hdmaprnlem7N - Mathbox for Norm Megill

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Theorem hdmaprnlem7N 41874

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 2735	. . 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 41611	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
8		lmodabl 20866	. . . 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 41849	. . 3 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
15		hdmaprnlem1.se	. . . 4 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
16	15	eldifad 3938	. . 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 41871	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
27	4, 10, 11, 5, 1, 12, 6, 26	hdmapcl 41849	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
28	1, 2, 3, 9, 14, 16, 27, 9, 14, 16, 27	ablpnpcan 19800	. 2 ⊢ (𝜑 → (((𝑆‘𝑢) ✚ 𝑠)(-_g‘𝐶)((𝑆‘𝑢) ✚ (𝑆‘𝑡))) = (𝑠(-_g‘𝐶)(𝑆‘𝑡)))
29	1, 2	lmodvacl 20832	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
30	7, 14, 16, 29	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
31		eqid 2735	. . . . 5 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
32	1, 31, 18	lspsncl 20934	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
33	7, 30, 32	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
34	1, 18	lspsnid 20950	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
35	7, 30, 34	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
36	1, 2	lmodvacl 20832	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ (𝑆‘𝑡) ∈ 𝐷) → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ 𝐷)
37	7, 14, 27, 36	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ 𝐷)
38	1, 18	lspsnid 20950	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ 𝐷) → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))
39	7, 37, 38	syl2anc 584	. . . 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 41873	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))
43	39, 42	eleqtrrd 2837	. . 3 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
44	3, 31	lssvsubcl 20901	. . 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 838	. 2 ⊢ (𝜑 → (((𝑆‘𝑢) ✚ 𝑠)(-_g‘𝐶)((𝑆‘𝑢) ✚ (𝑆‘𝑡))) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
46	28, 45	eqeltrrd 2835	1 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3923 {csn 4601 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 +_gcplusg 17271 0_gc0g 17453 -_gcsg 18918 Abelcabl 19762 LModclmod 20817 LSubSpclss 20888 LSpanclspn 20928 HLchlt 39368 LHypclh 40003 DVecHcdvh 41097 LCDualclcd 41605 mapdcmpd 41643 HDMapchdma 41811

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-riotaBAD 38971

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-ot 4610 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-0g 17455 df-mre 17598 df-mrc 17599 df-acs 17601 df-proset 18306 df-poset 18325 df-plt 18340 df-lub 18356 df-glb 18357 df-join 18358 df-meet 18359 df-p0 18435 df-p1 18436 df-lat 18442 df-clat 18509 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-cntz 19300 df-oppg 19329 df-lsm 19617 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-nzr 20473 df-rlreg 20654 df-domn 20655 df-drng 20691 df-lmod 20819 df-lss 20889 df-lsp 20929 df-lvec 21061 df-lsatoms 38994 df-lshyp 38995 df-lcv 39037 df-lfl 39076 df-lkr 39104 df-ldual 39142 df-oposet 39194 df-ol 39196 df-oml 39197 df-covers 39284 df-ats 39285 df-atl 39316 df-cvlat 39340 df-hlat 39369 df-llines 39517 df-lplanes 39518 df-lvols 39519 df-lines 39520 df-psubsp 39522 df-pmap 39523 df-padd 39815 df-lhyp 40007 df-laut 40008 df-ldil 40123 df-ltrn 40124 df-trl 40178 df-tgrp 40762 df-tendo 40774 df-edring 40776 df-dveca 41022 df-disoa 41048 df-dvech 41098 df-dib 41158 df-dic 41192 df-dih 41248 df-doch 41367 df-djh 41414 df-lcdual 41606 df-mapd 41644 df-hvmap 41776 df-hdmap1 41812 df-hdmap 41813

This theorem is referenced by: hdmaprnlem9N 41876

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