hdmaprnlem7N - Mathbox for Norm Megill

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Theorem hdmaprnlem7N 41458

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 2725	. . 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 41195	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
8		lmodabl 20804	. . . 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 41433	. . 3 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
15		hdmaprnlem1.se	. . . 4 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
16	15	eldifad 3956	. . 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 41455	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
27	4, 10, 11, 5, 1, 12, 6, 26	hdmapcl 41433	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
28	1, 2, 3, 9, 14, 16, 27, 9, 14, 16, 27	ablpnpcan 19786	. 2 ⊢ (𝜑 → (((𝑆‘𝑢) ✚ 𝑠)(-_g‘𝐶)((𝑆‘𝑢) ✚ (𝑆‘𝑡))) = (𝑠(-_g‘𝐶)(𝑆‘𝑡)))
29	1, 2	lmodvacl 20770	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
30	7, 14, 16, 29	syl3anc 1368	. . . 4 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
31		eqid 2725	. . . . 5 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
32	1, 31, 18	lspsncl 20873	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
33	7, 30, 32	syl2anc 582	. . 3 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
34	1, 18	lspsnid 20889	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
35	7, 30, 34	syl2anc 582	. . 3 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
36	1, 2	lmodvacl 20770	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ (𝑆‘𝑡) ∈ 𝐷) → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ 𝐷)
37	7, 14, 27, 36	syl3anc 1368	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ 𝐷)
38	1, 18	lspsnid 20889	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ 𝐷) → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))
39	7, 37, 38	syl2anc 582	. . . 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 41457	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))
43	39, 42	eleqtrrd 2828	. . 3 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
44	3, 31	lssvsubcl 20840	. . 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 837	. 2 ⊢ (𝜑 → (((𝑆‘𝑢) ✚ 𝑠)(-_g‘𝐶)((𝑆‘𝑢) ✚ (𝑆‘𝑡))) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
46	28, 45	eqeltrrd 2826	1 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∖ cdif 3941 {csn 4630 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 +_gcplusg 17236 0_gc0g 17424 -_gcsg 18900 Abelcabl 19748 LModclmod 20755 LSubSpclss 20827 LSpanclspn 20867 HLchlt 38952 LHypclh 39587 DVecHcdvh 40681 LCDualclcd 41189 mapdcmpd 41227 HDMapchdma 41395

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-riotaBAD 38555

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-0g 17426 df-mre 17569 df-mrc 17570 df-acs 17572 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-clat 18494 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cntz 19280 df-oppg 19309 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lvec 21000 df-lsatoms 38578 df-lshyp 38579 df-lcv 38621 df-lfl 38660 df-lkr 38688 df-ldual 38726 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-llines 39101 df-lplanes 39102 df-lvols 39103 df-lines 39104 df-psubsp 39106 df-pmap 39107 df-padd 39399 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762 df-tgrp 40346 df-tendo 40358 df-edring 40360 df-dveca 40606 df-disoa 40632 df-dvech 40682 df-dib 40742 df-dic 40776 df-dih 40832 df-doch 40951 df-djh 40998 df-lcdual 41190 df-mapd 41228 df-hvmap 41360 df-hdmap1 41396 df-hdmap 41397

This theorem is referenced by: hdmaprnlem9N 41460

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