Theorem hdmap1val 37088

Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 37013.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 37078 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 4186	. . . 4 ⊢ ⟨𝑋, 𝐹, 𝑌⟩ = ⟨⟨𝑋, 𝐹⟩, 𝑌⟩
16		hdmap1val.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		hdmap1val.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
18		opelxp 5146	. . . . . 6 ⊢ (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ↔ (𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷))
19	16, 17, 18	sylanbrc 698	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷))
20		hdmap1val.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
21		opelxp 5146	. . . . 5 ⊢ (⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉) ↔ (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ∧ 𝑌 ∈ 𝑉))
22	19, 20, 21	sylanbrc 698	. . . 4 ⊢ (𝜑 → ⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉))
23	15, 22	syl5eqel 2705	. . 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 37087	. 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 7184	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
26	20, 25	syl 17	. . . 4 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
27	26	eqeq1d 2624	. . 3 ⊢ (𝜑 → ((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 ↔ 𝑌 = 0 ))
28	26	sneqd 4189	. . . . . . . 8 ⊢ (𝜑 → {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)} = {𝑌})
29	28	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}) = (𝑁‘{𝑌}))
30	29	fveq2d 6195	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝑀‘(𝑁‘{𝑌})))
31	30	eqeq1d 2624	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ})))
32		ot1stg 7182	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑌 ∈ 𝑉) → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
33	16, 17, 20, 32	syl3anc 1326	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
34	33, 26	oveq12d 6668	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)) = (𝑋 − 𝑌))
35	34	sneqd 4189	. . . . . . . 8 ⊢ (𝜑 → {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))} = {(𝑋 − 𝑌)})
36	35	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}) = (𝑁‘{(𝑋 − 𝑌)}))
37	36	fveq2d 6195	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) − (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝑀‘(𝑁‘{(𝑋 − 𝑌)})))
38		ot2ndg 7183	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑌 ∈ 𝑉) → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
39	16, 17, 20, 38	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
40	39	oveq1d 6665	. . . . . . . 8 ⊢ (𝜑 → ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ) = (𝐹𝑅ℎ))
41	40	sneqd 4189	. . . . . . 7 ⊢ (𝜑 → {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)} = {(𝐹𝑅ℎ)})
42	41	fveq2d 6195	. . . . . 6 ⊢ (𝜑 → (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅ℎ)}) = (𝐽‘{(𝐹𝑅ℎ)}))
43	37, 42	eqeq12d 2637	. . . . 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)})))
44	31, 43	anbi12d 747	. . . 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)}))))
45	44	riotabidv 6613	. . 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)}))))
46	27, 45	ifbieq2d 4111	. 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)})))))
47	24, 46	eqtrd 2656	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 384   = wceq 1483   ∈ wcel 1990  ifcif 4086  {csn 4177  ⟨cop 4183  ⟨cotp 4185   × cxp 5112  ‘cfv 5888  ℩crio 6610  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  Basecbs 15857  0_gc0g 16100  -_gcsg 17424  LSpanclspn 18971  LHypclh 35270  DVecHcdvh 36367  LCDualclcd 36875  mapdcmpd 36913  HDMap1chdma1 37081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169  df-hdmap1 37083

This theorem is referenced by:  hdmap1val0  37089  hdmap1val2  37090  hdmap1valc  37093