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Theorem hdmap1valc 31812
 Description: Connect the value of the preliminary map from vectors to functionals to the hypothesis used by earlier theorems. Note: the hypothesis could be the more general but the former will be easier to use. TODO: use the function directly in those theorems, so this theorem becomes unnecessary? TODO: The hdmap1cbv 31811 is probably unnecessary, but it would mean different \$d's later on. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmap1valc.h
hdmap1valc.u
hdmap1valc.v
hdmap1valc.s
hdmap1valc.o
hdmap1valc.n
hdmap1valc.c LCDual
hdmap1valc.d
hdmap1valc.r
hdmap1valc.q
hdmap1valc.j
hdmap1valc.m mapd
hdmap1valc.i HDMap1
hdmap1valc.k
hdmap1valc.x
hdmap1valc.f
hdmap1valc.y
hdmap1valc.l
Assertion
Ref Expression
hdmap1valc
Distinct variable groups:   ,   ,,   ,,   ,,   ,,   ,,   ,,   ,
Allowed substitution hints:   (,)   (,)   ()   (,)   (,)   (,)   (,)   (,)   (,)   (,)   (,)   (,)   (,)   ()

Proof of Theorem hdmap1valc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmap1valc.h . . 3
2 hdmap1valc.u . . 3
3 hdmap1valc.v . . 3
4 hdmap1valc.s . . 3
5 hdmap1valc.o . . 3
6 hdmap1valc.n . . 3
7 hdmap1valc.c . . 3 LCDual
8 hdmap1valc.d . . 3
9 hdmap1valc.r . . 3
10 hdmap1valc.q . . 3
11 hdmap1valc.j . . 3
12 hdmap1valc.m . . 3 mapd
13 hdmap1valc.i . . 3 HDMap1
14 hdmap1valc.k . . 3
15 hdmap1valc.x . . . 4
16 eldifi 3332 . . . 4
1715, 16syl 15 . . 3
18 hdmap1valc.f . . 3
19 hdmap1valc.y . . 3
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19hdmap1val 31807 . 2
21 hdmap1valc.l . . . 4
2221hdmap1cbv 31811 . . 3
2310, 22, 17, 18, 19mapdhval 31732 . 2
2420, 23eqtr4d 2351 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1633   wcel 1701  cvv 2822   cdif 3183  cif 3599  csn 3674  cotp 3678   cmpt 4114  cfv 5292  (class class class)co 5900  c1st 6162  c2nd 6163  crio 6339  cbs 13195  c0g 13449  csg 14414  clspn 15777  chlt 29358  clh 29991  cdvh 31086  LCDualclcd 31594  mapdcmpd 31632  HDMap1chdma1 31800 This theorem is referenced by:  hdmap1cl  31813  hdmap1eq2  31814  hdmap1eq4N  31815  hdmap1eulem  31832  hdmap1eulemOLDN  31833 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-ot 3684  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-1st 6164  df-2nd 6165  df-riota 6346  df-hdmap1 31802
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