hdmaprnlem3N - Mathbox for Norm Megill

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Theorem hdmaprnlem3N 37806

Description: Part of proof of part 12 in [Baer] p. 49 line 15, T ≠ P. Our (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) is Baer's P, where P* = G(u'+s). (Contributed by NM, 27-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hdmaprnlem1.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmaprnlem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmaprnlem1.v	⊢ 𝑉 = (Base‘𝑈)
hdmaprnlem1.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmaprnlem1.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmaprnlem1.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmaprnlem1.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmaprnlem1.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmaprnlem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmaprnlem1.se	⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
hdmaprnlem1.ve	⊢ (𝜑 → 𝑣 ∈ 𝑉)
hdmaprnlem1.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
hdmaprnlem1.ue	⊢ (𝜑 → 𝑢 ∈ 𝑉)
hdmaprnlem1.un	⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
hdmaprnlem1.d	⊢ 𝐷 = (Base‘𝐶)
hdmaprnlem1.q	⊢ 𝑄 = (0_g‘𝐶)
hdmaprnlem1.o	⊢ 0 = (0_g‘𝑈)
hdmaprnlem1.a	⊢ ✚ = (+_g‘𝐶)

**Assertion**
Ref	Expression
hdmaprnlem3N	⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))

**Proof of Theorem hdmaprnlem3N**
Step	Hyp	Ref	Expression
1		hdmaprnlem1.d	. . . . 5 ⊢ 𝐷 = (Base‘𝐶)
2		hdmaprnlem1.l	. . . . 5 ⊢ 𝐿 = (LSpan‘𝐶)
3		hdmaprnlem1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
4		hdmaprnlem1.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
5		hdmaprnlem1.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	3, 4, 5	lcdlmod 37548	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod)
7		hdmaprnlem1.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		hdmaprnlem1.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
9		hdmaprnlem1.s	. . . . . . 7 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
10		hdmaprnlem1.ue	. . . . . . 7 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
11	3, 7, 8, 4, 1, 9, 5, 10	hdmapcl 37786	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
12		hdmaprnlem1.se	. . . . . . 7 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
13	12	eldifad 3744	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
14		hdmaprnlem1.a	. . . . . . 7 ⊢ ✚ = (+_g‘𝐶)
15	1, 14	lmodvacl 19146	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
16	6, 11, 13, 15	syl3anc 1490	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
17		eqid 2765	. . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
18	1, 17, 2	lspsncl 19249	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
19	6, 13, 18	syl2anc 579	. . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
20	1, 2	lspsnid 19265	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠}))
21	6, 13, 20	syl2anc 579	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐿‘{𝑠}))
22		hdmaprnlem1.q	. . . . . . 7 ⊢ 𝑄 = (0_g‘𝐶)
23	3, 4, 5	lcdlvec 37547	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ LVec)
24		hdmaprnlem1.o	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
25		eqid 2765	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
26	3, 7, 5	dvhlmod 37066	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
27		hdmaprnlem1.ve	. . . . . . . . . 10 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
28		hdmaprnlem1.n	. . . . . . . . . . 11 ⊢ 𝑁 = (LSpan‘𝑈)
29	8, 25, 28	lspsncl 19249	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
30	26, 27, 29	syl2anc 579	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
31		hdmaprnlem1.un	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
32	24, 25, 26, 30, 10, 31	lssneln0 19222	. . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))
33	3, 7, 8, 24, 4, 22, 1, 9, 5, 32	hdmapnzcl 37801	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝑢) ∈ (𝐷 ∖ {𝑄}))
34		hdmaprnlem1.m	. . . . . . . 8 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
35		hdmaprnlem1.e	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
36	3, 7, 8, 28, 4, 2, 34, 9, 5, 12, 27, 35, 10, 31	hdmaprnlem1N 37805	. . . . . . 7 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠}))
37	1, 22, 2, 23, 33, 13, 36	lspsnne1 19389	. . . . . 6 ⊢ (𝜑 → ¬ (𝑆‘𝑢) ∈ (𝐿‘{𝑠}))
38	1, 14, 17, 6, 19, 21, 11, 37	lssvancl2 19215	. . . . 5 ⊢ (𝜑 → ¬ ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{𝑠}))
39	1, 2, 6, 16, 13, 38	lspsnne2 19390	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠}))
40	39	necomd 2992	. . 3 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
41	1, 17, 2	lspsncl 19249	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
42	6, 16, 41	syl2anc 579	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
43	3, 34, 4, 17, 5	mapdrn2 37607	. . . . 5 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
44	42, 43	eleqtrrd 2847	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
45	3, 34, 5, 44	mapdcnvid2 37613	. . 3 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
46	40, 35, 45	3netr4d 3014	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
47	3, 34, 7, 25, 5, 44	mapdcnvcl 37608	. . . 4 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ∈ (LSubSp‘𝑈))
48	3, 7, 25, 34, 5, 30, 47	mapd11 37595	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) = (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
49	48	necon3bid 2981	. 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
50	46, 49	mpbid 223	1 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1652 ∈ wcel 2155 ≠ wne 2937 ∖ cdif 3729 {csn 4334 ^◡ccnv 5276 ran crn 5278 ‘cfv 6068 (class class class)co 6842 Basecbs 16130 +_gcplusg 16214 0_gc0g 16366 LModclmod 19132 LSubSpclss 19201 LSpanclspn 19243 HLchlt 35306 LHypclh 35940 DVecHcdvh 37034 LCDualclcd 37542 mapdcmpd 37580 HDMapchdma 37748

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2070 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2352 ax-ext 2743 ax-rep 4930 ax-sep 4941 ax-nul 4949 ax-pow 5001 ax-pr 5062 ax-un 7147 ax-cnex 10245 ax-resscn 10246 ax-1cn 10247 ax-icn 10248 ax-addcl 10249 ax-addrcl 10250 ax-mulcl 10251 ax-mulrcl 10252 ax-mulcom 10253 ax-addass 10254 ax-mulass 10255 ax-distr 10256 ax-i2m1 10257 ax-1ne0 10258 ax-1rid 10259 ax-rnegex 10260 ax-rrecex 10261 ax-cnre 10262 ax-pre-lttri 10263 ax-pre-lttrn 10264 ax-pre-ltadd 10265 ax-pre-mulgt0 10266 ax-riotaBAD 34909

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-fal 1666 df-ex 1875 df-nf 1879 df-sb 2063 df-mo 2565 df-eu 2582 df-clab 2752 df-cleq 2758 df-clel 2761 df-nfc 2896 df-ne 2938 df-nel 3041 df-ral 3060 df-rex 3061 df-reu 3062 df-rmo 3063 df-rab 3064 df-v 3352 df-sbc 3597 df-csb 3692 df-dif 3735 df-un 3737 df-in 3739 df-ss 3746 df-pss 3748 df-nul 4080 df-if 4244 df-pw 4317 df-sn 4335 df-pr 4337 df-tp 4339 df-op 4341 df-ot 4343 df-uni 4595 df-int 4634 df-iun 4678 df-iin 4679 df-br 4810 df-opab 4872 df-mpt 4889 df-tr 4912 df-id 5185 df-eprel 5190 df-po 5198 df-so 5199 df-fr 5236 df-we 5238 df-xp 5283 df-rel 5284 df-cnv 5285 df-co 5286 df-dm 5287 df-rn 5288 df-res 5289 df-ima 5290 df-pred 5865 df-ord 5911 df-on 5912 df-lim 5913 df-suc 5914 df-iota 6031 df-fun 6070 df-fn 6071 df-f 6072 df-f1 6073 df-fo 6074 df-f1o 6075 df-fv 6076 df-riota 6803 df-ov 6845 df-oprab 6846 df-mpt2 6847 df-of 7095 df-om 7264 df-1st 7366 df-2nd 7367 df-tpos 7555 df-undef 7602 df-wrecs 7610 df-recs 7672 df-rdg 7710 df-1o 7764 df-oadd 7768 df-er 7947 df-map 8062 df-en 8161 df-dom 8162 df-sdom 8163 df-fin 8164 df-pnf 10330 df-mnf 10331 df-xr 10332 df-ltxr 10333 df-le 10334 df-sub 10522 df-neg 10523 df-nn 11275 df-2 11335 df-3 11336 df-4 11337 df-5 11338 df-6 11339 df-n0 11539 df-z 11625 df-uz 11887 df-fz 12534 df-struct 16132 df-ndx 16133 df-slot 16134 df-base 16136 df-sets 16137 df-ress 16138 df-plusg 16227 df-mulr 16228 df-sca 16230 df-vsca 16231 df-0g 16368 df-mre 16512 df-mrc 16513 df-acs 16515 df-proset 17194 df-poset 17212 df-plt 17224 df-lub 17240 df-glb 17241 df-join 17242 df-meet 17243 df-p0 17305 df-p1 17306 df-lat 17312 df-clat 17374 df-mgm 17508 df-sgrp 17550 df-mnd 17561 df-submnd 17602 df-grp 17692 df-minusg 17693 df-sbg 17694 df-subg 17855 df-cntz 18013 df-oppg 18039 df-lsm 18315 df-cmn 18461 df-abl 18462 df-mgp 18757 df-ur 18769 df-ring 18816 df-oppr 18890 df-dvdsr 18908 df-unit 18909 df-invr 18939 df-dvr 18950 df-drng 19018 df-lmod 19134 df-lss 19202 df-lsp 19244 df-lvec 19375 df-lsatoms 34932 df-lshyp 34933 df-lcv 34975 df-lfl 35014 df-lkr 35042 df-ldual 35080 df-oposet 35132 df-ol 35134 df-oml 35135 df-covers 35222 df-ats 35223 df-atl 35254 df-cvlat 35278 df-hlat 35307 df-llines 35454 df-lplanes 35455 df-lvols 35456 df-lines 35457 df-psubsp 35459 df-pmap 35460 df-padd 35752 df-lhyp 35944 df-laut 35945 df-ldil 36060 df-ltrn 36061 df-trl 36115 df-tgrp 36699 df-tendo 36711 df-edring 36713 df-dveca 36959 df-disoa 36985 df-dvech 37035 df-dib 37095 df-dic 37129 df-dih 37185 df-doch 37304 df-djh 37351 df-lcdual 37543 df-mapd 37581 df-hvmap 37713 df-hdmap1 37749 df-hdmap 37750

This theorem is referenced by: hdmaprnlem9N 37813 hdmaprnlem3eN 37814

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