mapdpglem5N - Mathbox for Norm Megill

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Theorem mapdpglem5N 35984

Description: Lemma for mapdpg 36013. (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 2610	. . . 4 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
6		mapdpglem.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
7		eqid 2610	. . . 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 35983	. . . . 5 ⊢ (𝜑 → (𝑋 − 𝑌) ≠ 𝑄)
27	2, 4, 8	dvhlmod 35417	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
28	9, 10	lmodvsubcl 18731	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
29	27, 12, 13, 28	syl3anc 1318	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
30	9, 11, 24, 5, 27, 29	lsatspn0 33305	. . . . 5 ⊢ (𝜑 → ((𝑁‘{(𝑋 − 𝑌)}) ∈ (LSAtoms‘𝑈) ↔ (𝑋 − 𝑌) ≠ 𝑄))
31	26, 30	mpbird 246	. . . 4 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSAtoms‘𝑈))
32	2, 3, 4, 5, 6, 7, 8, 31	mapdat 35974	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ∈ (LSAtoms‘𝐶))
33	1, 32	eqeltrrd 2689	. 2 ⊢ (𝜑 → (𝐽‘{𝑡}) ∈ (LSAtoms‘𝐶))
34		eqid 2610	. . 3 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
35	2, 6, 8	lcdlmod 35899	. . 3 ⊢ (𝜑 → 𝐶 ∈ LMod)
36	2, 3, 4, 9, 10, 11, 6, 8, 12, 13, 14, 15, 16, 17	mapdpglem2a 35981	. . 3 ⊢ (𝜑 → 𝑡 ∈ 𝐹)
37	16, 15, 34, 7, 35, 36	lsatspn0 33305	. 2 ⊢ (𝜑 → ((𝐽‘{𝑡}) ∈ (LSAtoms‘𝐶) ↔ 𝑡 ≠ (0_g‘𝐶)))
38	33, 37	mpbid 221	1 ⊢ (𝜑 → 𝑡 ≠ (0_g‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {csn 4125 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·_𝑠 cvsca 15772 0_gc0g 15923 -_gcsg 17247 LSSumclsm 17872 LModclmod 18686 LSpanclspn 18792 LSAtomsclsa 33279 HLchlt 33655 LHypclh 34288 DVecHcdvh 35385 LCDualclcd 35893 mapdcmpd 35931

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-mre 16069 df-mrc 16070 df-acs 16072 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-oppg 17599 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-lshyp 33282 df-lcv 33324 df-lfl 33363 df-lkr 33391 df-ldual 33429 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tgrp 35049 df-tendo 35061 df-edring 35063 df-dveca 35309 df-disoa 35336 df-dvech 35386 df-dib 35446 df-dic 35480 df-dih 35536 df-doch 35655 df-djh 35702 df-lcdual 35894 df-mapd 35932

This theorem is referenced by: (None)

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