hdmaprnlem7N - Mathbox for Norm Megill

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Theorem hdmaprnlem7N 41838

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 41575	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
8		lmodabl 20924	. . . 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 41813	. . 3 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
15		hdmaprnlem1.se	. . . 4 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
16	15	eldifad 3975	. . 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 41835	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
27	4, 10, 11, 5, 1, 12, 6, 26	hdmapcl 41813	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
28	1, 2, 3, 9, 14, 16, 27, 9, 14, 16, 27	ablpnpcan 19852	. 2 ⊢ (𝜑 → (((𝑆‘𝑢) ✚ 𝑠)(-_g‘𝐶)((𝑆‘𝑢) ✚ (𝑆‘𝑡))) = (𝑠(-_g‘𝐶)(𝑆‘𝑡)))
29	1, 2	lmodvacl 20890	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
30	7, 14, 16, 29	syl3anc 1370	. . . 4 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
31		eqid 2735	. . . . 5 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
32	1, 31, 18	lspsncl 20993	. . . 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 21009	. . . 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 20890	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ (𝑆‘𝑡) ∈ 𝐷) → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ 𝐷)
37	7, 14, 27, 36	syl3anc 1370	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ 𝐷)
38	1, 18	lspsnid 21009	. . . . 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 41837	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))
43	39, 42	eleqtrrd 2842	. . 3 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ (𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
44	3, 31	lssvsubcl 20960	. . 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 839	. 2 ⊢ (𝜑 → (((𝑆‘𝑢) ✚ 𝑠)(-_g‘𝐶)((𝑆‘𝑢) ✚ (𝑆‘𝑡))) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
46	28, 45	eqeltrrd 2840	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 1537 ∈ wcel 2106 ∖ cdif 3960 {csn 4631 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +_gcplusg 17298 0_gc0g 17486 -_gcsg 18966 Abelcabl 19814 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 LCDualclcd 41569 mapdcmpd 41607 HDMapchdma 41775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lcv 39001 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378 df-lcdual 41570 df-mapd 41608 df-hvmap 41740 df-hdmap1 41776 df-hdmap 41777

This theorem is referenced by: hdmaprnlem9N 41840

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