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 42731

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 42513 and mapdcnv11N 42533. 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 42729	. . . . 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 42730	. . . . . 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 42727	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
25	23, 24	eleqtrd 2862	. . . . 5 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{𝑠}))
26	22, 25	elind 4146	. . . 4 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})))
27	1, 5, 9	lcdlvec 42465	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LVec)
28	1, 5, 9	lcdlmod 42466	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ LMod)
29	1, 2, 3, 5, 15, 8, 9, 13	hdmapcl 42704	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
30	10	eldifad 3911	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
31	15, 18	lmodvacl 21060	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
32	28, 29, 30, 31	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
33		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
34	15, 33, 6	lspsncl 21162	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
35	28, 30, 34	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
36	1, 7, 5, 33, 9	mapdrn2 42525	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
37	35, 36	eleqtrrd 2863	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ ran 𝑀)
38	1, 7, 9, 37	mapdcnvid2 42531	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) = (𝐿‘{𝑠}))
39	12, 38	eqtr4d 2798	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))))
40		eqid 2760	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
41	1, 2, 9	dvhlmod 41984	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
42	3, 40, 4	lspsncl 21162	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
43	41, 11, 42	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
44	1, 7, 2, 40, 9, 37	mapdcnvcl 42526	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{𝑠})) ∈ (LSubSp‘𝑈))
45	1, 2, 40, 7, 9, 43, 44	mapd11 42513	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{𝑠}))) ↔ (𝑁‘{𝑣}) = (^◡𝑀‘(𝐿‘{𝑠}))))
46	39, 45	mpbid 235	. . . . . . . 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 42724	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
48	46, 47	eqnetrrd 3023	. . . . . . 7 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{𝑠})) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
49	15, 33, 6	lspsncl 21162	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
50	28, 32, 49	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
51	50, 36	eleqtrrd 2863	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
52	1, 7, 9, 37, 51	mapdcnv11N 42533	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝑀‘(𝐿‘{𝑠})) = (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
53	52	necon3bid 2999	. . . . . . 7 ⊢ (𝜑 → ((^◡𝑀‘(𝐿‘{𝑠})) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ↔ (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
54	48, 53	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
55	54	necomd 3010	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠}))
56	15, 16, 6, 27, 32, 30, 55	lspdisj2 21315	. . . 4 ⊢ (𝜑 → ((𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∩ (𝐿‘{𝑠})) = {𝑄})
57	26, 56	eleqtrd 2862	. . 3 ⊢ (𝜑 → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ {𝑄})
58		elsni 4601	. . 3 ⊢ ((𝑠(-_g‘𝐶)(𝑆‘𝑡)) ∈ {𝑄} → (𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄)
59	57, 58	syl 18	. 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 42726	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
61	1, 2, 3, 5, 15, 8, 9, 60	hdmapcl 42704	. . 3 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
62		eqid 2760	. . . 4 ⊢ (-_g‘𝐶) = (-_g‘𝐶)
63	15, 16, 62	lmodsubeq0 21106	. . 3 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷 ∧ (𝑆‘𝑡) ∈ 𝐷) → ((𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡)))
64	28, 30, 61, 63	syl3anc 1398	. 2 ⊢ (𝜑 → ((𝑠(-_g‘𝐶)(𝑆‘𝑡)) = 𝑄 ↔ 𝑠 = (𝑆‘𝑡)))
65	59, 64	mpbid 235	1 ⊢ (𝜑 → 𝑠 = (𝑆‘𝑡))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∖ cdif 3896 ∩ cin 3898 {csn 4584 ^◡ccnv 5654 ran crn 5656 ‘cfv 6533 (class class class)co 7414 Basecbs 17302 +_gcplusg 17343 0_gc0g 17525 -_gcsg 19060 LModclmod 21045 LSubSpclss 21116 LSpanclspn 21156 HLchlt 40224 LHypclh 40858 DVecHcdvh 41952 LCDualclcd 42460 mapdcmpd 42498 HDMapchdma 42666

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-riotaBAD 39827

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-er 8697 df-map 8829 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-n0 12530 df-z 12617 df-uz 12889 df-fz 13563 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-sca 17359 df-vsca 17360 df-0g 17527 df-mre 17671 df-mrc 17672 df-acs 17674 df-proset 18383 df-poset 18402 df-plt 18417 df-lub 18433 df-glb 18434 df-join 18435 df-meet 18436 df-p0 18512 df-p1 18513 df-lat 18521 df-clat 18588 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-submnd 18893 df-grp 19061 df-minusg 19062 df-sbg 19063 df-subg 19247 df-cntz 19445 df-oppg 19474 df-lsm 19764 df-cmn 19910 df-abl 19911 df-mgp 20275 df-rng 20289 df-ur 20322 df-ring 20375 df-oppr 20479 df-dvdsr 20499 df-unit 20500 df-invr 20530 df-dvr 20543 df-nzr 20674 df-rlreg 20857 df-domn 20858 df-drng 20893 df-lmod 21047 df-lss 21117 df-lsp 21157 df-lvec 21288 df-lsatoms 39850 df-lshyp 39851 df-lcv 39893 df-lfl 39932 df-lkr 39960 df-ldual 39998 df-oposet 40050 df-ol 40052 df-oml 40053 df-covers 40140 df-ats 40141 df-atl 40172 df-cvlat 40196 df-hlat 40225 df-llines 40372 df-lplanes 40373 df-lvols 40374 df-lines 40375 df-psubsp 40377 df-pmap 40378 df-padd 40670 df-lhyp 40862 df-laut 40863 df-ldil 40978 df-ltrn 40979 df-trl 41033 df-tgrp 41617 df-tendo 41629 df-edring 41631 df-dveca 41877 df-disoa 41903 df-dvech 41953 df-dib 42013 df-dic 42047 df-dih 42103 df-doch 42222 df-djh 42269 df-lcdual 42461 df-mapd 42499 df-hvmap 42631 df-hdmap1 42667 df-hdmap 42668

This theorem is used by: hdmaprnlem10N 42733

W3C validator