mapdpglem5N - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem5N		Structured version Visualization version GIF version

Theorem mapdpglem5N 41004

Description: Lemma for mapdpg 41033. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)

**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}))

**Assertion**
Ref	Expression
mapdpglem5N	⊢ (𝜑 → 𝑡 ≠ (0_g‘𝐶))

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

**Proof of Theorem mapdpglem5N**
Step	Hyp	Ref	Expression
1		mapdpglem4.jt	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
2		mapdpglem.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
3		mapdpglem.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
4		mapdpglem.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		eqid 2724	. . . 4 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
6		mapdpglem.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
7		eqid 2724	. . . 4 ⊢ (LSAtoms‘𝐶) = (LSAtoms‘𝐶)
8		mapdpglem.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		mapdpglem.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
10		mapdpglem.s	. . . . . 6 ⊢ − = (-_g‘𝑈)
11		mapdpglem.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
12		mapdpglem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13		mapdpglem.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
14		mapdpglem1.p	. . . . . 6 ⊢ ⊕ = (LSSum‘𝐶)
15		mapdpglem2.j	. . . . . 6 ⊢ 𝐽 = (LSpan‘𝐶)
16		mapdpglem3.f	. . . . . 6 ⊢ 𝐹 = (Base‘𝐶)
17		mapdpglem3.te	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
18		mapdpglem3.a	. . . . . 6 ⊢ 𝐴 = (Scalar‘𝑈)
19		mapdpglem3.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
20		mapdpglem3.t	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝐶)
21		mapdpglem3.r	. . . . . 6 ⊢ 𝑅 = (-_g‘𝐶)
22		mapdpglem3.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
23		mapdpglem3.e	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
24		mapdpglem4.q	. . . . . 6 ⊢ 𝑄 = (0_g‘𝑈)
25		mapdpglem.ne	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
26	2, 3, 4, 9, 10, 11, 6, 8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25	mapdpglem4N 41003	. . . . 5 ⊢ (𝜑 → (𝑋 − 𝑌) ≠ 𝑄)
27	2, 4, 8	dvhlmod 40437	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
28	9, 10	lmodvsubcl 20742	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
29	27, 12, 13, 28	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
30	9, 11, 24, 5, 27, 29	lsatspn0 38326	. . . . 5 ⊢ (𝜑 → ((𝑁‘{(𝑋 − 𝑌)}) ∈ (LSAtoms‘𝑈) ↔ (𝑋 − 𝑌) ≠ 𝑄))
31	26, 30	mpbird 257	. . . 4 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSAtoms‘𝑈))
32	2, 3, 4, 5, 6, 7, 8, 31	mapdat 40994	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ∈ (LSAtoms‘𝐶))
33	1, 32	eqeltrrd 2826	. 2 ⊢ (𝜑 → (𝐽‘{𝑡}) ∈ (LSAtoms‘𝐶))
34		eqid 2724	. . 3 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
35	2, 6, 8	lcdlmod 40919	. . 3 ⊢ (𝜑 → 𝐶 ∈ LMod)
36	2, 3, 4, 9, 10, 11, 6, 8, 12, 13, 14, 15, 16, 17	mapdpglem2a 41001	. . 3 ⊢ (𝜑 → 𝑡 ∈ 𝐹)
37	16, 15, 34, 7, 35, 36	lsatspn0 38326	. 2 ⊢ (𝜑 → ((𝐽‘{𝑡}) ∈ (LSAtoms‘𝐶) ↔ 𝑡 ≠ (0_g‘𝐶)))
38	33, 37	mpbid 231	1 ⊢ (𝜑 → 𝑡 ≠ (0_g‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 {csn 4620 ‘cfv 6533 (class class class)co 7401 Basecbs 17142 Scalarcsca 17198 ·_𝑠 cvsca 17199 0_gc0g 17383 -_gcsg 18854 LSSumclsm 19543 LModclmod 20695 LSpanclspn 20807 LSAtomsclsa 38300 HLchlt 38676 LHypclh 39311 DVecHcdvh 40405 LCDualclcd 40913 mapdcmpd 40951

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-riotaBAD 38279

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-undef 8253 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-sca 17211 df-vsca 17212 df-0g 17385 df-mre 17528 df-mrc 17529 df-acs 17531 df-proset 18249 df-poset 18267 df-plt 18284 df-lub 18300 df-glb 18301 df-join 18302 df-meet 18303 df-p0 18379 df-p1 18380 df-lat 18386 df-clat 18453 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-grp 18855 df-minusg 18856 df-sbg 18857 df-subg 19039 df-cntz 19222 df-oppg 19251 df-lsm 19545 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-drng 20578 df-lmod 20697 df-lss 20768 df-lsp 20808 df-lvec 20940 df-lsatoms 38302 df-lshyp 38303 df-lcv 38345 df-lfl 38384 df-lkr 38412 df-ldual 38450 df-oposet 38502 df-ol 38504 df-oml 38505 df-covers 38592 df-ats 38593 df-atl 38624 df-cvlat 38648 df-hlat 38677 df-llines 38825 df-lplanes 38826 df-lvols 38827 df-lines 38828 df-psubsp 38830 df-pmap 38831 df-padd 39123 df-lhyp 39315 df-laut 39316 df-ldil 39431 df-ltrn 39432 df-trl 39486 df-tgrp 40070 df-tendo 40082 df-edring 40084 df-dveca 40330 df-disoa 40356 df-dvech 40406 df-dib 40466 df-dic 40500 df-dih 40556 df-doch 40675 df-djh 40722 df-lcdual 40914 df-mapd 40952

This theorem is referenced by: (None)

W3C validator