mapdpglem5N - Mathbox for Norm Megill

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Theorem mapdpglem5N 41959

Description: Lemma for mapdpg 41988. (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 2736	. . . 4 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
6		mapdpglem.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
7		eqid 2736	. . . 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 41958	. . . . 5 ⊢ (𝜑 → (𝑋 − 𝑌) ≠ 𝑄)
27	2, 4, 8	dvhlmod 41392	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
28	9, 10	lmodvsubcl 20860	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
29	27, 12, 13, 28	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
30	9, 11, 24, 5, 27, 29	lsatspn0 39282	. . . . 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 41949	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ∈ (LSAtoms‘𝐶))
33	1, 32	eqeltrrd 2837	. 2 ⊢ (𝜑 → (𝐽‘{𝑡}) ∈ (LSAtoms‘𝐶))
34		eqid 2736	. . 3 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
35	2, 6, 8	lcdlmod 41874	. . 3 ⊢ (𝜑 → 𝐶 ∈ LMod)
36	2, 3, 4, 9, 10, 11, 6, 8, 12, 13, 14, 15, 16, 17	mapdpglem2a 41956	. . 3 ⊢ (𝜑 → 𝑡 ∈ 𝐹)
37	16, 15, 34, 7, 35, 36	lsatspn0 39282	. 2 ⊢ (𝜑 → ((𝐽‘{𝑡}) ∈ (LSAtoms‘𝐶) ↔ 𝑡 ≠ (0_g‘𝐶)))
38	33, 37	mpbid 232	1 ⊢ (𝜑 → 𝑡 ≠ (0_g‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 {csn 4580 ‘cfv 6492 (class class class)co 7358 Basecbs 17138 Scalarcsca 17182 ·_𝑠 cvsca 17183 0_gc0g 17361 -_gcsg 18867 LSSumclsm 19565 LModclmod 20813 LSpanclspn 20924 LSAtomsclsa 39256 HLchlt 39632 LHypclh 40266 DVecHcdvh 41360 LCDualclcd 41868 mapdcmpd 41906

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-riotaBAD 39235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-n0 12404 df-z 12491 df-uz 12754 df-fz 13426 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-0g 17363 df-mre 17507 df-mrc 17508 df-acs 17510 df-proset 18219 df-poset 18238 df-plt 18253 df-lub 18269 df-glb 18270 df-join 18271 df-meet 18272 df-p0 18348 df-p1 18349 df-lat 18357 df-clat 18424 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cntz 19248 df-oppg 19277 df-lsm 19567 df-cmn 19713 df-abl 19714 df-mgp 20078 df-rng 20090 df-ur 20119 df-ring 20172 df-oppr 20275 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-nzr 20448 df-rlreg 20629 df-domn 20630 df-drng 20666 df-lmod 20815 df-lss 20885 df-lsp 20925 df-lvec 21057 df-lsatoms 39258 df-lshyp 39259 df-lcv 39301 df-lfl 39340 df-lkr 39368 df-ldual 39406 df-oposet 39458 df-ol 39460 df-oml 39461 df-covers 39548 df-ats 39549 df-atl 39580 df-cvlat 39604 df-hlat 39633 df-llines 39780 df-lplanes 39781 df-lvols 39782 df-lines 39783 df-psubsp 39785 df-pmap 39786 df-padd 40078 df-lhyp 40270 df-laut 40271 df-ldil 40386 df-ltrn 40387 df-trl 40441 df-tgrp 41025 df-tendo 41037 df-edring 41039 df-dveca 41285 df-disoa 41311 df-dvech 41361 df-dib 41421 df-dic 41455 df-dih 41511 df-doch 41630 df-djh 41677 df-lcdual 41869 df-mapd 41907

This theorem is referenced by: (None)

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