hdmaprnlem3N - Mathbox for Norm Megill

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Theorem hdmaprnlem3N 37663

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 37402	. . . . 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 37643	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
12		hdmaprnlem1.se	. . . . . . 7 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
13	12	eldifad 3728	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
14		hdmaprnlem1.a	. . . . . . 7 ⊢ ✚ = (+_g‘𝐶)
15	1, 14	lmodvacl 19100	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
16	6, 11, 13, 15	syl3anc 1477	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
17		eqid 2761	. . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
18	1, 17, 2	lspsncl 19200	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
19	6, 13, 18	syl2anc 696	. . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
20	1, 2	lspsnid 19216	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠}))
21	6, 13, 20	syl2anc 696	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐿‘{𝑠}))
22		hdmaprnlem1.q	. . . . . . 7 ⊢ 𝑄 = (0_g‘𝐶)
23	3, 4, 5	lcdlvec 37401	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ LVec)
24		hdmaprnlem1.o	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
25		eqid 2761	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
26	3, 7, 5	dvhlmod 36920	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
27		hdmaprnlem1.ve	. . . . . . . . . 10 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
28		hdmaprnlem1.n	. . . . . . . . . . 11 ⊢ 𝑁 = (LSpan‘𝑈)
29	8, 25, 28	lspsncl 19200	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
30	26, 27, 29	syl2anc 696	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
31		hdmaprnlem1.un	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
32	8, 24, 25, 26, 30, 10, 31	lssneln0 19175	. . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))
33	3, 7, 8, 24, 4, 22, 1, 9, 5, 32	hdmapnzcl 37658	. . . . . . 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 37662	. . . . . . 7 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠}))
37	1, 22, 2, 23, 33, 13, 36	lspsnne1 19340	. . . . . 6 ⊢ (𝜑 → ¬ (𝑆‘𝑢) ∈ (𝐿‘{𝑠}))
38	1, 14, 17, 6, 19, 21, 11, 37	lssvancl2 19169	. . . . 5 ⊢ (𝜑 → ¬ ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{𝑠}))
39	1, 2, 6, 16, 13, 38	lspsnne2 19341	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠}))
40	39	necomd 2988	. . 3 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
41	1, 17, 2	lspsncl 19200	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
42	6, 16, 41	syl2anc 696	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
43	3, 34, 4, 17, 5	mapdrn2 37461	. . . . 5 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
44	42, 43	eleqtrrd 2843	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
45	3, 34, 5, 44	mapdcnvid2 37467	. . 3 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
46	40, 35, 45	3netr4d 3010	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
47	3, 34, 7, 25, 5, 44	mapdcnvcl 37462	. . . 4 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ∈ (LSubSp‘𝑈))
48	3, 7, 25, 34, 5, 30, 47	mapd11 37449	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) = (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
49	48	necon3bid 2977	. 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
50	46, 49	mpbid 222	1 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2140 ≠ wne 2933 ∖ cdif 3713 {csn 4322 ^◡ccnv 5266 ran crn 5268 ‘cfv 6050 (class class class)co 6815 Basecbs 16080 +_gcplusg 16164 0_gc0g 16323 LModclmod 19086 LSubSpclss 19155 LSpanclspn 19194 HLchlt 35159 LHypclh 35792 DVecHcdvh 36888 LCDualclcd 37396 mapdcmpd 37434 HDMapchdma 37603

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-rep 4924 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226 ax-riotaBAD 34761

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-ot 4331 df-uni 4590 df-int 4629 df-iun 4675 df-iin 4676 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-of 7064 df-om 7233 df-1st 7335 df-2nd 7336 df-tpos 7523 df-undef 7570 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-oadd 7735 df-er 7914 df-map 8028 df-en 8125 df-dom 8126 df-sdom 8127 df-fin 8128 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-nn 11234 df-2 11292 df-3 11293 df-4 11294 df-5 11295 df-6 11296 df-n0 11506 df-z 11591 df-uz 11901 df-fz 12541 df-struct 16082 df-ndx 16083 df-slot 16084 df-base 16086 df-sets 16087 df-ress 16088 df-plusg 16177 df-mulr 16178 df-sca 16180 df-vsca 16181 df-0g 16325 df-mre 16469 df-mrc 16470 df-acs 16472 df-preset 17150 df-poset 17168 df-plt 17180 df-lub 17196 df-glb 17197 df-join 17198 df-meet 17199 df-p0 17261 df-p1 17262 df-lat 17268 df-clat 17330 df-mgm 17464 df-sgrp 17506 df-mnd 17517 df-submnd 17558 df-grp 17647 df-minusg 17648 df-sbg 17649 df-subg 17813 df-cntz 17971 df-oppg 17997 df-lsm 18272 df-cmn 18416 df-abl 18417 df-mgp 18711 df-ur 18723 df-ring 18770 df-oppr 18844 df-dvdsr 18862 df-unit 18863 df-invr 18893 df-dvr 18904 df-drng 18972 df-lmod 19088 df-lss 19156 df-lsp 19195 df-lvec 19326 df-lsatoms 34785 df-lshyp 34786 df-lcv 34828 df-lfl 34867 df-lkr 34895 df-ldual 34933 df-oposet 34985 df-ol 34987 df-oml 34988 df-covers 35075 df-ats 35076 df-atl 35107 df-cvlat 35131 df-hlat 35160 df-llines 35306 df-lplanes 35307 df-lvols 35308 df-lines 35309 df-psubsp 35311 df-pmap 35312 df-padd 35604 df-lhyp 35796 df-laut 35797 df-ldil 35912 df-ltrn 35913 df-trl 35968 df-tgrp 36552 df-tendo 36564 df-edring 36566 df-dveca 36812 df-disoa 36839 df-dvech 36889 df-dib 36949 df-dic 36983 df-dih 37039 df-doch 37158 df-djh 37205 df-lcdual 37397 df-mapd 37435 df-hvmap 37567 df-hdmap1 37604 df-hdmap 37605

This theorem is referenced by: hdmaprnlem9N 37670 hdmaprnlem3eN 37671

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