Theorem hdmap1valc 41841

Description: Connect the value of the preliminary map from vectors to functionals 𝐼 to the hypothesis 𝐿 used by earlier theorems. Note: the 𝑋 ∈ (𝑉 ∖ { 0 }) hypothesis could be the more general 𝑋 ∈ 𝑉 but the former will be easier to use. TODO: use the 𝐼 function directly in those theorems, so this theorem becomes unnecessary? TODO: The hdmap1cbv 41840 is probably unnecessary, but it would mean different $d's later on. (Contributed by NM, 15-May-2015.)

Hypotheses

Ref	Expression
hdmap1valc.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap1valc.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1valc.v	⊢ 𝑉 = (Base‘𝑈)
hdmap1valc.s	⊢ − = (-g‘𝑈)
hdmap1valc.o	⊢ 0 = (0g‘𝑈)
hdmap1valc.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap1valc.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1valc.d	⊢ 𝐷 = (Base‘𝐶)
hdmap1valc.r	⊢ 𝑅 = (-g‘𝐶)
hdmap1valc.q	⊢ 𝑄 = (0g‘𝐶)
hdmap1valc.j	⊢ 𝐽 = (LSpan‘𝐶)
hdmap1valc.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1valc.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1valc.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap1valc.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap1valc.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
hdmap1valc.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
hdmap1valc.l	⊢ 𝐿 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))

Assertion

Ref	Expression
hdmap1valc	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐿‘⟨𝑋, 𝐹, 𝑌⟩))

Distinct variable groups:   𝑥, 0   𝑥,ℎ,𝐷   ℎ,𝐽,𝑥   ℎ,𝑀,𝑥   − ,ℎ,𝑥   ℎ,𝑁,𝑥   𝑅,ℎ,𝑥   𝑥,𝑄

Allowed substitution hints:   𝜑(𝑥,ℎ)   𝐶(𝑥,ℎ)   𝑄(ℎ)   𝑈(𝑥,ℎ)   𝐹(𝑥,ℎ)   𝐻(𝑥,ℎ)   𝐼(𝑥,ℎ)   𝐾(𝑥,ℎ)   𝐿(𝑥,ℎ)   𝑉(𝑥,ℎ)   𝑊(𝑥,ℎ)   𝑋(𝑥,ℎ)   𝑌(𝑥,ℎ)   0 (ℎ)

Proof of Theorem hdmap1valc

Dummy variables 𝑤 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmap1valc.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1valc.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1valc.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap1valc.s	. . 3 ⊢ − = (-g‘𝑈)
5		hdmap1valc.o	. . 3 ⊢ 0 = (0g‘𝑈)
6		hdmap1valc.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
7		hdmap1valc.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		hdmap1valc.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
9		hdmap1valc.r	. . 3 ⊢ 𝑅 = (-g‘𝐶)
10		hdmap1valc.q	. . 3 ⊢ 𝑄 = (0g‘𝐶)
11		hdmap1valc.j	. . 3 ⊢ 𝐽 = (LSpan‘𝐶)
12		hdmap1valc.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		hdmap1valc.i	. . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
14		hdmap1valc.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		hdmap1valc.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
16	15	eldifad 3914	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		hdmap1valc.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
18		hdmap1valc.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18	hdmap1val 41836	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (℩𝑔 ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑔}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝑔)})))))
20		hdmap1valc.l	. . . 4 ⊢ 𝐿 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))
21	20	hdmap1cbv 41840	. . 3 ⊢ 𝐿 = (𝑤 ∈ V ↦ if((2nd ‘𝑤) = 0 , 𝑄, (℩𝑔 ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑤)})) = (𝐽‘{𝑔}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑤)) − (2nd ‘𝑤))})) = (𝐽‘{((2nd ‘(1st ‘𝑤))𝑅𝑔)})))))
22	10, 21, 16, 17, 18	mapdhval 41762	. 2 ⊢ (𝜑 → (𝐿‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (℩𝑔 ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑔}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝑔)})))))
23	19, 22	eqtr4d 2769	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐿‘⟨𝑋, 𝐹, 𝑌⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3899  ifcif 4475  {csn 4576  ⟨cotp 4584   ↦ cmpt 5172  ‘cfv 6481  ℩crio 7302  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17117  0_gc0g 17340  -_gcsg 18845  LSpanclspn 20902  HLchlt 39388  LHypclh 40022  DVecHcdvh 41116  LCDualclcd 41624  mapdcmpd 41662  HDMap1chdma1 41829

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-ot 4585  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-1st 7921  df-2nd 7922  df-hdmap1 41831

This theorem is referenced by:  hdmap1cl  41842  hdmap1eq2  41843  hdmap1eq4N  41844  hdmap1eulem  41860  hdmap1eulemOLDN  41861