mapdh7fN - Mathbox for Norm Megill

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Theorem mapdh7fN 41463

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 41462	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑣, 𝐺, 𝑤⟩) = 𝐸)
25	18	eldifad 3958	. . . . 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 41439	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) ∈ 𝐷)
27	22, 26	eqeltrrd 2827	. . 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 41440	. . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑣})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑢 − 𝑣)})) = (𝐽‘{(𝐹𝑅𝐺)}))))
29	22, 28	mpbid 231	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑢 − 𝑣)})) = (𝐽‘{(𝐹𝑅𝐺)})))
30	29	simpld 493	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐽‘{𝐺}))
31	19	eldifad 3958	. . . . 5 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
32	1, 2, 14	dvhlvec 40821	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
33	17	eldifad 3958	. . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
34	3, 6, 32, 31, 33, 25, 21	lspindpi 21109	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑢}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑣})))
35	34	simpld 493	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑢}))
36	35	necomd 2986	. . . . 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 41439	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) ∈ 𝐷)
38	23, 37	eqeltrrd 2827	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐷)
39	34	simprd 494	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑣}))
40	39	necomd 2986	. . 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 41441	. 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 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 ∖ cdif 3943 ifcif 4523 {csn 4623 {cpr 4625 ⟨cotp 4631 ↦ cmpt 5228 ‘cfv 6546 ℩crio 7371 (class class class)co 7416 1^st c1st 7993 2^nd c2nd 7994 Basecbs 17208 0_gc0g 17449 -_gcsg 18925 LSpanclspn 20944 HLchlt 39061 LHypclh 39696 DVecHcdvh 40790 LCDualclcd 41298 mapdcmpd 41336

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-riotaBAD 38664

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-tpos 8233 df-undef 8280 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-0g 17451 df-mre 17594 df-mrc 17595 df-acs 17597 df-proset 18315 df-poset 18333 df-plt 18350 df-lub 18366 df-glb 18367 df-join 18368 df-meet 18369 df-p0 18445 df-p1 18446 df-lat 18452 df-clat 18519 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19113 df-cntz 19307 df-oppg 19336 df-lsm 19630 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20312 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-nzr 20491 df-rlreg 20668 df-domn 20669 df-drng 20705 df-lmod 20834 df-lss 20905 df-lsp 20945 df-lvec 21077 df-lsatoms 38687 df-lshyp 38688 df-lcv 38730 df-lfl 38769 df-lkr 38797 df-ldual 38835 df-oposet 38887 df-ol 38889 df-oml 38890 df-covers 38977 df-ats 38978 df-atl 39009 df-cvlat 39033 df-hlat 39062 df-llines 39210 df-lplanes 39211 df-lvols 39212 df-lines 39213 df-psubsp 39215 df-pmap 39216 df-padd 39508 df-lhyp 39700 df-laut 39701 df-ldil 39816 df-ltrn 39817 df-trl 39871 df-tgrp 40455 df-tendo 40467 df-edring 40469 df-dveca 40715 df-disoa 40741 df-dvech 40791 df-dib 40851 df-dic 40885 df-dih 40941 df-doch 41060 df-djh 41107 df-lcdual 41299 df-mapd 41337

This theorem is referenced by: (None)

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