mapdpglem14 - Mathbox for Norm Megill

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Theorem mapdpglem14 38853

Description: Lemma for mapdpg 38874. (Contributed by NM, 20-Mar-2015.)

**Hypotheses**
Ref	Expression
mapdpglem.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdpglem.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdpglem.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdpglem.v	⊢ 𝑉 = (Base‘𝑈)
mapdpglem.s	⊢ − = (-_g‘𝑈)
mapdpglem.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdpglem.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdpglem.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdpglem.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
mapdpglem.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mapdpglem1.p	⊢ ⊕ = (LSSum‘𝐶)
mapdpglem2.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdpglem3.f	⊢ 𝐹 = (Base‘𝐶)
mapdpglem3.te	⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
mapdpglem3.a	⊢ 𝐴 = (Scalar‘𝑈)
mapdpglem3.b	⊢ 𝐵 = (Base‘𝐴)
mapdpglem3.t	⊢ · = ( ·_𝑠 ‘𝐶)
mapdpglem3.r	⊢ 𝑅 = (-_g‘𝐶)
mapdpglem3.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
mapdpglem3.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
mapdpglem4.q	⊢ 𝑄 = (0_g‘𝑈)
mapdpglem.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdpglem4.jt	⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
mapdpglem4.z	⊢ 0 = (0_g‘𝐴)
mapdpglem4.g4	⊢ (𝜑 → 𝑔 ∈ 𝐵)
mapdpglem4.z4	⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
mapdpglem4.t4	⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
mapdpglem4.xn	⊢ (𝜑 → 𝑋 ≠ 𝑄)
mapdpglem12.yn	⊢ (𝜑 → 𝑌 ≠ 𝑄)
mapdpglem12.g0	⊢ (𝜑 → 𝑧 = (0_g‘𝐶))

**Assertion**
Ref	Expression
mapdpglem14	⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋}))

Distinct variable groups: 𝑡, − 𝑡,𝐶 𝑡,𝐽 𝑡,𝑀 𝑡,𝑁 𝑡,𝑋 𝑡,𝑌 𝐵,𝑔 𝑧,𝑔,𝐶 𝑔,𝐹 𝑔,𝐺,𝑧 𝑔,𝐽,𝑧 𝑔,𝑀,𝑧 𝑔,𝑁,𝑧 𝑅,𝑔,𝑧 · ,𝑔,𝑧 𝑔,𝑌,𝑧,𝑡 Allowed substitution hints: 𝜑(𝑧,𝑡,𝑔) 𝐴(𝑧,𝑡,𝑔) 𝐵(𝑧,𝑡) ⊕ (𝑧,𝑡,𝑔) 𝑄(𝑧,𝑡,𝑔) 𝑅(𝑡) · (𝑡) 𝑈(𝑧,𝑡,𝑔) 𝐹(𝑧,𝑡) 𝐺(𝑡) 𝐻(𝑧,𝑡,𝑔) 𝐾(𝑧,𝑡,𝑔) − (𝑧,𝑔) 𝑉(𝑧,𝑡,𝑔) 𝑊(𝑧,𝑡,𝑔) 𝑋(𝑧,𝑔) 0 (𝑧,𝑡,𝑔)

**Proof of Theorem mapdpglem14**
Step	Hyp	Ref	Expression
1		mapdpglem.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdpglem.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdpglem.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 38278	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		mapdpglem.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
6		mapdpglem.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
7		mapdpglem.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
8		eqid 2821	. . . 4 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
9		mapdpglem.s	. . . 4 ⊢ − = (-_g‘𝑈)
10	7, 8, 9	lmodvnpcan 19671	. . 3 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ((𝑌 − 𝑋)(+_g‘𝑈)𝑋) = 𝑌)
11	4, 5, 6, 10	syl3anc 1367	. 2 ⊢ (𝜑 → ((𝑌 − 𝑋)(+_g‘𝑈)𝑋) = 𝑌)
12		eqid 2821	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
13		mapdpglem.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
14	7, 12, 13	lspsncl 19732	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
15	4, 6, 14	syl2anc 586	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
16		lmodgrp 19624	. . . . . 6 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp)
17	4, 16	syl 17	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Grp)
18		eqid 2821	. . . . . 6 ⊢ (inv_g‘𝑈) = (inv_g‘𝑈)
19	7, 9, 18	grpinvsub 18164	. . . . 5 ⊢ ((𝑈 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((inv_g‘𝑈)‘(𝑋 − 𝑌)) = (𝑌 − 𝑋))
20	17, 6, 5, 19	syl3anc 1367	. . . 4 ⊢ (𝜑 → ((inv_g‘𝑈)‘(𝑋 − 𝑌)) = (𝑌 − 𝑋))
21		mapdpglem.m	. . . . . . 7 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
22		mapdpglem.c	. . . . . . 7 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
23		mapdpglem1.p	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝐶)
24		mapdpglem2.j	. . . . . . 7 ⊢ 𝐽 = (LSpan‘𝐶)
25		mapdpglem3.f	. . . . . . 7 ⊢ 𝐹 = (Base‘𝐶)
26		mapdpglem3.te	. . . . . . 7 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
27		mapdpglem3.a	. . . . . . 7 ⊢ 𝐴 = (Scalar‘𝑈)
28		mapdpglem3.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
29		mapdpglem3.t	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝐶)
30		mapdpglem3.r	. . . . . . 7 ⊢ 𝑅 = (-_g‘𝐶)
31		mapdpglem3.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
32		mapdpglem3.e	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
33		mapdpglem4.q	. . . . . . 7 ⊢ 𝑄 = (0_g‘𝑈)
34		mapdpglem.ne	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
35		mapdpglem4.jt	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
36		mapdpglem4.z	. . . . . . 7 ⊢ 0 = (0_g‘𝐴)
37		mapdpglem4.g4	. . . . . . 7 ⊢ (𝜑 → 𝑔 ∈ 𝐵)
38		mapdpglem4.z4	. . . . . . 7 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
39		mapdpglem4.t4	. . . . . . 7 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
40		mapdpglem4.xn	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 𝑄)
41		mapdpglem12.yn	. . . . . . 7 ⊢ (𝜑 → 𝑌 ≠ 𝑄)
42		mapdpglem12.g0	. . . . . . 7 ⊢ (𝜑 → 𝑧 = (0_g‘𝐶))
43	1, 21, 2, 7, 9, 13, 22, 3, 6, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42	mapdpglem13 38852	. . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑋}))
44	7, 9	lmodvsubcl 19662	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
45	4, 6, 5, 44	syl3anc 1367	. . . . . . 7 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
46	7, 13	lspsnid 19748	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑋 − 𝑌) ∈ (𝑁‘{(𝑋 − 𝑌)}))
47	4, 45, 46	syl2anc 586	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝑁‘{(𝑋 − 𝑌)}))
48	43, 47	sseldd 3956	. . . . 5 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝑁‘{𝑋}))
49	12, 18	lssvnegcl 19711	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈) ∧ (𝑋 − 𝑌) ∈ (𝑁‘{𝑋})) → ((inv_g‘𝑈)‘(𝑋 − 𝑌)) ∈ (𝑁‘{𝑋}))
50	4, 15, 48, 49	syl3anc 1367	. . . 4 ⊢ (𝜑 → ((inv_g‘𝑈)‘(𝑋 − 𝑌)) ∈ (𝑁‘{𝑋}))
51	20, 50	eqeltrrd 2914	. . 3 ⊢ (𝜑 → (𝑌 − 𝑋) ∈ (𝑁‘{𝑋}))
52	7, 13	lspsnid 19748	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋}))
53	4, 6, 52	syl2anc 586	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋}))
54	8, 12	lssvacl 19709	. . 3 ⊢ (((𝑈 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) ∧ ((𝑌 − 𝑋) ∈ (𝑁‘{𝑋}) ∧ 𝑋 ∈ (𝑁‘{𝑋}))) → ((𝑌 − 𝑋)(+_g‘𝑈)𝑋) ∈ (𝑁‘{𝑋}))
55	4, 15, 51, 53, 54	syl22anc 836	. 2 ⊢ (𝜑 → ((𝑌 − 𝑋)(+_g‘𝑈)𝑋) ∈ (𝑁‘{𝑋}))
56	11, 55	eqeltrrd 2914	1 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 {csn 4553 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 +_gcplusg 16548 Scalarcsca 16551 ·_𝑠 cvsca 16552 0_gc0g 16696 Grpcgrp 18086 inv_gcminusg 18087 -_gcsg 18088 LSSumclsm 18742 LModclmod 19617 LSubSpclss 19686 LSpanclspn 19726 HLchlt 36518 LHypclh 37152 DVecHcdvh 38246 LCDualclcd 38754 mapdcmpd 38792

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-riotaBAD 36121

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-tpos 7878 df-undef 7925 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-n0 11885 df-z 11969 df-uz 12231 df-fz 12883 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-0g 16698 df-mre 16840 df-mrc 16841 df-acs 16843 df-proset 17521 df-poset 17539 df-plt 17551 df-lub 17567 df-glb 17568 df-join 17569 df-meet 17570 df-p0 17632 df-p1 17633 df-lat 17639 df-clat 17701 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-submnd 17940 df-grp 18089 df-minusg 18090 df-sbg 18091 df-subg 18259 df-cntz 18430 df-oppg 18457 df-lsm 18744 df-cmn 18891 df-abl 18892 df-mgp 19223 df-ur 19235 df-ring 19282 df-oppr 19356 df-dvdsr 19374 df-unit 19375 df-invr 19405 df-dvr 19416 df-drng 19487 df-lmod 19619 df-lss 19687 df-lsp 19727 df-lvec 19858 df-lsatoms 36144 df-lshyp 36145 df-lcv 36187 df-lfl 36226 df-lkr 36254 df-ldual 36292 df-oposet 36344 df-ol 36346 df-oml 36347 df-covers 36434 df-ats 36435 df-atl 36466 df-cvlat 36490 df-hlat 36519 df-llines 36666 df-lplanes 36667 df-lvols 36668 df-lines 36669 df-psubsp 36671 df-pmap 36672 df-padd 36964 df-lhyp 37156 df-laut 37157 df-ldil 37272 df-ltrn 37273 df-trl 37327 df-tgrp 37911 df-tendo 37923 df-edring 37925 df-dveca 38171 df-disoa 38197 df-dvech 38247 df-dib 38307 df-dic 38341 df-dih 38397 df-doch 38516 df-djh 38563 df-lcdual 38755 df-mapd 38793

This theorem is referenced by: mapdpglem15 38854

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