Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.44-1 Structured version   Unicode version

Theorem tz7.44-1 7132
 Description: The value of at . Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypotheses
Ref Expression
tz7.44.1
tz7.44.2
tz7.44-1.3
Assertion
Ref Expression
tz7.44-1
Distinct variable groups:   ,   ,,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem tz7.44-1
StepHypRef Expression
1 fveq2 5881 . . . 4
2 reseq2 5120 . . . . . 6
3 res0 5129 . . . . . 6
42, 3syl6eq 2486 . . . . 5
54fveq2d 5885 . . . 4
61, 5eqeq12d 2451 . . 3
7 tz7.44.2 . . 3
86, 7vtoclga 3151 . 2
9 0ex 4557 . . 3
10 iftrue 3921 . . . 4
11 tz7.44.1 . . . 4
12 tz7.44-1.3 . . . 4
1310, 11, 12fvmpt 5964 . . 3
149, 13ax-mp 5 . 2
158, 14syl6eq 2486 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1437   wcel 1870  cvv 3087  c0 3767  cif 3915  cuni 4222   cmpt 4484   cdm 4854   crn 4855   cres 4856   wlim 5443  cfv 5601 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-iota 5565  df-fun 5603  df-fv 5609 This theorem is referenced by:  rdg0  7147
 Copyright terms: Public domain W3C validator