Theorem hdmap1val 39812

Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 39738.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 39802 to shorten proofs with no $d on 𝑥. (Contributed by NM, 15-May-2015.)

Hypotheses

Ref	Expression
hdmap1val.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap1fval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1fval.v	⊢ 𝑉 = (Base‘𝑈)
hdmap1fval.s	⊢ − = (-g‘𝑈)
hdmap1fval.o	⊢ 0 = (0g‘𝑈)
hdmap1fval.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap1fval.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1fval.d	⊢ 𝐷 = (Base‘𝐶)
hdmap1fval.r	⊢ 𝑅 = (-g‘𝐶)
hdmap1fval.q	⊢ 𝑄 = (0g‘𝐶)
hdmap1fval.j	⊢ 𝐽 = (LSpan‘𝐶)
hdmap1fval.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1fval.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1fval.k	⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))
hdmap1val.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hdmap1val.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
hdmap1val.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)

Assertion

Ref	Expression
hdmap1val	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))))

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

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

Proof of Theorem hdmap1val

Step	Hyp	Ref	Expression
1		hdmap1val.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1fval.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1fval.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap1fval.s	. . 3 ⊢ − = (-g‘𝑈)
5		hdmap1fval.o	. . 3 ⊢ 0 = (0g‘𝑈)
6		hdmap1fval.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
7		hdmap1fval.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		hdmap1fval.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
9		hdmap1fval.r	. . 3 ⊢ 𝑅 = (-g‘𝐶)
10		hdmap1fval.q	. . 3 ⊢ 𝑄 = (0g‘𝐶)
11		hdmap1fval.j	. . 3 ⊢ 𝐽 = (LSpan‘𝐶)
12		hdmap1fval.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		hdmap1fval.i	. . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
14		hdmap1fval.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))
15		df-ot 4570	. . . 4 ⊢ ⟨𝑋, 𝐹, 𝑌⟩ = ⟨⟨𝑋, 𝐹⟩, 𝑌⟩
16		hdmap1val.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		hdmap1val.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
18		opelxp 5625	. . . . . 6 ⊢ (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ↔ (𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷))
19	16, 17, 18	sylanbrc 583	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷))
20		hdmap1val.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
21		opelxp 5625	. . . . 5 ⊢ (⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉) ↔ (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ∧ 𝑌 ∈ 𝑉))
22	19, 20, 21	sylanbrc 583	. . . 4 ⊢ (𝜑 → ⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉))
23	15, 22	eqeltrid 2843	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝐹, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉))
24	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23	hdmap1vallem 39811	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})))))
25		ot3rdg 7847	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
26	20, 25	syl 17	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
27	26	eqeq1d 2740	. . 3 ⊢ (𝜑 → ((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 ↔ 𝑌 = 0 ))
28	26	sneqd 4573	. . . . . . 7 ⊢ (𝜑 → {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)} = {𝑌})
29	28	fveq2d 6778	. . . . . 6 ⊢ (𝜑 → (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}) = (𝑁‘{𝑌}))
30	29	fveqeq2d 6782	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ})))
31		ot1stg 7845	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑌 ∈ 𝑉) → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
32	16, 17, 20, 31	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
33	32, 26	oveq12d 7293	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)) = (𝑋 − 𝑌))
34	33	sneqd 4573	. . . . . . . 8 ⊢ (𝜑 → {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))} = {(𝑋 − 𝑌)})
35	34	fveq2d 6778	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}) = (𝑁‘{(𝑋 − 𝑌)}))
36	35	fveq2d 6778	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝑀‘(𝑁‘{(𝑋 − 𝑌)})))
37		ot2ndg 7846	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑌 ∈ 𝑉) → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
38	16, 17, 20, 37	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
39	38	oveq1d 7290	. . . . . . . 8 ⊢ (𝜑 → ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ) = (𝐹𝑅ℎ))
40	39	sneqd 4573	. . . . . . 7 ⊢ (𝜑 → {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)} = {(𝐹𝑅ℎ)})
41	40	fveq2d 6778	. . . . . 6 ⊢ (𝜑 → (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}) = (𝐽‘{(𝐹𝑅ℎ)}))
42	36, 41	eqeq12d 2754	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))
43	30, 42	anbi12d 631	. . . 4 ⊢ (𝜑 → (((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})) ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))
44	43	riotabidv 7234	. . 3 ⊢ (𝜑 → (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))
45	27, 44	ifbieq2d 4485	. 2 ⊢ (𝜑 → if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)})))) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))))
46	24, 45	eqtrd 2778	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ifcif 4459  {csn 4561  ⟨cop 4567  ⟨cotp 4569   × cxp 5587  ‘cfv 6433  ℩crio 7231  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  Basecbs 16912  0_gc0g 17150  -_gcsg 18579  LSpanclspn 20233  LHypclh 37998  DVecHcdvh 39092  LCDualclcd 39600  mapdcmpd 39638  HDMap1chdma1 39805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-ot 4570  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-1st 7831  df-2nd 7832  df-hdmap1 39807

This theorem is referenced by:  hdmap1val0  39813  hdmap1val2  39814  hdmap1valc  39817