⊢ 0 = (0_g‘𝑈)
25		hdmaprnlem1.t2	. . . . 5 ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))
26	4, 10, 11, 17, 5, 18, 19, 12, 6, 15, 20, 21, 13, 22, 1, 23, 24, 2, 25	hdmaprnlem4tN 37928	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
27	4, 10, 11, 5, 1, 12, 6, 26	hdmapcl 37906	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
28	1, 2, 3, 9, 14, 16, 27, 9, 14, 16, 27	ablpnpcan 18579	. 2 ⊢ (𝜑 → (((𝑆‘𝑢) ✚ 𝑠)(-_g‘𝐶)((𝑆‘𝑢) ✚ (𝑆‘𝑡))) = (𝑠(-_g‘𝐶)(𝑆‘𝑡)))
29	1, 2	lmodvacl 19234	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
30	7, 14, 16, 29	syl3anc 1496	. . . 4 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
31		eqid 2826	. . . . 5 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
32	1, 31, 18	lspsncl 19337	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
33	7, 30, 32	syl2anc 581	. . 3 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
34	1, 18	lspsnid 19353	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
35	7, 30, 34	syl2anc 581	. . 3 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
36	1, 2	lmodvacl 19234	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ (𝑆‘𝑡) ∈ 𝐷) → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ 𝐷)
37	7, 14, 27, 36	syl3anc 1496	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ 𝐷)
38	1, 18	lspsnid 19353	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ 𝐷) → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))
39	7, 37, 38	syl2anc 581	. . . 4 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))
40		hdmaprnlem1.p	. . . . 5 ⊢ + = (+_g‘𝑈)
41		hdmaprnlem1.pt	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))
42	4, 10, 11, 17, 5, 18, 19, 12, 6, 15, 20, 21, 13, 22, 1, 23, 24, 2, 25, 40, 41	hdmaprnlem6N 37930	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))
43	39, 42	eleqtrrd 2910	. . 3 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
44	3, 31	lssvsubcl 19301	. . 3 ⊢ (((𝐶 ∈ LMod ∧ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶)) ∧ (((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∧ ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) → (((𝑆‘𝑢) ✚ 𝑠)(-_g‘𝐶)((𝑆‘𝑢) ✚ (𝑆‘𝑡))) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
45	7, 33, 35, 43, 44	syl22anc 874	. 2 ⊢ (𝜑 → (((𝑆‘𝑢) ✚ 𝑠)(-_g‘𝐶)((𝑆‘𝑢) ✚ (𝑆‘𝑡))) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
46	28, 45	eqeltrrd 2908	1 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∖ cdif 3796 {csn 4398 ‘cfv 6124 (class class class)co 6906 Basecbs 16223 +_gcplusg 16306 0_gc0g 16454 -_gcsg 17779 Abelcabl 18548 LModclmod 19220 LSubSpclss 19289 LSpanclspn 19331 HLchlt 35426 LHypclh 36060 DVecHcdvh 37154 LCDualclcd 37662 mapdcmpd 37700 HDMapchdma 37868

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-riotaBAD 35029

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 df-om 7328 df-1st 7429 df-2nd 7430 df-tpos 7618 df-undef 7665 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-n0 11620 df-z 11706 df-uz 11970 df-fz 12621 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-sca 16322 df-vsca 16323 df-0g 16456 df-mre 16600 df-mrc 16601 df-acs 16603 df-proset 17282 df-poset 17300 df-plt 17312 df-lub 17328 df-glb 17329 df-join 17330 df-meet 17331 df-p0 17393 df-p1 17394 df-lat 17400 df-clat 17462 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-submnd 17690 df-grp 17780 df-minusg 17781 df-sbg 17782 df-subg 17943 df-cntz 18101 df-oppg 18127 df-lsm 18403 df-cmn 18549 df-abl 18550 df-mgp 18845 df-ur 18857 df-ring 18904 df-oppr 18978 df-dvdsr 18996 df-unit 18997 df-invr 19027 df-dvr 19038 df-drng 19106 df-lmod 19222 df-lss 19290 df-lsp 19332 df-lvec 19463 df-lsatoms 35052 df-lshyp 35053 df-lcv 35095 df-lfl 35134 df-lkr 35162 df-ldual 35200 df-oposet 35252 df-ol 35254 df-oml 35255 df-covers 35342 df-ats 35343 df-atl 35374 df-cvlat 35398 df-hlat 35427 df-llines 35574 df-lplanes 35575 df-lvols 35576 df-lines 35577 df-psubsp 35579 df-pmap 35580 df-padd 35872 df-lhyp 36064 df-laut 36065 df-ldil 36180 df-ltrn 36181 df-trl 36235 df-tgrp 36819 df-tendo 36831 df-edring 36833 df-dveca 37079 df-disoa 37105 df-dvech 37155 df-dib 37215 df-dic 37249 df-dih 37305 df-doch 37424 df-djh 37471 df-lcdual 37663 df-mapd 37701 df-hvmap 37833 df-hdmap1 37869 df-hdmap 37870

This theorem is referenced by: hdmaprnlem9N 37933

W3C validator