Theorem hdmap1valc 42391

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 42390 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 3916	. . 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 42386	. 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 42390	. . 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 42312	. 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 2799	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐿‘⟨𝑋, 𝐹, 𝑌⟩))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∖ cdif 3901  ifcif 4479  {csn 4581  ⟨cotp 4589   ↦ cmpt 5180  ‘cfv 6517  ℩crio 7348  (class class class)co 7392  1^st c1st 7964  2^nd c2nd 7965  Basecbs 17228  0_gc0g 17451  -_gcsg 18960  LSpanclspn 21018  HLchlt 39938  LHypclh 40572  DVecHcdvh 41666  LCDualclcd 42174  mapdcmpd 42212  HDMap1chdma1 42379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-ot 4590  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-1st 7966  df-2nd 7967  df-hdmap1 42381

This theorem is referenced by:  hdmap1cl  42392  hdmap1eq2  42393  hdmap1eq4N  42394  hdmap1eulem  42410  hdmap1eulemOLDN  42411