Theorem hdmap1val2 42260

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 2737	. . 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 3902	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18	hdmap1val 42258	. 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 4734	. . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
21	20	neneqd 2938	. . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
22		iffalse 4476	. . 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 2772	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 1542   ∈ wcel 2114   ∖ cdif 3887  ifcif 4467  {csn 4568  ⟨cotp 4576  ‘cfv 6492  ℩crio 7316  (class class class)co 7360  Basecbs 17170  0_gc0g 17393  -_gcsg 18902  LSpanclspn 20957  HLchlt 39810  LHypclh 40444  DVecHcdvh 41538  LCDualclcd 42046  mapdcmpd 42084  HDMap1chdma1 42251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-ot 4577  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-1st 7935  df-2nd 7936  df-hdmap1 42253

This theorem is referenced by:  hdmap1eq  42261