Theorem hdmap1val2 39741

Description: Value of preliminary map from vectors to functionals in the closed kernel dual space, for nonzero 𝑌. (Contributed by NM, 16-May-2015.)

Hypotheses

Ref	Expression
hdmap1val2.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap1val2.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1val2.v	⊢ 𝑉 = (Base‘𝑈)
hdmap1val2.s	⊢ − = (-g‘𝑈)
hdmap1val2.o	⊢ 0 = (0g‘𝑈)
hdmap1val2.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap1val2.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1val2.d	⊢ 𝐷 = (Base‘𝐶)
hdmap1val2.r	⊢ 𝑅 = (-g‘𝐶)
hdmap1val2.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap1val2.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1val2.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1val2.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap1val2.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hdmap1val2.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
hdmap1val2.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))

Assertion

Ref	Expression
hdmap1val2	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)}))))

Distinct variable groups:   𝐶,ℎ   𝐷,ℎ   ℎ,𝐹   ℎ,𝐿   ℎ,𝑀   ℎ,𝑁   𝑈,ℎ   ℎ,𝑉   ℎ,𝑋   ℎ,𝑌   𝜑,ℎ

Allowed substitution hints:   𝑅(ℎ)   𝐻(ℎ)   𝐼(ℎ)   𝐾(ℎ)   − (ℎ)   𝑊(ℎ)   0 (ℎ)

Proof of Theorem hdmap1val2

Step	Hyp	Ref	Expression
1		hdmap1val2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1val2.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1val2.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap1val2.s	. . 3 ⊢ − = (-g‘𝑈)
5		hdmap1val2.o	. . 3 ⊢ 0 = (0g‘𝑈)
6		hdmap1val2.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
7		hdmap1val2.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		hdmap1val2.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
9		hdmap1val2.r	. . 3 ⊢ 𝑅 = (-g‘𝐶)
10		eqid 2738	. . 3 ⊢ (0g‘𝐶) = (0g‘𝐶)
11		hdmap1val2.l	. . 3 ⊢ 𝐿 = (LSpan‘𝐶)
12		hdmap1val2.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		hdmap1val2.i	. . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
14		hdmap1val2.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		hdmap1val2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
16		hdmap1val2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
17		hdmap1val2.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
18	17	eldifad 3895	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18	hdmap1val 39739	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)})))))
20		eldifsni 4720	. . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
21	20	neneqd 2947	. . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
22		iffalse 4465	. . 3 ⊢ (¬ 𝑌 = 0 → if(𝑌 = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)})))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)}))))
23	17, 21, 22	3syl 18	. 2 ⊢ (𝜑 → if(𝑌 = 0 , (0g‘𝐶), (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)})))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)}))))
24	19, 23	eqtrd 2778	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)}))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880  ifcif 4456  {csn 4558  ⟨cotp 4566  ‘cfv 6418  ℩crio 7211  (class class class)co 7255  Basecbs 16840  0_gc0g 17067  -_gcsg 18494  LSpanclspn 20148  HLchlt 37291  LHypclh 37925  DVecHcdvh 39019  LCDualclcd 39527  mapdcmpd 39565  HDMap1chdma1 39732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-ot 4567  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-1st 7804  df-2nd 7805  df-hdmap1 39734

This theorem is referenced by:  hdmap1eq  39742