mapdh7fN - Mathbox for Norm Megill

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Theorem mapdh7fN 40027

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 40026	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑣, 𝐺, 𝑤⟩) = 𝐸)
25	18	eldifad 3910	. . . . 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 40003	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) ∈ 𝐷)
27	22, 26	eqeltrrd 2838	. . 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 40004	. . . . 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 495	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐽‘{𝐺}))
31	19	eldifad 3910	. . . . 5 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
32	1, 2, 14	dvhlvec 39385	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
33	17	eldifad 3910	. . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
34	3, 6, 32, 31, 33, 25, 21	lspindpi 20500	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑢}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑣})))
35	34	simpld 495	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑢}))
36	35	necomd 2996	. . . . 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 40003	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) ∈ 𝐷)
38	23, 37	eqeltrrd 2838	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐷)
39	34	simprd 496	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑣}))
40	39	necomd 2996	. . 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 40005	. 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 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 Vcvv 3441 ∖ cdif 3895 ifcif 4473 {csn 4573 {cpr 4575 ⟨cotp 4581 ↦ cmpt 5175 ‘cfv 6479 ℩crio 7292 (class class class)co 7337 1^st c1st 7897 2^nd c2nd 7898 Basecbs 17009 0_gc0g 17247 -_gcsg 18675 LSpanclspn 20339 HLchlt 37625 LHypclh 38260 DVecHcdvh 39354 LCDualclcd 39862 mapdcmpd 39900

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-riotaBAD 37228

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-ot 4582 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-of 7595 df-om 7781 df-1st 7899 df-2nd 7900 df-tpos 8112 df-undef 8159 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-n0 12335 df-z 12421 df-uz 12684 df-fz 13341 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-sca 17075 df-vsca 17076 df-0g 17249 df-mre 17392 df-mrc 17393 df-acs 17395 df-proset 18110 df-poset 18128 df-plt 18145 df-lub 18161 df-glb 18162 df-join 18163 df-meet 18164 df-p0 18240 df-p1 18241 df-lat 18247 df-clat 18314 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-submnd 18528 df-grp 18676 df-minusg 18677 df-sbg 18678 df-subg 18848 df-cntz 19019 df-oppg 19046 df-lsm 19337 df-cmn 19483 df-abl 19484 df-mgp 19816 df-ur 19833 df-ring 19880 df-oppr 19957 df-dvdsr 19978 df-unit 19979 df-invr 20009 df-dvr 20020 df-drng 20095 df-lmod 20231 df-lss 20300 df-lsp 20340 df-lvec 20471 df-lsatoms 37251 df-lshyp 37252 df-lcv 37294 df-lfl 37333 df-lkr 37361 df-ldual 37399 df-oposet 37451 df-ol 37453 df-oml 37454 df-covers 37541 df-ats 37542 df-atl 37573 df-cvlat 37597 df-hlat 37626 df-llines 37774 df-lplanes 37775 df-lvols 37776 df-lines 37777 df-psubsp 37779 df-pmap 37780 df-padd 38072 df-lhyp 38264 df-laut 38265 df-ldil 38380 df-ltrn 38381 df-trl 38435 df-tgrp 39019 df-tendo 39031 df-edring 39033 df-dveca 39279 df-disoa 39305 df-dvech 39355 df-dib 39415 df-dic 39449 df-dih 39505 df-doch 39624 df-djh 39671 df-lcdual 39863 df-mapd 39901

This theorem is referenced by: (None)

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