mapdpglem5N - Mathbox for Norm Megill

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Theorem mapdpglem5N 42615

Description: Lemma for mapdpg 42644. (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 2760	. . . 4 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
6		mapdpglem.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
7		eqid 2760	. . . 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 42614	. . . . 5 ⊢ (𝜑 → (𝑋 − 𝑌) ≠ 𝑄)
27	2, 4, 8	dvhlmod 42048	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
28	9, 10	lmodvsubcl 21121	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
29	27, 12, 13, 28	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
30	9, 11, 24, 5, 27, 29	lsatspn0 39938	. . . . 5 ⊢ (𝜑 → ((𝑁‘{(𝑋 − 𝑌)}) ∈ (LSAtoms‘𝑈) ↔ (𝑋 − 𝑌) ≠ 𝑄))
31	26, 30	mpbird 260	. . . 4 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSAtoms‘𝑈))
32	2, 3, 4, 5, 6, 7, 8, 31	mapdat 42605	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ∈ (LSAtoms‘𝐶))
33	1, 32	eqeltrrd 2861	. 2 ⊢ (𝜑 → (𝐽‘{𝑡}) ∈ (LSAtoms‘𝐶))
34		eqid 2760	. . 3 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
35	2, 6, 8	lcdlmod 42530	. . 3 ⊢ (𝜑 → 𝐶 ∈ LMod)
36	2, 3, 4, 9, 10, 11, 6, 8, 12, 13, 14, 15, 16, 17	mapdpglem2a 42612	. . 3 ⊢ (𝜑 → 𝑡 ∈ 𝐹)
37	16, 15, 34, 7, 35, 36	lsatspn0 39938	. 2 ⊢ (𝜑 → ((𝐽‘{𝑡}) ∈ (LSAtoms‘𝐶) ↔ 𝑡 ≠ (0_g‘𝐶)))
38	33, 37	mpbid 235	1 ⊢ (𝜑 → 𝑡 ≠ (0_g‘𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 {csn 4584 ‘cfv 6535 (class class class)co 7416 Basecbs 17326 Scalarcsca 17370 ·_𝑠 cvsca 17371 0_gc0g 17549 -_gcsg 19085 LSSumclsm 19787 LModclmod 21074 LSpanclspn 21185 LSAtomsclsa 39912 HLchlt 40288 LHypclh 40922 DVecHcdvh 42016 LCDualclcd 42524 mapdcmpd 42562

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 7742 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-riotaBAD 39891

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-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 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8229 df-undef 8276 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-n0 12554 df-z 12641 df-uz 12913 df-fz 13587 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-sca 17383 df-vsca 17384 df-0g 17551 df-mre 17695 df-mrc 17696 df-acs 17698 df-proset 18407 df-poset 18426 df-plt 18441 df-lub 18457 df-glb 18458 df-join 18459 df-meet 18460 df-p0 18536 df-p1 18537 df-lat 18545 df-clat 18612 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-submnd 18918 df-grp 19086 df-minusg 19087 df-sbg 19088 df-subg 19272 df-cntz 19470 df-oppg 19499 df-lsm 19789 df-cmn 19935 df-abl 19936 df-mgp 20300 df-rng 20314 df-ur 20347 df-ring 20400 df-oppr 20506 df-dvdsr 20526 df-unit 20527 df-invr 20557 df-dvr 20570 df-nzr 20702 df-rlreg 20885 df-domn 20886 df-drng 20921 df-lmod 21076 df-lss 21146 df-lsp 21186 df-lvec 21317 df-lsatoms 39914 df-lshyp 39915 df-lcv 39957 df-lfl 39996 df-lkr 40024 df-ldual 40062 df-oposet 40114 df-ol 40116 df-oml 40117 df-covers 40204 df-ats 40205 df-atl 40236 df-cvlat 40260 df-hlat 40289 df-llines 40436 df-lplanes 40437 df-lvols 40438 df-lines 40439 df-psubsp 40441 df-pmap 40442 df-padd 40734 df-lhyp 40926 df-laut 40927 df-ldil 41042 df-ltrn 41043 df-trl 41097 df-tgrp 41681 df-tendo 41693 df-edring 41695 df-dveca 41941 df-disoa 41967 df-dvech 42017 df-dib 42077 df-dic 42111 df-dih 42167 df-doch 42286 df-djh 42333 df-lcdual 42525 df-mapd 42563

This theorem is used by: (None)

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