hdmaprnlem9N - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaprnlem9N		Structured version Visualization version GIF version

Theorem hdmaprnlem9N 41966

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 41748 and mapdcnv11N 41768. 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 41964	. . . . 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 41965	. . . . . 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 41962	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
25	23, 24	eleqtrd 2835	. . . . 5 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{𝑠}))
26	22, 25	elind 4151	. . . 4 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})))
27	1, 5, 9	lcdlvec 41700	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LVec)
28	1, 5, 9	lcdlmod 41701	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LMod)
29	1, 2, 3, 5, 15, 8, 9, 13	hdmapcl 41939	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
30	10	eldifad 3911	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
31	15, 18	lmodvacl 20818	. . . . . 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 2733	. . . . . . . . . . . . . 14 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
34	15, 33, 6	lspsncl 20920	. . . . . . . . . . . . 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 41760	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
37	35, 36	eleqtrrd 2836	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ ran 𝑀)
38	1, 7, 9, 37	mapdcnvid2 41766	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝐿‘{𝑠}))
39	12, 38	eqtr4d 2771	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))))
40		eqid 2733	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
41	1, 2, 9	dvhlmod 41219	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
42	3, 40, 4	lspsncl 20920	. . . . . . . . . . 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 41761	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{𝑠})) ∈ (LSubSp‘𝑈))
45	1, 2, 40, 7, 9, 43, 44	mapd11 41748	. . . . . . . . 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 41959	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
48	46, 47	eqnetrrd 2998	. . . . . . 7 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{𝑠})) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
49	15, 33, 6	lspsncl 20920	. . . . . . . . . . 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 2836	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
52	1, 7, 9, 37, 51	mapdcnv11N 41768	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝑀‘(𝐿‘{𝑠})) = (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
53	52	necon3bid 2974	. . . . . . 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 2985	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠}))
56	15, 16, 6, 27, 32, 30, 55	lspdisj2 21074	. . . 4 ⊢ (𝜑 → ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})) = {𝑄})
57	26, 56	eleqtrd 2835	. . 3 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ {𝑄})
58		elsni 4594	. . 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 41961	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
61	1, 2, 3, 5, 15, 8, 9, 60	hdmapcl 41939	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
62		eqid 2733	. . . 4 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
63	15, 16, 62	lmodsubeq0 20864	. . 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 2113 ≠ wne 2930 ∖ cdif 3896 ∩ cin 3898 {csn 4577 ^◡ccnv 5620 ran crn 5622 ‘cfv 6489 (class class class)co 7355 Basecbs 17130 +_gcplusg 17171 0_gc0g 17353 -_gcsg 18858 LModclmod 20803 LSubSpclss 20874 LSpanclspn 20914 HLchlt 39459 LHypclh 40093 DVecHcdvh 41187 LCDualclcd 41695 mapdcmpd 41733 HDMapchdma 41901

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093 ax-riotaBAD 39062

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-tpos 8165 df-undef 8212 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-nn 12136 df-2 12198 df-3 12199 df-4 12200 df-5 12201 df-6 12202 df-n0 12392 df-z 12479 df-uz 12743 df-fz 13418 df-struct 17068 df-sets 17085 df-slot 17103 df-ndx 17115 df-base 17131 df-ress 17152 df-plusg 17184 df-mulr 17185 df-sca 17187 df-vsca 17188 df-0g 17355 df-mre 17498 df-mrc 17499 df-acs 17501 df-proset 18210 df-poset 18229 df-plt 18244 df-lub 18260 df-glb 18261 df-join 18262 df-meet 18263 df-p0 18339 df-p1 18340 df-lat 18348 df-clat 18415 df-mgm 18558 df-sgrp 18637 df-mnd 18653 df-submnd 18702 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19046 df-cntz 19239 df-oppg 19268 df-lsm 19558 df-cmn 19704 df-abl 19705 df-mgp 20069 df-rng 20081 df-ur 20110 df-ring 20163 df-oppr 20265 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-nzr 20438 df-rlreg 20619 df-domn 20620 df-drng 20656 df-lmod 20805 df-lss 20875 df-lsp 20915 df-lvec 21047 df-lsatoms 39085 df-lshyp 39086 df-lcv 39128 df-lfl 39167 df-lkr 39195 df-ldual 39233 df-oposet 39285 df-ol 39287 df-oml 39288 df-covers 39375 df-ats 39376 df-atl 39407 df-cvlat 39431 df-hlat 39460 df-llines 39607 df-lplanes 39608 df-lvols 39609 df-lines 39610 df-psubsp 39612 df-pmap 39613 df-padd 39905 df-lhyp 40097 df-laut 40098 df-ldil 40213 df-ltrn 40214 df-trl 40268 df-tgrp 40852 df-tendo 40864 df-edring 40866 df-dveca 41112 df-disoa 41138 df-dvech 41188 df-dib 41248 df-dic 41282 df-dih 41338 df-doch 41457 df-djh 41504 df-lcdual 41696 df-mapd 41734 df-hvmap 41866 df-hdmap1 41902 df-hdmap 41903

This theorem is referenced by: hdmaprnlem10N 41968

W3C validator