mapdpglem9 - Mathbox for Norm Megill

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Theorem mapdpglem9 39894

Description: Lemma for mapdpg 39920. Baer p. 45, line 4: "...so that x would consequently belong to Fy." (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)
mapdpglem4.g0	⊢ (𝜑 → 𝑔 = 0 )

**Assertion**
Ref	Expression
mapdpglem9	⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌}))

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 mapdpglem9**
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 39324	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		mapdpglem.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
6		mapdpglem.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
7		mapdpglem.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
8		eqid 2736	. . . 4 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
9		mapdpglem.s	. . . 4 ⊢ − = (-_g‘𝑈)
10	7, 8, 9	lmodvnpcan 20226	. . 3 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 − 𝑌)(+_g‘𝑈)𝑌) = 𝑋)
11	4, 5, 6, 10	syl3anc 1371	. 2 ⊢ (𝜑 → ((𝑋 − 𝑌)(+_g‘𝑈)𝑌) = 𝑋)
12		eqid 2736	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
13		mapdpglem.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
14	7, 12, 13	lspsncl 20288	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
15	4, 6, 14	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
16		mapdpglem.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
17		mapdpglem.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
18		mapdpglem1.p	. . . . 5 ⊢ ⊕ = (LSSum‘𝐶)
19		mapdpglem2.j	. . . . 5 ⊢ 𝐽 = (LSpan‘𝐶)
20		mapdpglem3.f	. . . . 5 ⊢ 𝐹 = (Base‘𝐶)
21		mapdpglem3.te	. . . . 5 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
22		mapdpglem3.a	. . . . 5 ⊢ 𝐴 = (Scalar‘𝑈)
23		mapdpglem3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
24		mapdpglem3.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝐶)
25		mapdpglem3.r	. . . . 5 ⊢ 𝑅 = (-_g‘𝐶)
26		mapdpglem3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
27		mapdpglem3.e	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
28		mapdpglem4.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝑈)
29		mapdpglem.ne	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30		mapdpglem4.jt	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
31		mapdpglem4.z	. . . . 5 ⊢ 0 = (0_g‘𝐴)
32		mapdpglem4.g4	. . . . 5 ⊢ (𝜑 → 𝑔 ∈ 𝐵)
33		mapdpglem4.z4	. . . . 5 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
34		mapdpglem4.t4	. . . . 5 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
35		mapdpglem4.xn	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑄)
36		mapdpglem4.g0	. . . . 5 ⊢ (𝜑 → 𝑔 = 0 )
37	1, 16, 2, 7, 9, 13, 17, 3, 5, 6, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36	mapdpglem8 39893	. . . 4 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑌}))
38	7, 9	lmodvsubcl 20217	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
39	4, 5, 6, 38	syl3anc 1371	. . . . 5 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
40	7, 13	lspsnid 20304	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑋 − 𝑌) ∈ (𝑁‘{(𝑋 − 𝑌)}))
41	4, 39, 40	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝑁‘{(𝑋 − 𝑌)}))
42	37, 41	sseldd 3927	. . 3 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝑁‘{𝑌}))
43	7, 13	lspsnid 20304	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
44	4, 6, 43	syl2anc 585	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌}))
45	8, 12	lssvacl 20265	. . 3 ⊢ (((𝑈 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) ∧ ((𝑋 − 𝑌) ∈ (𝑁‘{𝑌}) ∧ 𝑌 ∈ (𝑁‘{𝑌}))) → ((𝑋 − 𝑌)(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑌}))
46	4, 15, 42, 44, 45	syl22anc 837	. 2 ⊢ (𝜑 → ((𝑋 − 𝑌)(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑌}))
47	11, 46	eqeltrrd 2838	1 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1539 ∈ wcel 2104 ≠ wne 2941 {csn 4565 ‘cfv 6458 (class class class)co 7307 Basecbs 16961 +_gcplusg 17011 Scalarcsca 17014 ·_𝑠 cvsca 17015 0_gc0g 17199 -_gcsg 18628 LSSumclsm 19288 LModclmod 20172 LSubSpclss 20242 LSpanclspn 20282 HLchlt 37564 LHypclh 38198 DVecHcdvh 39292 LCDualclcd 39800 mapdcmpd 39838

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10977 ax-resscn 10978 ax-1cn 10979 ax-icn 10980 ax-addcl 10981 ax-addrcl 10982 ax-mulcl 10983 ax-mulrcl 10984 ax-mulcom 10985 ax-addass 10986 ax-mulass 10987 ax-distr 10988 ax-i2m1 10989 ax-1ne0 10990 ax-1rid 10991 ax-rnegex 10992 ax-rrecex 10993 ax-cnre 10994 ax-pre-lttri 10995 ax-pre-lttrn 10996 ax-pre-ltadd 10997 ax-pre-mulgt0 10998 ax-riotaBAD 37167

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3304 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-tpos 8073 df-undef 8120 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-pnf 11061 df-mnf 11062 df-xr 11063 df-ltxr 11064 df-le 11065 df-sub 11257 df-neg 11258 df-nn 12024 df-2 12086 df-3 12087 df-4 12088 df-5 12089 df-6 12090 df-n0 12284 df-z 12370 df-uz 12633 df-fz 13290 df-struct 16897 df-sets 16914 df-slot 16932 df-ndx 16944 df-base 16962 df-ress 16991 df-plusg 17024 df-mulr 17025 df-sca 17027 df-vsca 17028 df-0g 17201 df-mre 17344 df-mrc 17345 df-acs 17347 df-proset 18062 df-poset 18080 df-plt 18097 df-lub 18113 df-glb 18114 df-join 18115 df-meet 18116 df-p0 18192 df-p1 18193 df-lat 18199 df-clat 18266 df-mgm 18375 df-sgrp 18424 df-mnd 18435 df-submnd 18480 df-grp 18629 df-minusg 18630 df-sbg 18631 df-subg 18801 df-cntz 18972 df-oppg 18999 df-lsm 19290 df-cmn 19437 df-abl 19438 df-mgp 19770 df-ur 19787 df-ring 19834 df-oppr 19911 df-dvdsr 19932 df-unit 19933 df-invr 19963 df-dvr 19974 df-drng 20042 df-lmod 20174 df-lss 20243 df-lsp 20283 df-lvec 20414 df-lsatoms 37190 df-lshyp 37191 df-lcv 37233 df-lfl 37272 df-lkr 37300 df-ldual 37338 df-oposet 37390 df-ol 37392 df-oml 37393 df-covers 37480 df-ats 37481 df-atl 37512 df-cvlat 37536 df-hlat 37565 df-llines 37712 df-lplanes 37713 df-lvols 37714 df-lines 37715 df-psubsp 37717 df-pmap 37718 df-padd 38010 df-lhyp 38202 df-laut 38203 df-ldil 38318 df-ltrn 38319 df-trl 38373 df-tgrp 38957 df-tendo 38969 df-edring 38971 df-dveca 39217 df-disoa 39243 df-dvech 39293 df-dib 39353 df-dic 39387 df-dih 39443 df-doch 39562 df-djh 39609 df-lcdual 39801 df-mapd 39839

This theorem is referenced by: mapdpglem10 39895

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