Theorem hdmap1valc 41751

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 41750 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 3936	. . 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 41746	. 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 41750	. . 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 41672	. 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 2772	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐿‘⟨𝑋, 𝐹, 𝑌⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3457   ∖ cdif 3921  ifcif 4498  {csn 4599  ⟨cotp 4607   ↦ cmpt 5199  ‘cfv 6528  ℩crio 7356  (class class class)co 7400  1^st c1st 7981  2^nd c2nd 7982  Basecbs 17215  0_gc0g 17440  -_gcsg 18905  LSpanclspn 20915  HLchlt 39297  LHypclh 39932  DVecHcdvh 41026  LCDualclcd 41534  mapdcmpd 41572  HDMap1chdma1 41739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-ot 4608  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-riota 7357  df-ov 7403  df-1st 7983  df-2nd 7984  df-hdmap1 41741

This theorem is referenced by:  hdmap1cl  41752  hdmap1eq2  41753  hdmap1eq4N  41754  hdmap1eulem  41770  hdmap1eulemOLDN  41771