mapdh7dN - Mathbox for Norm Megill

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Theorem mapdh7dN 42626

Description: Part (7) of [Baer] p. 48 line 10 (4 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
mapdh7dN	⊢ (𝜑 → (𝐼‘⟨𝑣, 𝐺, 𝑤⟩) = 𝐸)

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 mapdh7dN**
Step	Hyp	Ref	Expression
1		mapdh7.q	. 2 ⊢ 𝑄 = (0_g‘𝐶)
2		mapdh7.i	. 2 ⊢ 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)})))))
3		mapdh7.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
4		mapdh7.m	. 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
5		mapdh7.u	. 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		mapdh7.v	. 2 ⊢ 𝑉 = (Base‘𝑈)
7		mapdh7.s	. 2 ⊢ − = (-_g‘𝑈)
8		mapdh7.o	. 2 ⊢ 0 = (0_g‘𝑈)
9		mapdh7.n	. 2 ⊢ 𝑁 = (LSpan‘𝑈)
10		mapdh7.c	. 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
11		mapdh7.d	. 2 ⊢ 𝐷 = (Base‘𝐶)
12		mapdh7.r	. 2 ⊢ 𝑅 = (-_g‘𝐶)
13		mapdh7.j	. 2 ⊢ 𝐽 = (LSpan‘𝐶)
14		mapdh7.k	. 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		mapdh7.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
16		mapdh7.mn	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹}))
17		mapdh7.x	. 2 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))
18		mapdh7.y	. 2 ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 }))
19		mapdh7.z	. 2 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
20	3, 5, 14	dvhlvec 41985	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
21	18	eldifad 3911	. . . . 5 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
22	19	eldifad 3911	. . . . 5 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
23		mapdh7.ne	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣}))
24		mapdh7.wn	. . . . 5 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣}))
25	6, 8, 9, 20, 17, 21, 22, 23, 24	lspindp1 21323	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑣}) ∧ ¬ 𝑢 ∈ (𝑁‘{𝑤, 𝑣})))
26	25	simprd 501	. . 3 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑤, 𝑣}))
27		prcom 4693	. . . . 5 ⊢ {𝑣, 𝑤} = {𝑤, 𝑣}
28	27	fveq2i 6882	. . . 4 ⊢ (𝑁‘{𝑣, 𝑤}) = (𝑁‘{𝑤, 𝑣})
29	28	eleq2i 2852	. . 3 ⊢ (𝑢 ∈ (𝑁‘{𝑣, 𝑤}) ↔ 𝑢 ∈ (𝑁‘{𝑤, 𝑣}))
30	26, 29	sylnibr 332	. 2 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣, 𝑤}))
31	17	eldifad 3911	. . . . 5 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
32	6, 9, 20, 22, 31, 21, 24	lspindpi 21322	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑢}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑣})))
33	32	simprd 501	. . 3 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑣}))
34	33	necomd 3010	. 2 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (𝑁‘{𝑤}))
35		mapdh7a	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺)
36		mapdh7.b	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) = 𝐸)
37	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 34, 35, 36	mapdheq4 42608	1 ⊢ (𝜑 → (𝐼‘⟨𝑣, 𝐺, 𝑤⟩) = 𝐸)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 Vcvv 3450 ∖ cdif 3896 ifcif 4482 {csn 4584 {cpr 4586 ⟨cotp 4592 ↦ cmpt 5186 ‘cfv 6533 ℩crio 7370 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 Basecbs 17304 0_gc0g 17527 -_gcsg 19062 LSpanclspn 21158 HLchlt 40226 LHypclh 40860 DVecHcdvh 41954 LCDualclcd 42462 mapdcmpd 42500

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 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-riotaBAD 39829

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12532 df-z 12619 df-uz 12891 df-fz 13565 df-struct 17242 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-sca 17361 df-vsca 17362 df-0g 17529 df-mre 17673 df-mrc 17674 df-acs 17676 df-proset 18385 df-poset 18404 df-plt 18419 df-lub 18435 df-glb 18436 df-join 18437 df-meet 18438 df-p0 18514 df-p1 18515 df-lat 18523 df-clat 18590 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-submnd 18895 df-grp 19063 df-minusg 19064 df-sbg 19065 df-subg 19249 df-cntz 19447 df-oppg 19476 df-lsm 19766 df-cmn 19912 df-abl 19913 df-mgp 20277 df-rng 20291 df-ur 20324 df-ring 20377 df-oppr 20481 df-dvdsr 20501 df-unit 20502 df-invr 20532 df-dvr 20545 df-nzr 20676 df-rlreg 20859 df-domn 20860 df-drng 20895 df-lmod 21049 df-lss 21119 df-lsp 21159 df-lvec 21290 df-lsatoms 39852 df-lshyp 39853 df-lcv 39895 df-lfl 39934 df-lkr 39962 df-ldual 40000 df-oposet 40052 df-ol 40054 df-oml 40055 df-covers 40142 df-ats 40143 df-atl 40174 df-cvlat 40198 df-hlat 40227 df-llines 40374 df-lplanes 40375 df-lvols 40376 df-lines 40377 df-psubsp 40379 df-pmap 40380 df-padd 40672 df-lhyp 40864 df-laut 40865 df-ldil 40980 df-ltrn 40981 df-trl 41035 df-tgrp 41619 df-tendo 41631 df-edring 41633 df-dveca 41879 df-disoa 41905 df-dvech 41955 df-dib 42015 df-dic 42049 df-dih 42105 df-doch 42224 df-djh 42271 df-lcdual 42463 df-mapd 42501

This theorem is used by: mapdh7fN 42627

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