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Theorem hdmap1val 41182
Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 41108.) TODO: change 𝐼 = (π‘₯ ∈ V ↦... to (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘Œ > ) =... in e.g. mapdh8 41172 to shorten proofs with no $d on π‘₯. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmap1val.h 𝐻 = (LHypβ€˜πΎ)
hdmap1fval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hdmap1fval.v 𝑉 = (Baseβ€˜π‘ˆ)
hdmap1fval.s βˆ’ = (-gβ€˜π‘ˆ)
hdmap1fval.o 0 = (0gβ€˜π‘ˆ)
hdmap1fval.n 𝑁 = (LSpanβ€˜π‘ˆ)
hdmap1fval.c 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
hdmap1fval.d 𝐷 = (Baseβ€˜πΆ)
hdmap1fval.r 𝑅 = (-gβ€˜πΆ)
hdmap1fval.q 𝑄 = (0gβ€˜πΆ)
hdmap1fval.j 𝐽 = (LSpanβ€˜πΆ)
hdmap1fval.m 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
hdmap1fval.i 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
hdmap1fval.k (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
hdmap1val.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
hdmap1val.f (πœ‘ β†’ 𝐹 ∈ 𝐷)
hdmap1val.y (πœ‘ β†’ π‘Œ ∈ 𝑉)
Assertion
Ref Expression
hdmap1val (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = if(π‘Œ = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)})))))
Distinct variable groups:   𝐢,β„Ž   𝐷,β„Ž   β„Ž,𝐽   β„Ž,𝑀   β„Ž,𝑁   π‘ˆ,β„Ž   β„Ž,𝑉   β„Ž,𝐹   β„Ž,𝑋   β„Ž,π‘Œ   πœ‘,β„Ž
Allowed substitution hints:   𝐴(β„Ž)   𝑄(β„Ž)   𝑅(β„Ž)   𝐻(β„Ž)   𝐼(β„Ž)   𝐾(β„Ž)   βˆ’ (β„Ž)   π‘Š(β„Ž)   0 (β„Ž)

Proof of Theorem hdmap1val
StepHypRef Expression
1 hdmap1val.h . . 3 𝐻 = (LHypβ€˜πΎ)
2 hdmap1fval.u . . 3 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
3 hdmap1fval.v . . 3 𝑉 = (Baseβ€˜π‘ˆ)
4 hdmap1fval.s . . 3 βˆ’ = (-gβ€˜π‘ˆ)
5 hdmap1fval.o . . 3 0 = (0gβ€˜π‘ˆ)
6 hdmap1fval.n . . 3 𝑁 = (LSpanβ€˜π‘ˆ)
7 hdmap1fval.c . . 3 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
8 hdmap1fval.d . . 3 𝐷 = (Baseβ€˜πΆ)
9 hdmap1fval.r . . 3 𝑅 = (-gβ€˜πΆ)
10 hdmap1fval.q . . 3 𝑄 = (0gβ€˜πΆ)
11 hdmap1fval.j . . 3 𝐽 = (LSpanβ€˜πΆ)
12 hdmap1fval.m . . 3 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
13 hdmap1fval.i . . 3 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
14 hdmap1fval.k . . 3 (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
15 df-ot 4632 . . . 4 βŸ¨π‘‹, 𝐹, π‘ŒβŸ© = βŸ¨βŸ¨π‘‹, 𝐹⟩, π‘ŒβŸ©
16 hdmap1val.x . . . . . 6 (πœ‘ β†’ 𝑋 ∈ 𝑉)
17 hdmap1val.f . . . . . 6 (πœ‘ β†’ 𝐹 ∈ 𝐷)
18 opelxp 5705 . . . . . 6 (βŸ¨π‘‹, 𝐹⟩ ∈ (𝑉 Γ— 𝐷) ↔ (𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷))
1916, 17, 18sylanbrc 582 . . . . 5 (πœ‘ β†’ βŸ¨π‘‹, 𝐹⟩ ∈ (𝑉 Γ— 𝐷))
20 hdmap1val.y . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝑉)
21 opelxp 5705 . . . . 5 (βŸ¨βŸ¨π‘‹, 𝐹⟩, π‘ŒβŸ© ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↔ (βŸ¨π‘‹, 𝐹⟩ ∈ (𝑉 Γ— 𝐷) ∧ π‘Œ ∈ 𝑉))
2219, 20, 21sylanbrc 582 . . . 4 (πœ‘ β†’ βŸ¨βŸ¨π‘‹, 𝐹⟩, π‘ŒβŸ© ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉))
2315, 22eqeltrid 2831 . . 3 (πœ‘ β†’ βŸ¨π‘‹, 𝐹, π‘ŒβŸ© ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉))
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23hdmap1vallem 41181 . 2 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = if((2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)})))))
25 ot3rdg 7990 . . . . 5 (π‘Œ ∈ 𝑉 β†’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = π‘Œ)
2620, 25syl 17 . . . 4 (πœ‘ β†’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = π‘Œ)
2726eqeq1d 2728 . . 3 (πœ‘ β†’ ((2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 0 ↔ π‘Œ = 0 ))
2826sneqd 4635 . . . . . . 7 (πœ‘ β†’ {(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)} = {π‘Œ})
2928fveq2d 6889 . . . . . 6 (πœ‘ β†’ (π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)}) = (π‘β€˜{π‘Œ}))
3029fveqeq2d 6893 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)})) = (π½β€˜{β„Ž}) ↔ (π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž})))
31 ot1stg 7988 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ π‘Œ ∈ 𝑉) β†’ (1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) = 𝑋)
3216, 17, 20, 31syl3anc 1368 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) = 𝑋)
3332, 26oveq12d 7423 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) = (𝑋 βˆ’ π‘Œ))
3433sneqd 4635 . . . . . . . 8 (πœ‘ β†’ {((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))} = {(𝑋 βˆ’ π‘Œ)})
3534fveq2d 6889 . . . . . . 7 (πœ‘ β†’ (π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))}) = (π‘β€˜{(𝑋 βˆ’ π‘Œ)}))
3635fveq2d 6889 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})))
37 ot2ndg 7989 . . . . . . . . . 10 ((𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ π‘Œ ∈ 𝑉) β†’ (2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) = 𝐹)
3816, 17, 20, 37syl3anc 1368 . . . . . . . . 9 (πœ‘ β†’ (2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) = 𝐹)
3938oveq1d 7420 . . . . . . . 8 (πœ‘ β†’ ((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž) = (πΉπ‘…β„Ž))
4039sneqd 4635 . . . . . . 7 (πœ‘ β†’ {((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)} = {(πΉπ‘…β„Ž)})
4140fveq2d 6889 . . . . . 6 (πœ‘ β†’ (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)}) = (π½β€˜{(πΉπ‘…β„Ž)}))
4236, 41eqeq12d 2742 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)}) ↔ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)})))
4330, 42anbi12d 630 . . . 4 (πœ‘ β†’ (((π‘€β€˜(π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)})) ↔ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)}))))
4443riotabidv 7363 . . 3 (πœ‘ β†’ (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)}))) = (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)}))))
4527, 44ifbieq2d 4549 . 2 (πœ‘ β†’ if((2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)})))) = if(π‘Œ = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)})))))
4624, 45eqtrd 2766 1 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = if(π‘Œ = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)})))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ifcif 4523  {csn 4623  βŸ¨cop 4629  βŸ¨cotp 4631   Γ— cxp 5667  β€˜cfv 6537  β„©crio 7360  (class class class)co 7405  1st c1st 7972  2nd c2nd 7973  Basecbs 17153  0gc0g 17394  -gcsg 18865  LSpanclspn 20818  LHypclh 39368  DVecHcdvh 40462  LCDualclcd 40970  mapdcmpd 41008  HDMap1chdma1 41175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-ot 4632  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-1st 7974  df-2nd 7975  df-hdmap1 41177
This theorem is referenced by:  hdmap1val0  41183  hdmap1val2  41184  hdmap1valc  41187
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