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Theorem hdmapfval 32020
 Description: Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmapval.h
hdmapfval.e
hdmapfval.u
hdmapfval.v
hdmapfval.n
hdmapfval.c LCDual
hdmapfval.d
hdmapfval.j HVMap
hdmapfval.i HDMap1
hdmapfval.s HDMap
hdmapfval.k
Assertion
Ref Expression
hdmapfval
Distinct variable groups:   ,,,   ,   ,,,   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)   (,,)   (,)   (,,)   (,,)   (,,)   (,,)

Proof of Theorem hdmapfval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmapfval.k . 2
2 hdmapfval.s . . . 4 HDMap
3 hdmapval.h . . . . . 6
43hdmapffval 32019 . . . . 5 HDMap HDMap1 LCDual HVMap
54fveq1d 5527 . . . 4 HDMap HDMap1 LCDual HVMap
62, 5syl5eq 2327 . . 3 HDMap1 LCDual HVMap
7 fveq2 5525 . . . . . . . . . 10
87reseq2d 4955 . . . . . . . . 9
98opeq2d 3803 . . . . . . . 8
10 dfsbcq 2993 . . . . . . . 8 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
119, 10syl 15 . . . . . . 7 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
12 fveq2 5525 . . . . . . . . . 10
13 dfsbcq 2993 . . . . . . . . . 10 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
1412, 13syl 15 . . . . . . . . 9 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
15 fveq2 5525 . . . . . . . . . . . . 13 HDMap1 HDMap1
16 dfsbcq 2993 . . . . . . . . . . . . 13 HDMap1 HDMap1 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
1715, 16syl 15 . . . . . . . . . . . 12 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
18 fveq2 5525 . . . . . . . . . . . . . . . . 17 LCDual LCDual
1918fveq2d 5529 . . . . . . . . . . . . . . . 16 LCDual LCDual
20 fveq2 5525 . . . . . . . . . . . . . . . . . . . . . . . 24 HVMap HVMap
2120fveq1d 5527 . . . . . . . . . . . . . . . . . . . . . . 23 HVMap HVMap
22 oteq2 3806 . . . . . . . . . . . . . . . . . . . . . . 23 HVMap HVMap HVMap HVMap
2321, 22syl 15 . . . . . . . . . . . . . . . . . . . . . 22 HVMap HVMap
2423fveq2d 5529 . . . . . . . . . . . . . . . . . . . . 21 HVMap HVMap
25 oteq2 3806 . . . . . . . . . . . . . . . . . . . . 21 HVMap HVMap HVMap HVMap
2624, 25syl 15 . . . . . . . . . . . . . . . . . . . 20 HVMap HVMap
2726fveq2d 5529 . . . . . . . . . . . . . . . . . . 19 HVMap HVMap
2827eqeq2d 2294 . . . . . . . . . . . . . . . . . 18 HVMap HVMap
2928imbi2d 307 . . . . . . . . . . . . . . . . 17 HVMap HVMap
3029ralbidv 2563 . . . . . . . . . . . . . . . 16 HVMap HVMap
3119, 30riotaeqbidv 6307 . . . . . . . . . . . . . . 15 LCDual HVMap LCDual HVMap
3231mpteq2dv 4107 . . . . . . . . . . . . . 14 LCDual HVMap LCDual HVMap
3332eleq2d 2350 . . . . . . . . . . . . 13 LCDual HVMap LCDual HVMap
3433sbcbidv 3045 . . . . . . . . . . . 12 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
3517, 34bitrd 244 . . . . . . . . . . 11 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
3635sbcbidv 3045 . . . . . . . . . 10 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
3736sbcbidv 3045 . . . . . . . . 9 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
3814, 37bitrd 244 . . . . . . . 8 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
3938sbcbidv 3045 . . . . . . 7 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
4011, 39bitrd 244 . . . . . 6 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
41 opex 4237 . . . . . . 7
42 fvex 5539 . . . . . . 7
43 fvex 5539 . . . . . . 7
44 simp1 955 . . . . . . . . 9
45 hdmapfval.e . . . . . . . . 9
4644, 45syl6eqr 2333 . . . . . . . 8
47 simp2 956 . . . . . . . . 9
48 hdmapfval.u . . . . . . . . 9
4947, 48syl6eqr 2333 . . . . . . . 8
50 simp3 957 . . . . . . . . . 10
5149fveq2d 5529 . . . . . . . . . 10
5250, 51eqtrd 2315 . . . . . . . . 9
53 hdmapfval.v . . . . . . . . 9
5452, 53syl6eqr 2333 . . . . . . . 8
55 fvex 5539 . . . . . . . . . 10 HDMap1
56 id 19 . . . . . . . . . . . 12 HDMap1 HDMap1
57 hdmapfval.i . . . . . . . . . . . 12 HDMap1
5856, 57syl6eqr 2333 . . . . . . . . . . 11 HDMap1
59 fveq1 5524 . . . . . . . . . . . . . . . . . 18 HVMap HVMap
60 fveq1 5524 . . . . . . . . . . . . . . . . . . . 20 HVMap HVMap
61 oteq2 3806 . . . . . . . . . . . . . . . . . . . 20 HVMap HVMap HVMap HVMap
6260, 61syl 15 . . . . . . . . . . . . . . . . . . 19 HVMap HVMap
6362fveq2d 5529 . . . . . . . . . . . . . . . . . 18 HVMap HVMap
6459, 63eqtrd 2315 . . . . . . . . . . . . . . . . 17 HVMap HVMap
6564eqeq2d 2294 . . . . . . . . . . . . . . . 16 HVMap HVMap
6665imbi2d 307 . . . . . . . . . . . . . . 15 HVMap HVMap
6766ralbidv 2563 . . . . . . . . . . . . . 14 HVMap HVMap
6867riotabidv 6306 . . . . . . . . . . . . 13 LCDual HVMap LCDual HVMap
6968mpteq2dv 4107 . . . . . . . . . . . 12 LCDual HVMap LCDual HVMap
7069eleq2d 2350 . . . . . . . . . . 11 LCDual HVMap LCDual HVMap
7158, 70syl 15 . . . . . . . . . 10 HDMap1 LCDual HVMap LCDual HVMap
7255, 71sbcie 3025 . . . . . . . . 9 HDMap1 LCDual HVMap LCDual HVMap
73 simp3 957 . . . . . . . . . . 11
74 hdmapfval.d . . . . . . . . . . . . . 14
75 hdmapfval.c . . . . . . . . . . . . . . 15 LCDual
7675fveq2i 5528 . . . . . . . . . . . . . 14 LCDual
7774, 76eqtr2i 2304 . . . . . . . . . . . . 13 LCDual
7877a1i 10 . . . . . . . . . . . 12 LCDual
79 simp2 956 . . . . . . . . . . . . . . . . . . . 20
8079fveq2d 5529 . . . . . . . . . . . . . . . . . . 19
81 hdmapfval.n . . . . . . . . . . . . . . . . . . 19
8280, 81syl6eqr 2333 . . . . . . . . . . . . . . . . . 18
83 simp1 955 . . . . . . . . . . . . . . . . . . 19
8483sneqd 3653 . . . . . . . . . . . . . . . . . 18
8582, 84fveq12d 5531 . . . . . . . . . . . . . . . . 17
8682fveq1d 5527 . . . . . . . . . . . . . . . . 17
8785, 86uneq12d 3330 . . . . . . . . . . . . . . . 16
8887eleq2d 2350 . . . . . . . . . . . . . . 15
8988notbid 285 . . . . . . . . . . . . . 14
90 oteq1 3805 . . . . . . . . . . . . . . . . . . . 20 HVMap HVMap
9183, 90syl 15 . . . . . . . . . . . . . . . . . . 19 HVMap HVMap
9283fveq2d 5529 . . . . . . . . . . . . . . . . . . . . 21 HVMap HVMap
93 hdmapfval.j . . . . . . . . . . . . . . . . . . . . . 22 HVMap
9493fveq1i 5526 . . . . . . . . . . . . . . . . . . . . 21 HVMap
9592, 94syl6eqr 2333 . . . . . . . . . . . . . . . . . . . 20 HVMap
96 oteq2 3806 . . . . . . . . . . . . . . . . . . . 20 HVMap HVMap
9795, 96syl 15 . . . . . . . . . . . . . . . . . . 19 HVMap
9891, 97eqtrd 2315 . . . . . . . . . . . . . . . . . 18 HVMap
9998fveq2d 5529 . . . . . . . . . . . . . . . . 17 HVMap
100 oteq2 3806 . . . . . . . . . . . . . . . . 17 HVMap HVMap
10199, 100syl 15 . . . . . . . . . . . . . . . 16 HVMap
102101fveq2d 5529 . . . . . . . . . . . . . . 15 HVMap
103102eqeq2d 2294 . . . . . . . . . . . . . 14 HVMap
10489, 103imbi12d 311 . . . . . . . . . . . . 13 HVMap
10573, 104raleqbidv 2748 . . . . . . . . . . . 12 HVMap
10678, 105riotaeqbidv 6307 . . . . . . . . . . 11 LCDual HVMap
10773, 106mpteq12dv 4098 . . . . . . . . . 10 LCDual HVMap
108107eleq2d 2350 . . . . . . . . 9 LCDual HVMap
10972, 108syl5bb 248 . . . . . . . 8 HDMap1 LCDual HVMap
11046, 49, 54, 109syl3anc 1182 . . . . . . 7 HDMap1 LCDual HVMap
11141, 42, 43, 110sbc3ie 3060 . . . . . 6 HDMap1 LCDual HVMap
11240, 111syl6bb 252 . . . . 5 HDMap1 LCDual HVMap
113112abbi1dv 2399 . . . 4 HDMap1 LCDual HVMap
114 eqid 2283 . . . 4 HDMap1 LCDual HVMap HDMap1 LCDual HVMap
115 fvex 5539 . . . . . 6
11653, 115eqeltri 2353 . . . . 5
117116mptex 5746 . . . 4
118113, 114, 117fvmpt 5602 . . 3 HDMap1 LCDual HVMap
1196, 118sylan9eq 2335 . 2
1201, 119syl 15 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wa 358   w3a 934   wceq 1623   wcel 1684  cab 2269  wral 2543  cvv 2788  wsbc 2991   cun 3150  csn 3640  cop 3643  cotp 3644   cmpt 4077   cid 4304   cres 4691  cfv 5255  crio 6297  cbs 13148  clspn 15728  clh 30173  cltrn 30290  cdvh 31268  LCDualclcd 31776  HVMapchvm 31946  HDMap1chdma1 31982  HDMapchdma 31983 This theorem is referenced by:  hdmapval  32021  hdmapfnN  32022 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 6304  df-hdmap 31985
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