hdmaprnlem3N - Mathbox for Norm Megill

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Theorem hdmaprnlem3N 42731

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 42473	. . . . 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 42711	. . . . . 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 21065	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
16	6, 11, 13, 15	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
17		eqid 2762	. . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
18	1, 17, 2	lspsncl 21167	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
19	6, 13, 18	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶))
20	1, 2	lspsnid 21183	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠}))
21	6, 13, 20	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐿‘{𝑠}))
22		hdmaprnlem1.q	. . . . . . 7 ⊢ 𝑄 = (0_g‘𝐶)
23	3, 4, 5	lcdlvec 42472	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ LVec)
24		hdmaprnlem1.o	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
25		eqid 2762	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
26	3, 7, 5	dvhlmod 41991	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
27		hdmaprnlem1.ve	. . . . . . . . . 10 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
28		hdmaprnlem1.n	. . . . . . . . . . 11 ⊢ 𝑁 = (LSpan‘𝑈)
29	8, 25, 28	lspsncl 21167	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
30	26, 27, 29	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
31		hdmaprnlem1.un	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
32	24, 25, 26, 30, 10, 31	lssneln0 21143	. . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))
33	3, 7, 8, 24, 4, 22, 1, 9, 5, 32	hdmapnzcl 42726	. . . . . . 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 42730	. . . . . . 7 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠}))
37	1, 22, 2, 23, 33, 13, 36	lspsnne1 21310	. . . . . 6 ⊢ (𝜑 → ¬ (𝑆‘𝑢) ∈ (𝐿‘{𝑠}))
38	1, 14, 17, 6, 19, 21, 11, 37	lssvancl2 21136	. . . . 5 ⊢ (𝜑 → ¬ ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{𝑠}))
39	1, 2, 6, 16, 13, 38	lspsnne2 21311	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠}))
40	39	necomd 3012	. . 3 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
41	1, 17, 2	lspsncl 21167	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
42	6, 16, 41	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
43	3, 34, 4, 17, 5	mapdrn2 42532	. . . . 5 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
44	42, 43	eleqtrrd 2865	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
45	3, 34, 5, 44	mapdcnvid2 42538	. . 3 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
46	40, 35, 45	3netr4d 3034	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
47	3, 34, 7, 25, 5, 44	mapdcnvcl 42533	. . . 4 ⊢ (𝜑 → (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ∈ (LSubSp‘𝑈))
48	3, 7, 25, 34, 5, 30, 47	mapd11 42520	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) = (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
49	48	necon3bid 3001	. 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))))
50	46, 49	mpbid 235	1 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (^◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∖ cdif 3899 {csn 4587 ^◡ccnv 5658 ran crn 5660 ‘cfv 6537 (class class class)co 7417 Basecbs 17307 +_gcplusg 17348 0_gc0g 17530 LModclmod 21050 LSubSpclss 21121 LSpanclspn 21161 HLchlt 40231 LHypclh 40865 DVecHcdvh 41959 LCDualclcd 42467 mapdcmpd 42505 HDMapchdma 42673

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-riotaBAD 39834

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8228 df-undef 8275 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-n0 12533 df-z 12620 df-uz 12892 df-fz 13566 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-sca 17364 df-vsca 17365 df-0g 17532 df-mre 17676 df-mrc 17677 df-acs 17679 df-proset 18388 df-poset 18407 df-plt 18422 df-lub 18438 df-glb 18439 df-join 18440 df-meet 18441 df-p0 18517 df-p1 18518 df-lat 18526 df-clat 18593 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-submnd 18898 df-grp 19066 df-minusg 19067 df-sbg 19068 df-subg 19252 df-cntz 19450 df-oppg 19479 df-lsm 19769 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-ring 20380 df-oppr 20484 df-dvdsr 20504 df-unit 20505 df-invr 20535 df-dvr 20548 df-nzr 20679 df-rlreg 20862 df-domn 20863 df-drng 20898 df-lmod 21052 df-lss 21122 df-lsp 21162 df-lvec 21293 df-lsatoms 39857 df-lshyp 39858 df-lcv 39900 df-lfl 39939 df-lkr 39967 df-ldual 40005 df-oposet 40057 df-ol 40059 df-oml 40060 df-covers 40147 df-ats 40148 df-atl 40179 df-cvlat 40203 df-hlat 40232 df-llines 40379 df-lplanes 40380 df-lvols 40381 df-lines 40382 df-psubsp 40384 df-pmap 40385 df-padd 40677 df-lhyp 40869 df-laut 40870 df-ldil 40985 df-ltrn 40986 df-trl 41040 df-tgrp 41624 df-tendo 41636 df-edring 41638 df-dveca 41884 df-disoa 41910 df-dvech 41960 df-dib 42020 df-dic 42054 df-dih 42110 df-doch 42229 df-djh 42276 df-lcdual 42468 df-mapd 42506 df-hvmap 42638 df-hdmap1 42674 df-hdmap 42675

This theorem is used by: hdmaprnlem9N 42738 hdmaprnlem3eN 42739

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