Theorem hdmap1val2 42246

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 2736	. . 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 3901	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18	hdmap1val 42244	. 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 4735	. . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
21	20	neneqd 2937	. . 3 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
22		iffalse 4475	. . 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 2771	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 3886  ifcif 4466  {csn 4567  ⟨cotp 4575  ‘cfv 6498  ℩crio 7323  (class class class)co 7367  Basecbs 17179  0_gc0g 17402  -_gcsg 18911  LSpanclspn 20966  HLchlt 39796  LHypclh 40430  DVecHcdvh 41524  LCDualclcd 42032  mapdcmpd 42070  HDMap1chdma1 42237

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-ot 4576  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-1st 7942  df-2nd 7943  df-hdmap1 42239

This theorem is referenced by:  hdmap1eq  42247