hdmaprnlem3N - Mathbox for Norm Megill

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Theorem hdmaprnlem3N 41895

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 41637	. . . . 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 41875	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
12		hdmaprnlem1.se	. . . . . . 7 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
13	12	eldifad 3914	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
14		hdmaprnlem1.a	. . . . . . 7 ⊢ ✚ = (+_g‘𝐶)
15	1, 14	lmodvacl 20809	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
16	6, 11, 13, 15	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
17		eqid 2731	. . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
18	1, 17, 2	lspsncl 20911	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
19	6, 13, 18	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
20	1, 2	lspsnid 20927	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠}))
21	6, 13, 20	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐿‘{𝑠}))
22		hdmaprnlem1.q	. . . . . . 7 ⊢ 𝑄 = (0_g‘𝐶)
23	3, 4, 5	lcdlvec 41636	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ LVec)
24		hdmaprnlem1.o	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
25		eqid 2731	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
26	3, 7, 5	dvhlmod 41155	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
27		hdmaprnlem1.ve	. . . . . . . . . 10 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
28		hdmaprnlem1.n	. . . . . . . . . . 11 ⊢ 𝑁 = (LSpan‘𝑈)
29	8, 25, 28	lspsncl 20911	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
30	26, 27, 29	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
31		hdmaprnlem1.un	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
32	24, 25, 26, 30, 10, 31	lssneln0 20887	. . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))
33	3, 7, 8, 24, 4, 22, 1, 9, 5, 32	hdmapnzcl 41890	. . . . . . 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 41894	. . . . . . 7 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠}))
37	1, 22, 2, 23, 33, 13, 36	lspsnne1 21055	. . . . . 6 ⊢ (𝜑 → ¬ (𝑆‘𝑢) ∈ (𝐿‘{𝑠}))
38	1, 14, 17, 6, 19, 21, 11, 37	lssvancl2 20880	. . . . 5 ⊢ (𝜑 → ¬ ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{𝑠}))
39	1, 2, 6, 16, 13, 38	lspsnne2 21056	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠}))
40	39	necomd 2983	. . 3 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
41	1, 17, 2	lspsncl 20911	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
42	6, 16, 41	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
43	3, 34, 4, 17, 5	mapdrn2 41696	. . . . 5 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
44	42, 43	eleqtrrd 2834	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
45	3, 34, 5, 44	mapdcnvid2 41702	. . 3 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
46	40, 35, 45	3netr4d 3005	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
47	3, 34, 7, 25, 5, 44	mapdcnvcl 41697	. . . 4 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ∈ (LSubSp‘𝑈))
48	3, 7, 25, 34, 5, 30, 47	mapd11 41684	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) = (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
49	48	necon3bid 2972	. 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
50	46, 49	mpbid 232	1 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3899 {csn 4576 ^◡ccnv 5615 ran crn 5617 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 LModclmod 20794 LSubSpclss 20865 LSpanclspn 20905 HLchlt 39395 LHypclh 40029 DVecHcdvh 41123 LCDualclcd 41631 mapdcmpd 41669 HDMapchdma 41837

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-riotaBAD 38998

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-ot 4585 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-cntz 19230 df-oppg 19259 df-lsm 19549 df-cmn 19695 df-abl 19696 df-mgp 20060 df-rng 20072 df-ur 20101 df-ring 20154 df-oppr 20256 df-dvdsr 20276 df-unit 20277 df-invr 20307 df-dvr 20320 df-nzr 20429 df-rlreg 20610 df-domn 20611 df-drng 20647 df-lmod 20796 df-lss 20866 df-lsp 20906 df-lvec 21038 df-lsatoms 39021 df-lshyp 39022 df-lcv 39064 df-lfl 39103 df-lkr 39131 df-ldual 39169 df-oposet 39221 df-ol 39223 df-oml 39224 df-covers 39311 df-ats 39312 df-atl 39343 df-cvlat 39367 df-hlat 39396 df-llines 39543 df-lplanes 39544 df-lvols 39545 df-lines 39546 df-psubsp 39548 df-pmap 39549 df-padd 39841 df-lhyp 40033 df-laut 40034 df-ldil 40149 df-ltrn 40150 df-trl 40204 df-tgrp 40788 df-tendo 40800 df-edring 40802 df-dveca 41048 df-disoa 41074 df-dvech 41124 df-dib 41184 df-dic 41218 df-dih 41274 df-doch 41393 df-djh 41440 df-lcdual 41632 df-mapd 41670 df-hvmap 41802 df-hdmap1 41838 df-hdmap 41839

This theorem is referenced by: hdmaprnlem9N 41902 hdmaprnlem3eN 41903

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