hdmaprnlem9N - Mathbox for Norm Megill

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Theorem hdmaprnlem9N 41904

Description: Part of proof of part 12 in [Baer] p. 49 line 21, s=S(t). TODO: we seem to be going back and forth with mapd11 41686 and mapdcnv11N 41706. Use better hypotheses and/or theorems? (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
hdmaprnlem9N	⊢ (𝜑 → 𝑠 = (𝑆‘𝑡))

**Proof of Theorem hdmaprnlem9N**
Step	Hyp	Ref	Expression
1		hdmaprnlem1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmaprnlem1.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmaprnlem1.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
4		hdmaprnlem1.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
5		hdmaprnlem1.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
6		hdmaprnlem1.l	. . . . . 6 ⊢ 𝐿 = (LSpan‘𝐶)
7		hdmaprnlem1.m	. . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
8		hdmaprnlem1.s	. . . . . 6 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
9		hdmaprnlem1.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		hdmaprnlem1.se	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
11		hdmaprnlem1.ve	. . . . . 6 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
12		hdmaprnlem1.e	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
13		hdmaprnlem1.ue	. . . . . 6 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
14		hdmaprnlem1.un	. . . . . 6 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
15		hdmaprnlem1.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐶)
16		hdmaprnlem1.q	. . . . . 6 ⊢ 𝑄 = (0_g‘𝐶)
17		hdmaprnlem1.o	. . . . . 6 ⊢ 0 = (0_g‘𝑈)
18		hdmaprnlem1.a	. . . . . 6 ⊢ ✚ = (+_g‘𝐶)
19		hdmaprnlem1.t2	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))
20		hdmaprnlem1.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
21		hdmaprnlem1.pt	. . . . . 6 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))
22	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21	hdmaprnlem7N 41902	. . . . 5 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
23	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21	hdmaprnlem8N 41903	. . . . . 6 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝑀‘(𝑁‘{𝑡})))
24	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	hdmaprnlem4N 41900	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
25	23, 24	eleqtrd 2833	. . . . 5 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{𝑠}))
26	22, 25	elind 4147	. . . 4 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})))
27	1, 5, 9	lcdlvec 41638	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LVec)
28	1, 5, 9	lcdlmod 41639	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LMod)
29	1, 2, 3, 5, 15, 8, 9, 13	hdmapcl 41877	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
30	10	eldifad 3909	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
31	15, 18	lmodvacl 20808	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
32	28, 29, 30, 31	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
33		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
34	15, 33, 6	lspsncl 20910	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
35	28, 30, 34	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
36	1, 7, 5, 33, 9	mapdrn2 41698	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
37	35, 36	eleqtrrd 2834	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ ran 𝑀)
38	1, 7, 9, 37	mapdcnvid2 41704	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝐿‘{𝑠}))
39	12, 38	eqtr4d 2769	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))))
40		eqid 2731	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
41	1, 2, 9	dvhlmod 41157	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
42	3, 40, 4	lspsncl 20910	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
43	41, 11, 42	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
44	1, 7, 2, 40, 9, 37	mapdcnvcl 41699	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{𝑠})) ∈ (LSubSp‘𝑈))
45	1, 2, 40, 7, 9, 43, 44	mapd11 41686	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) ↔ (𝑁‘{𝑣}) = (^◡𝑀‘(𝐿‘{𝑠}))))
46	39, 45	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑣}) = (^◡𝑀‘(𝐿‘{𝑠})))
47	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	hdmaprnlem3N 41897	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
48	46, 47	eqnetrrd 2996	. . . . . . 7 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{𝑠})) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
49	15, 33, 6	lspsncl 20910	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
50	28, 32, 49	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
51	50, 36	eleqtrrd 2834	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
52	1, 7, 9, 37, 51	mapdcnv11N 41706	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝑀‘(𝐿‘{𝑠})) = (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
53	52	necon3bid 2972	. . . . . . 7 ⊢ (𝜑 → ((^◡𝑀‘(𝐿‘{𝑠})) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
54	48, 53	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
55	54	necomd 2983	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠}))
56	15, 16, 6, 27, 32, 30, 55	lspdisj2 21064	. . . 4 ⊢ (𝜑 → ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})) = {𝑄})
57	26, 56	eleqtrd 2833	. . 3 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ {𝑄})
58		elsni 4590	. . 3 ⊢ ((𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ {𝑄} → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄)
59	57, 58	syl 17	. 2 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄)
60	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	hdmaprnlem4tN 41899	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
61	1, 2, 3, 5, 15, 8, 9, 60	hdmapcl 41877	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
62		eqid 2731	. . . 4 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
63	15, 16, 62	lmodsubeq0 20854	. . 3 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷 ∧ (𝑆‘𝑡) ∈ 𝐷) → ((𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡)))
64	28, 30, 61, 63	syl3anc 1373	. 2 ⊢ (𝜑 → ((𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡)))
65	59, 64	mpbid 232	1 ⊢ (𝜑 → 𝑠 = (𝑆‘𝑡))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3894 ∩ cin 3896 {csn 4573 ^◡ccnv 5613 ran crn 5615 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 -_gcsg 18848 LModclmod 20793 LSubSpclss 20864 LSpanclspn 20904 HLchlt 39397 LHypclh 40031 DVecHcdvh 41125 LCDualclcd 41633 mapdcmpd 41671 HDMapchdma 41839

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-riotaBAD 39000

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-ot 4582 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-cntz 19229 df-oppg 19258 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-dvr 20319 df-nzr 20428 df-rlreg 20609 df-domn 20610 df-drng 20646 df-lmod 20795 df-lss 20865 df-lsp 20905 df-lvec 21037 df-lsatoms 39023 df-lshyp 39024 df-lcv 39066 df-lfl 39105 df-lkr 39133 df-ldual 39171 df-oposet 39223 df-ol 39225 df-oml 39226 df-covers 39313 df-ats 39314 df-atl 39345 df-cvlat 39369 df-hlat 39398 df-llines 39545 df-lplanes 39546 df-lvols 39547 df-lines 39548 df-psubsp 39550 df-pmap 39551 df-padd 39843 df-lhyp 40035 df-laut 40036 df-ldil 40151 df-ltrn 40152 df-trl 40206 df-tgrp 40790 df-tendo 40802 df-edring 40804 df-dveca 41050 df-disoa 41076 df-dvech 41126 df-dib 41186 df-dic 41220 df-dih 41276 df-doch 41395 df-djh 41442 df-lcdual 41634 df-mapd 41672 df-hvmap 41804 df-hdmap1 41840 df-hdmap 41841

This theorem is referenced by: hdmaprnlem10N 41906

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