hdmaprnlem3N - Mathbox for Norm Megill

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Theorem hdmaprnlem3N 38431

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 38173	. . . . 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 38411	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
12		hdmaprnlem1.se	. . . . . . 7 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
13	12	eldifad 3843	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
14		hdmaprnlem1.a	. . . . . . 7 ⊢ ✚ = (+_g‘𝐶)
15	1, 14	lmodvacl 19373	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
16	6, 11, 13, 15	syl3anc 1351	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
17		eqid 2778	. . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
18	1, 17, 2	lspsncl 19474	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
19	6, 13, 18	syl2anc 576	. . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
20	1, 2	lspsnid 19490	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠}))
21	6, 13, 20	syl2anc 576	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐿‘{𝑠}))
22		hdmaprnlem1.q	. . . . . . 7 ⊢ 𝑄 = (0_g‘𝐶)
23	3, 4, 5	lcdlvec 38172	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ LVec)
24		hdmaprnlem1.o	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
25		eqid 2778	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
26	3, 7, 5	dvhlmod 37691	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
27		hdmaprnlem1.ve	. . . . . . . . . 10 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
28		hdmaprnlem1.n	. . . . . . . . . . 11 ⊢ 𝑁 = (LSpan‘𝑈)
29	8, 25, 28	lspsncl 19474	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
30	26, 27, 29	syl2anc 576	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
31		hdmaprnlem1.un	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
32	24, 25, 26, 30, 10, 31	lssneln0 19449	. . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))
33	3, 7, 8, 24, 4, 22, 1, 9, 5, 32	hdmapnzcl 38426	. . . . . . 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 38430	. . . . . . 7 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠}))
37	1, 22, 2, 23, 33, 13, 36	lspsnne1 19614	. . . . . 6 ⊢ (𝜑 → ¬ (𝑆‘𝑢) ∈ (𝐿‘{𝑠}))
38	1, 14, 17, 6, 19, 21, 11, 37	lssvancl2 19442	. . . . 5 ⊢ (𝜑 → ¬ ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{𝑠}))
39	1, 2, 6, 16, 13, 38	lspsnne2 19615	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠}))
40	39	necomd 3022	. . 3 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
41	1, 17, 2	lspsncl 19474	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
42	6, 16, 41	syl2anc 576	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
43	3, 34, 4, 17, 5	mapdrn2 38232	. . . . 5 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
44	42, 43	eleqtrrd 2869	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
45	3, 34, 5, 44	mapdcnvid2 38238	. . 3 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
46	40, 35, 45	3netr4d 3044	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
47	3, 34, 7, 25, 5, 44	mapdcnvcl 38233	. . . 4 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ∈ (LSubSp‘𝑈))
48	3, 7, 25, 34, 5, 30, 47	mapd11 38220	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) = (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
49	48	necon3bid 3011	. 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
50	46, 49	mpbid 224	1 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ∖ cdif 3828 {csn 4442 ^◡ccnv 5407 ran crn 5409 ‘cfv 6190 (class class class)co 6978 Basecbs 16342 +_gcplusg 16424 0_gc0g 16572 LModclmod 19359 LSubSpclss 19428 LSpanclspn 19468 HLchlt 35931 LHypclh 36565 DVecHcdvh 37659 LCDualclcd 38167 mapdcmpd 38205 HDMapchdma 38373

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-riotaBAD 35534

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-ot 4451 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-om 7399 df-1st 7503 df-2nd 7504 df-tpos 7697 df-undef 7744 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-map 8210 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-n0 11711 df-z 11797 df-uz 12062 df-fz 12712 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-sca 16440 df-vsca 16441 df-0g 16574 df-mre 16718 df-mrc 16719 df-acs 16721 df-proset 17399 df-poset 17417 df-plt 17429 df-lub 17445 df-glb 17446 df-join 17447 df-meet 17448 df-p0 17510 df-p1 17511 df-lat 17517 df-clat 17579 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-grp 17897 df-minusg 17898 df-sbg 17899 df-subg 18063 df-cntz 18221 df-oppg 18248 df-lsm 18525 df-cmn 18671 df-abl 18672 df-mgp 18966 df-ur 18978 df-ring 19025 df-oppr 19099 df-dvdsr 19117 df-unit 19118 df-invr 19148 df-dvr 19159 df-drng 19230 df-lmod 19361 df-lss 19429 df-lsp 19469 df-lvec 19600 df-lsatoms 35557 df-lshyp 35558 df-lcv 35600 df-lfl 35639 df-lkr 35667 df-ldual 35705 df-oposet 35757 df-ol 35759 df-oml 35760 df-covers 35847 df-ats 35848 df-atl 35879 df-cvlat 35903 df-hlat 35932 df-llines 36079 df-lplanes 36080 df-lvols 36081 df-lines 36082 df-psubsp 36084 df-pmap 36085 df-padd 36377 df-lhyp 36569 df-laut 36570 df-ldil 36685 df-ltrn 36686 df-trl 36740 df-tgrp 37324 df-tendo 37336 df-edring 37338 df-dveca 37584 df-disoa 37610 df-dvech 37660 df-dib 37720 df-dic 37754 df-dih 37810 df-doch 37929 df-djh 37976 df-lcdual 38168 df-mapd 38206 df-hvmap 38338 df-hdmap1 38374 df-hdmap 38375

This theorem is referenced by: hdmaprnlem9N 38438 hdmaprnlem3eN 38439

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