Theorem hdmap1val2 36108

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 2610	. . 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 3552	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18	hdmap1val 36106	. 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 4261	. . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
21	20	neneqd 2787	. . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
22		iffalse 4045	. . 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 2644	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 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ifcif 4036  {csn 4125  ⟨cotp 4133  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  0_gc0g 15923  -_gcsg 17247  LSpanclspn 18792  HLchlt 33655  LHypclh 34288  DVecHcdvh 35385  LCDualclcd 35893  mapdcmpd 35931  HDMap1chdma1 36099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-1st 7059  df-2nd 7060  df-hdmap1 36101

This theorem is referenced by:  hdmap1eq  36109