Theorem hdmap1val 41755

Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 41681.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 41745 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 4657	. . . 4 ⊢ ⟨𝑋, 𝐹, 𝑌⟩ = ⟨⟨𝑋, 𝐹⟩, 𝑌⟩
16		hdmap1val.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		hdmap1val.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
18		opelxp 5736	. . . . . 6 ⊢ (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ↔ (𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷))
19	16, 17, 18	sylanbrc 582	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷))
20		hdmap1val.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
21		opelxp 5736	. . . . 5 ⊢ (⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉) ↔ (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ∧ 𝑌 ∈ 𝑉))
22	19, 20, 21	sylanbrc 582	. . . 4 ⊢ (𝜑 → ⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉))
23	15, 22	eqeltrid 2848	. . 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 41754	. 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 8046	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
26	20, 25	syl 17	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
27	26	eqeq1d 2742	. . 3 ⊢ (𝜑 → ((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 ↔ 𝑌 = 0 ))
28	26	sneqd 4660	. . . . . . 7 ⊢ (𝜑 → {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)} = {𝑌})
29	28	fveq2d 6924	. . . . . 6 ⊢ (𝜑 → (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}) = (𝑁‘{𝑌}))
30	29	fveqeq2d 6928	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ})))
31		ot1stg 8044	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑌 ∈ 𝑉) → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
32	16, 17, 20, 31	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
33	32, 26	oveq12d 7466	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)) = (𝑋 − 𝑌))
34	33	sneqd 4660	. . . . . . . 8 ⊢ (𝜑 → {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))} = {(𝑋 − 𝑌)})
35	34	fveq2d 6924	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}) = (𝑁‘{(𝑋 − 𝑌)}))
36	35	fveq2d 6924	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝑀‘(𝑁‘{(𝑋 − 𝑌)})))
37		ot2ndg 8045	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑌 ∈ 𝑉) → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
38	16, 17, 20, 37	syl3anc 1371	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
39	38	oveq1d 7463	. . . . . . . 8 ⊢ (𝜑 → ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ) = (𝐹𝑅ℎ))
40	39	sneqd 4660	. . . . . . 7 ⊢ (𝜑 → {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)} = {(𝐹𝑅ℎ)})
41	40	fveq2d 6924	. . . . . 6 ⊢ (𝜑 → (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}) = (𝐽‘{(𝐹𝑅ℎ)}))
42	36, 41	eqeq12d 2756	. . . . 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 7406	. . 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 4574	. 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 2780	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 395   = wceq 1537   ∈ wcel 2108  ifcif 4548  {csn 4648  ⟨cop 4654  ⟨cotp 4656   × cxp 5698  ‘cfv 6573  ℩crio 7403  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  Basecbs 17258  0_gc0g 17499  -_gcsg 18975  LSpanclspn 20992  LHypclh 39941  DVecHcdvh 41035  LCDualclcd 41543  mapdcmpd 41581  HDMap1chdma1 41748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-ot 4657  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-1st 8030  df-2nd 8031  df-hdmap1 41750

This theorem is referenced by:  hdmap1val0  41756  hdmap1val2  41757  hdmap1valc  41760