mapdh7fN - Mathbox for Norm Megill

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Theorem mapdh7fN 39451

Description: Part (7) of [Baer] p. 48 line 10 (6 of 6 cases). (Contributed by NM, 2-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
mapdh7.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdh7.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdh7.v	⊢ 𝑉 = (Base‘𝑈)
mapdh7.s	⊢ − = (-_g‘𝑈)
mapdh7.o	⊢ 0 = (0_g‘𝑈)
mapdh7.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdh7.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdh7.d	⊢ 𝐷 = (Base‘𝐶)
mapdh7.r	⊢ 𝑅 = (-_g‘𝐶)
mapdh7.q	⊢ 𝑄 = (0_g‘𝐶)
mapdh7.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdh7.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdh7.i	⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
mapdh7.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdh7.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh7.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹}))
mapdh7.x	⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))
mapdh7.y	⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 }))
mapdh7.z	⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
mapdh7.ne	⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣}))
mapdh7.wn	⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣}))
mapdh7a	⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺)
mapdh7.b	⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) = 𝐸)

**Assertion**
Ref	Expression
mapdh7fN	⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑣⟩) = 𝐺)

Distinct variable groups: 𝑥,ℎ, − 𝐶,ℎ 𝐷,ℎ,𝑥 ℎ,𝐸,𝑥 ℎ,𝐹,𝑥 ℎ,𝐺,𝑥 0 ,ℎ,𝑥 ℎ,𝐽,𝑥 ℎ,𝑀,𝑥 ℎ,𝑁,𝑥 𝜑,ℎ 𝑥,𝑄 𝑢,ℎ,𝑣,𝑤,𝑥 𝑅,ℎ,𝑥 𝑈,ℎ Allowed substitution hints: 𝜑(𝑥,𝑤,𝑣,𝑢) 𝐶(𝑥,𝑤,𝑣,𝑢) 𝐷(𝑤,𝑣,𝑢) 𝑄(𝑤,𝑣,𝑢,ℎ) 𝑅(𝑤,𝑣,𝑢) 𝑈(𝑥,𝑤,𝑣,𝑢) 𝐸(𝑤,𝑣,𝑢) 𝐹(𝑤,𝑣,𝑢) 𝐺(𝑤,𝑣,𝑢) 𝐻(𝑥,𝑤,𝑣,𝑢,ℎ) 𝐼(𝑥,𝑤,𝑣,𝑢,ℎ) 𝐽(𝑤,𝑣,𝑢) 𝐾(𝑥,𝑤,𝑣,𝑢,ℎ) 𝑀(𝑤,𝑣,𝑢) − (𝑤,𝑣,𝑢) 𝑁(𝑤,𝑣,𝑢) 𝑉(𝑥,𝑤,𝑣,𝑢,ℎ) 𝑊(𝑥,𝑤,𝑣,𝑢,ℎ) 0 (𝑤,𝑣,𝑢)

**Proof of Theorem mapdh7fN**
Step	Hyp	Ref	Expression
1		mapdh7.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdh7.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdh7.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		mapdh7.s	. . 3 ⊢ − = (-_g‘𝑈)
5		mapdh7.o	. . 3 ⊢ 0 = (0_g‘𝑈)
6		mapdh7.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
7		mapdh7.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		mapdh7.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
9		mapdh7.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
10		mapdh7.q	. . 3 ⊢ 𝑄 = (0_g‘𝐶)
11		mapdh7.j	. . 3 ⊢ 𝐽 = (LSpan‘𝐶)
12		mapdh7.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		mapdh7.i	. . 3 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
14		mapdh7.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		mapdh7.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
16		mapdh7.mn	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹}))
17		mapdh7.x	. . 3 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))
18		mapdh7.y	. . 3 ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 }))
19		mapdh7.z	. . 3 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
20		mapdh7.ne	. . 3 ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣}))
21		mapdh7.wn	. . 3 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣}))
22		mapdh7a	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺)
23		mapdh7.b	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) = 𝐸)
24	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23	mapdh7dN 39450	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑣, 𝐺, 𝑤⟩) = 𝐸)
25	18	eldifad 3865	. . . . 5 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
26	10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 15, 16, 17, 25, 20	mapdhcl 39427	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) ∈ 𝐷)
27	22, 26	eqeltrrd 2832	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
28	10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 15, 16, 17, 18, 27, 20	mapdheq 39428	. . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑣})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑢 − 𝑣)})) = (𝐽‘{(𝐹𝑅𝐺)}))))
29	22, 28	mpbid 235	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑢 − 𝑣)})) = (𝐽‘{(𝐹𝑅𝐺)})))
30	29	simpld 498	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐽‘{𝐺}))
31	19	eldifad 3865	. . . . 5 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
32	1, 2, 14	dvhlvec 38809	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
33	17	eldifad 3865	. . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
34	3, 6, 32, 31, 33, 25, 21	lspindpi 20123	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑢}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑣})))
35	34	simpld 498	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑢}))
36	35	necomd 2987	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑤}))
37	10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 15, 16, 17, 31, 36	mapdhcl 39427	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) ∈ 𝐷)
38	23, 37	eqeltrrd 2832	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐷)
39	34	simprd 499	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑣}))
40	39	necomd 2987	. . 3 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (𝑁‘{𝑤}))
41	10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 27, 30, 18, 19, 38, 40	mapdheq2 39429	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑣, 𝐺, 𝑤⟩) = 𝐸 → (𝐼‘⟨𝑤, 𝐸, 𝑣⟩) = 𝐺))
42	24, 41	mpd 15	1 ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑣⟩) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 Vcvv 3398 ∖ cdif 3850 ifcif 4425 {csn 4527 {cpr 4529 ⟨cotp 4535 ↦ cmpt 5120 ‘cfv 6358 ℩crio 7147 (class class class)co 7191 1^st c1st 7737 2^nd c2nd 7738 Basecbs 16666 0_gc0g 16898 -_gcsg 18321 LSpanclspn 19962 HLchlt 37050 LHypclh 37684 DVecHcdvh 38778 LCDualclcd 39286 mapdcmpd 39324

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-riotaBAD 36653

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-ot 4536 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-tpos 7946 df-undef 7993 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-0g 16900 df-mre 17043 df-mrc 17044 df-acs 17046 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-p1 17886 df-lat 17892 df-clat 17959 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-subg 18494 df-cntz 18665 df-oppg 18692 df-lsm 18979 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-invr 19644 df-dvr 19655 df-drng 19723 df-lmod 19855 df-lss 19923 df-lsp 19963 df-lvec 20094 df-lsatoms 36676 df-lshyp 36677 df-lcv 36719 df-lfl 36758 df-lkr 36786 df-ldual 36824 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 df-llines 37198 df-lplanes 37199 df-lvols 37200 df-lines 37201 df-psubsp 37203 df-pmap 37204 df-padd 37496 df-lhyp 37688 df-laut 37689 df-ldil 37804 df-ltrn 37805 df-trl 37859 df-tgrp 38443 df-tendo 38455 df-edring 38457 df-dveca 38703 df-disoa 38729 df-dvech 38779 df-dib 38839 df-dic 38873 df-dih 38929 df-doch 39048 df-djh 39095 df-lcdual 39287 df-mapd 39325

This theorem is referenced by: (None)

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