Theorem hdmap1val2 42384

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 2761	. . 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 3914	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18	hdmap1val 42382	. 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 4747	. . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
21	20	neneqd 2961	. . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
22		iffalse 4486	. . 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 2796	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 399   = wceq 1559   ∈ wcel 2141   ∖ cdif 3899  ifcif 4477  {csn 4579  ⟨cotp 4587  ‘cfv 6515  ℩crio 7346  (class class class)co 7390  Basecbs 17235  0_gc0g 17458  -_gcsg 18967  LSpanclspn 21025  HLchlt 39934  LHypclh 40568  DVecHcdvh 41662  LCDualclcd 42170  mapdcmpd 42208  HDMap1chdma1 42375

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 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

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 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-ot 4588  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-1st 7964  df-2nd 7965  df-hdmap1 42377

This theorem is referenced by:  hdmap1eq  42385